# Time-Stepping Approximation of Rigid-Body Dynamics with Perfect Unilateral Constraints. II: The Partially Elastic Impact Case

Laetitia Paoli

► To cite this version:

Laetitia Paoli. Time-Stepping Approximation of Rigid-Body Dynamics with Perfect Unilateral Constraints. II: The Partially Elastic Impact Case. Archive for Rational Mechanics and Analysis, 2010, 198 (2), pp.505-568. 10.1090/S0002-9947-1932-1501655-2. hal-01564583

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Submitted on 18 Jul 2017

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# Time-Stepping Approximation of Rigid-Body Dynamics with Perfect Unilateral Constraints. II: The Partially Elastic Impact Case

L. PAOLI

As in Paoli (Arch Rational Mech Anal, 2010), we consider a discrete mechanical system with a non-trivial mass matrix subjected to perfect unilateral constraints described by geometrical inequalities $f_\alpha(q) \ge 0$, $\alpha \in \{1, \dots, v\}$ ($v \ge 1$), but we assume now that the transmission of the velocities at impacts is governed by Newton's Law with a coefficient of restitution $e \in (0, 1]$ (so that the impact is partially elastic). We generalize the time-discretization of the second order differential inclusion describing the dynamics proposed in Paoli (Arch Rational Mech Anal, 2010) to this case and, once again, we prove its convergence.

## 1. Introduction

As in [11], we consider a discrete mechanical system subjected to perfect unilateral constraints. More precisely, we denote by $u \in \mathbb{R}^d$ the generalized coordinates of a typical configuration of the system, and we assume that the set $K$ of admissible configurations is described by $v \ge 1$ geometrical inequalities

$$f_{\alpha}(u) \ge 0, \quad \alpha \in \{1, \dots, v\}$$

where $f_\alpha$ is a smooth function (at least $C^1$) such that $\nabla f_\alpha(u)$ does not vanish in a neighbourhood of $\{u \in \mathbb{R}^d; f_\alpha(u) = 0\}$.

Then, the dynamics is described by the following measure differential inclusion (see [7] for instance)

$$M(u)\ddot{u} - g(t, u, \dot{u}) \in -N_K(u) \quad (1)$$
where $M(u)$ is the mass matrix of the system and $N_K(u)$ is the normal cone to $K$ at $u$, given by

$$N_K(u) = \begin{cases} \{0\} & \text{if } u \in \text{Int}(K), \\ \left\{ \sum_{\alpha \in J(u)} \lambda_{\alpha} \nabla f_{\alpha}(u), \lambda_{\alpha} \le 0, \forall \alpha \in J(u) \right\} & \text{if } u \in \partial K, \\ \emptyset & \text{if } u \notin K, \end{cases}$$

with $J(u) = \{\alpha \in \{1, \dots, v\}; f_{\alpha}(u) \le 0\}$ for all $u \in \mathbb{R}^d$. We also define the tangent cone to $K$ at $u$ by

$$T_K(u) = \{w \in \mathbb{R}^d; (\nabla f_{\alpha}(w), w) \ge 0 \ \forall \alpha \in J(u)\}$$

where $(v, w)$ denotes the Euclidean scalar product of vectors $v$ and $w$ in $\mathbb{R}^d$. Since $u(s) \in K$ for all $s$ and we infer that

$$\dot{u}(t+0) \in T_K(u(t)), \quad \dot{u}(t-0) \in -T_K(u(t)) \quad (t > 0)$$

whenever $\dot{u}(t \pm 0)$ exists. It follows that the velocities are discontinuous at impacts if $\dot{u}(t-0) \notin T_K(u(t))$, and (1) implies that

$$M(u(t)) (\dot{u}(t+0) - \dot{u}(t-0)) \in -N_K(u(t)).$$

This relation does not uniquely determine $\dot{u}(t+0)$, so we should add an impact law. Following MOREAU [8] and BALLARD [1], we consider in this article a more general impact law, as in [11]: we assume that there exists a restitution coefficient $e \in [0, 1]$ such that

$$\dot{u}(t+0) = -e\dot{u}(t-0) + (1+e)\text{Proj}_{M(u(t))}(T_K(u(t)), \dot{u}(t-0)) \quad (2)$$

where $\text{Proj}_{M(u)}$ denotes the projection relatively to the Riemannian metric defined by the inertia operator $M(u)$.

It should be observed that this model is energetically consistent: indeed, the kinetic energy of the system, given by $E_c = \|\dot{u}\|_{M(u)}^2/2$, decreases at impacts if $e \in (0, 1]$ and is conserved if $e = 1$. Moreover, when $e = 0$, we recognize the case of “standard inelastic shocks” already studied in [11].

For admissible initial data $(u_0, v_0) \in K \times T_K(u_0)$ we consider the following initial-value problem:

**Problem (P)** Find $u: [0, \tau] \to \mathbb{R}^d$ ($\tau > 0$) such that:

(P1) $u$ is an absolutely continuous function from $[0, \tau]$ to $K$ and $\dot{u} \in BV(0, \tau; \mathbb{R}^d)$,

(P2) the differential inclusion

$$M(u)\ddot{u} - g(t, u, \dot{u}) \in -N_K(u)$$

is satisfied in the following sense: there exists a (non-unique) non-negative measure $\mu$ such that the Stieltjes measure $d\dot{u} = \ddot{u}$ and the usual Lebesgue measure $dt$ admit densities with respect to $d\mu$, that is, there exist two $d\mu$-integrable functions $v'_{\mu}$ and $t'_{\mu}$ such that $\ddot{u} = d\dot{u} = v'_{\mu}d\mu$, $dt = t'_{\mu}d\mu$ and

$$M(u(t)) v'_{\mu}(t) - g(t, u(t), \dot{u}(t)) t'_{\mu}(t) \in -N_K(u(t)) \quad d\mu \text{ almost everywhere, (3)}$$
(P3) for all $t \in (0, \tau)$

$$ \dot{u}(t+0) = -e\dot{u}(t-0) + (1+e)\mathrm{Proj}_{M(u(t))}(T_K(u(t)), \dot{u}(t-0)) $$

(P4) $u(0) = u_0$, $\dot{u}(0+0) = v_0$.

For this model of impact (for any value of the restitution coefficient $e \in [0, 1]$), the existence and uniqueness of a maximal solution for the initial-value problem has been proved by BALLARD [1], when the data are analytical. Uniqueness cannot be expected for less regular data (see [2, 14] or [1] for counter-examples) but existence results have still been established in the single constraint case (that is, $v=1$): see [5, 9, 12] for a trivial mass matrix (that is, $M(u) = \mathrm{Id}_{\mathbb{R}^d}$), and [13, 15] for a non-trivial mass matrix. All these results rely on the study of a sequence of approximate solutions constructed either by a penalty method [12, 15] or by a time-stepping scheme [5, 9, 13].

As in the inelastic shock case, these techniques encounter a new difficulty for the multi-constraint case since, in general, the motion is not continuous with respect to the data. Nevertheless, in the framework given by the sufficient conditions ensuring continuity on data established in [1, 10], we may expect some convergence results. So, in this article we propose a generalization to the partially elastic case of the time-stepping scheme introduced in [11] and we study its convergence.

More precisely, we assume the same kind of regularity for the data as in [10, 11], that is,

(H1) $g$ is a continuous function from $[0, T] \times \mathbb{R}^d \times \mathbb{R}^d$ ($T > 0$) to $\mathbb{R}^d$;

(H2) for all $\alpha \in \{1, \dots, v\}$, the function $f_\alpha$ belongs to $C^1(\mathbb{R}^d)$, $\nabla f_\alpha$ is locally Lipschitz continuous and does not vanish in a neighbourhood of $\{u \in \mathbb{R}^d; f_\alpha(u) = 0\}$;

(H3) the set $K$ is defined by

$$ K = \{u \in \mathbb{R}^d; f_\alpha(u) \ge 0, \alpha \in \{1, \dots, v\}\} $$

and the active constraints along $\partial K$ are functionally independent, that is, for all $u \in \partial K$ the vectors $(\nabla f_\alpha(u))_{j(u)}$ are linearly independent;

(H4) $M$ is a mapping of class $C^1$ from $\mathbb{R}^d$ to the set of symmetric positive definite $d \times d$ matrices.

With this last assumption we may define $M^{-1}(u)$, $M^{1/2}(u)$ and $M^{-1/2}(u)$ for all $u \in \mathbb{R}^d$ and the corresponding mappings are of class $C^1$ from $\mathbb{R}^d$ to the set of symmetric positive definite $d \times d$ matrices.

Let $F$ be a function such that

(H5) $F$ is continuous from $[0, T] \times \mathbb{R}^d \times \mathbb{R}^d \times [0, h^*]$ ($h^* > 0$) to $\mathbb{R}^d$ and is consistent with respect to $g$, that is,

$$ F(t, u, v, 0) = M^{-1}(u)g(t, u, v) \quad \forall (t, u, v) \in [0, T] \times \mathbb{R}^d \times \mathbb{R}^d. $$

For admissible initial data $(u_0, v_0) \in K \times T_K(u_0)$ we consider the initial-value problem (P). We define a time-stepping scheme as follows:
• the initial positions $U^0$ and $U^1$ are given by

$$U^0 = u_0, \quad U^1 \in \operatorname{Argmin}_{Z \in K} \|u_0 + hv_0 + hz(h) - Z\|_{M(u_0)} \quad (4)$$

with $\lim_{h \to 0} z(h) = 0$,

• for all $n \in \{1, \dots, \lfloor \frac{T}{h} \rfloor\}$, let

$$W^n = \frac{2U^n - (1-e)U^{n-1} + h^2 F^n}{1+e}, \quad F^n = F\left(nh, U^n, \frac{U^n - U^{n-1}}{h}, h\right) \quad (5)$$

and

$$U^{n+1} = -eU^{n-1} + (1+e)Z^n \quad (6)$$

with

$$Z^n \in \operatorname{Argmin}_{Z \in K} \|W^n - Z\|_{M(U^n)} \quad (7)$$

where $\| \cdot \|_{M(U)}$ is the norm associated to the kinetic metric at $U$ defined by $\|Z\|_{M(U)}^2 = (Z, Z)_{M(U)}$ with

$$(Z, Z')_{M(U)} = (Z, M(U)Z') = (M(U)Z, Z')$$

for all $(Z, Z', U) \in (\mathbb{R}^d)^3$.

For $e = 0$ we recover the scheme already studied in [11] and, if $M(u) = \operatorname{Id}_{\mathbb{R}^d}$ for all $u \in \mathbb{R}^d$ and $K$ is convex, we recognize the scheme introduced in [9] for the first time.

As in [11], we now define the approximate solutions $u_h$ by

$$u_h(t) = U^n + (t - nh) \frac{U^{n+1} - U^n}{h} \quad \forall t \in [nh, (n+1)h] \cap [0, T]$$

for all $n \in \{0, \dots, \lfloor T/h \rfloor\}$ and $h \in (0, h^*]$.

Since the impact law (2) leads to some discontinuity with respect to the data, as when the active constraints at impacts are not orthogonal when $e \neq 0$ (see [10]), we cannot expect convergence of the approximate motions unless we add some assumptions on the geometry of active constraints along $\partial K$.

So, for all $u \in K$ and for all $\alpha \in J(u)$, we define

$$e_{\alpha}(u) = \frac{M^{-1/2}(u) \nabla f_{\alpha}(u)}{|M^{-1/2}(u) \nabla f_{\alpha}(u)|}$$

where $| \cdot |$ denotes the Euclidean norm in $\mathbb{R}^d$, and we assume that

(H6) for all compact subset $B$ of $\mathbb{R}^d$, there exist $C_B > 0$ and $r_B > 0$ such that for all $(q_1, q_2) \in (K \cap B)^2$ such that $|q_1 - q_2| \le r_B$, and for all $(\alpha, \beta) \in J(q_1) \times J(q_2)$, such that $\alpha \neq \beta$, we have

$$|(e_{\alpha}(q_1), e_{\beta}(q_2))| \le C_B |q_1 - q_2|. \quad (8)$$
Let us observe that, for $q_1 = q_2$, (H6) reduces to the “angle condition” given in [10], which ensures continuity on data. In particular, for all $q \in K$

$$ (e_{\alpha}(q), e_{\beta}(q)) = 0 \quad \forall (\alpha, \beta) \in J(q)^2, \alpha \neq \beta $$

which means that the active constraints are orthogonal for the local momentum metric defined by $M^{-1}(q)$.

Then, under assumptions (H1)–(H6) we prove the convergence of a subsequence of the approximate solutions $(u_h)_{h^* \ge h > 0}$ to a solution of problem (P). The various steps of the proof are the same as in [11], but the non-trivial restitution coefficient $e \in (0, 1]$ leads to considerably more technicalities in the estimates of the velocity.

The paper is organized as follows. In the next sections we establish a priori estimates for the discrete velocities and accelerations on a non-trivial time interval $[0, \tau]$, with $0 < \tau \le T$. Then we pass to the limit when $h$ tends to zero on $[0, \tau]$; using Ascoli’s and Helly’s theorems we obtain the convergence of a subsequence of $(u_h)_{h^* \ge h > 0}$ to a limit $u$ which satisfies (P1) and (P2). Next, by a local study in the neighbourhood of any impact time, we prove that the limit $u$ also satisfies (P3) and (P4). Finally we conclude with some global results.

## 2. A priori estimates for the discrete velocities

As in [11], we begin the study with a priori estimates of the discrete velocities for a more general scheme in which the initialization procedure involves an initial time $t_{0h}$ depending on $h$. The purpose of this modification is to allow us, in the last section of the paper, to extend the estimates of the discrete velocities, by considering as “new” initial data, the already constructed approximate positions at some time steps $t_{0h}$ and $t_{0h} + h$.

More precisely, let $\mathcal{B}$ be a given convex compact subset of $\mathbb{R}^d$ such that $\mathcal{B} \cap K \neq \emptyset$. Possibly decreasing $h^*$, we may assume without loss of generality that

$$ |z(h)| \le 1 \quad \forall h \in (0, h^*]. \qquad (9) $$

For all $h \in (0, h^*]$, let $t_{0h} \in [0, T)$, and $U^0$ and $U^1$ be given in $\mathcal{B} \cap K$ and $K$, respectively. For all $n \in \{1, \dots, \lfloor \frac{T-t_{0h}}{h} \rfloor\}$ we define

$$ U^{n+1} = -eU^{n-1} + (1+e)Z^n $$

with

$$ Z^n \in \operatorname{Argmin}_{Z \in K} \|W^n - Z\|_{M(U^n)} $$

and

$$ W^n = \frac{2U^n - (1-e)U^{n-1} + h^2 F^n}{1+e}, \quad F^n = F\left(t_{0h}+nh, U^n, \frac{U^n-U^{n-1}}{h}, h\right). $$
For all $h \in (0, h^*]$ and $n \in \{0, \dots, \lfloor \frac{T-t_{0h}}{h} \rfloor\}$, let

$$V^n = \frac{U^{n+1} - U^n}{h}.$$

Then, we observe that

**Lemma 1.** For all $h \in (0, h^*]$ and $n \in \{1, \dots, \lfloor(T - t_{0h})/h\rfloor\}$, we have

$$M(U^n)(V^{n-1} - V^n + hF^n) \in N_K(Z^n).$$

**Proof.** The proof is the same as in Lemma 1 of [11], in which we simply replace $U^{n+1}$ by $Z^n$. More precisely, let $h \in (0, h^*]$ and $n \in \{1, \dots, \lfloor(T - t_{0h})/h\rfloor\}$. By definition of $Z^n$ we have

$$\|W^n - Z^n\|_{M(U^n)}^2 \le \|W^n - Z\|_{M(U^n)}^2$$

for all $Z \in K$, which yields

$$(W^n - Z^n, Z - Z^n)_{M(U^n)} \le \frac{1}{2} \|Z^n - Z\|_{M(U^n)}^2 \quad \forall Z \in K.$$

But, using formulae (5) and (6), we get

$$\begin{aligned} W^n - Z^n &= \frac{2U^n - (1-e)U^{n-1} + h^2 F^n}{1+e} - \frac{U^{n+1} + eU^{n-1}}{1+e} \\ &= h \frac{V^{n-1} - V^n + hF^n}{1+e}. \end{aligned} \quad (10)$$

Thus

$$\frac{2h}{1+e}(V^{n-1} - V^n + hF^n, Z - Z^n)_{M(U^n)} \le \|Z^n - Z\|_{M(U^n)}^2 \quad \forall Z \in K.$$

and the rest of the proof remains identical. □

Let us introduce the same notation as in [11]. We define

$$\lambda_{\max}(u) = \|M(u)\|, \quad \lambda_{\min}(u) = \frac{1}{\|M^{-1}(u)\|} \quad \forall u \in \mathbb{R}^d.$$

Since $u \mapsto M(u)$ is continuous with values in the set of symmetric positive definite matrices, the mappings $u \mapsto \lambda_{\max}(u)$ and $u \mapsto \lambda_{\min}(u)$ are well defined and continuous from $\mathbb{R}^d$ to $\mathbb{R}_+^*$. Moreover

$$\lambda_{\min}(u)|w|^2 \le \|w\|_{M(u)}^2 \le \lambda_{\max}(u)|w|^2 \quad \forall w \in \mathbb{R}^d, \quad \forall u \in \mathbb{R}^d.$$

Since $\mathcal{B}$ is compact, there exists $\delta > 0$ such that, for all $(q, q') \in \mathcal{B} \times \mathbb{R}^d$ such that $|q - q'| \le \delta$, we have:

$$|\lambda_{\min}(q) - \lambda_{\min}(q')| \le \frac{1}{2} \inf_{u \in \mathcal{B}} \lambda_{\min}(u),$$

$$|\lambda_{\max}(q) - \lambda_{\max}(q')| \le \frac{1}{2} \sup_{u \in \mathcal{B}} \lambda_{\max}(u).$$
We define

$$
B_0 = \{ u \in \mathbb{R}^d ; \mathrm{dist}(u, B) \le \delta \}. \qquad (11)
$$

Then $B_0$ is also a convex compact subset of $\mathbb{R}^d$, and we have

$$
\frac{1}{2} \inf_{u \in B} \lambda_{\min}(u) \le \inf_{u \in B_0} \lambda_{\min}(u), \quad \sup_{u \in B_0} \lambda_{\max}(u) \le \frac{3}{2} \sup_{u \in B} \lambda_{\max}(u).
$$

We let

$$
\lambda_{\min} = \frac{1}{2} \inf_{u \in \mathcal{B}} \lambda_{\min}(u), \quad \lambda_{\max} = \frac{3}{2} \sup_{u \in \mathcal{B}} \lambda_{\max}(u). \qquad (12)
$$

Of course, we have

$$
0 < \lambda_{\min} |w|^2 \le \|w\|^2_{M(u)} \le \lambda_{\max} |w|^2 \quad \forall w \in \mathbb{R}^d \setminus \{0\}, \quad \forall u \in B_0.
$$

Let $C_0 > 0$ and $C_F$ be given by

$$
C_F = \sup \{|F(t, u, v, h)| ; t \in [0, T], u \in B_0 \cup B_1, |v| \le C_0, h \in [0, h^*]\} \quad (13)
$$

with $B_1 = \bar{B}(u_0, C_0 T + 1)$. Since the mappings $M, M^{-1}, M^{1/2}$ and $M^{-1/2}$ are of class $C^1$ on $\mathbb{R}^d$, they are Lipschitz continuous on $B_0 \cup B_1$ and we denote by $L_M, L_{M^{-1}}, L_{M^{1/2}}$ and $L_{M^{-1/2}}$ the corresponding Lipschitz constants. Moreover the functions $\nabla f_\alpha$, $1 \le \alpha \le \nu$, are locally Lipschitzian. There exists also a positive real number $L_f$ such that

$$
|\nabla f_{\alpha}(Z) - \nabla f_{\alpha}(Z')| \le L_f |Z - Z'| \quad \forall (Z, Z') \in (B_0 \cup B_1)^2, \quad \forall \alpha \in \{1, \dots, \nu\}.
$$

We can obtain, as in [11], some rough estimates on the discrete velocities $V^n$. More
precisely:

**Proposition 1.** Let $C_0 > 0$ and $h_0^* \in (0, h^*]$ such that

$$
h_0^* \le \min \left( \frac{C_0}{2C_F}, \frac{\delta}{8C_0} \sqrt{\frac{\lambda_{\min}}{\lambda_{\max}}} \right)
$$

where $C_F$ is defined by (13) and $\lambda_{\min}$, $\lambda_{\max}$ are given by (12). Let $h \in (0, h_0^*]$,
$\tau_h = \min(\delta/(2C_0), T - t_{0h})$ and assume that there exists $n \in \{1, \dots, \lfloor \tau_h/h \rfloor\}$
such that

$$
|V^l| \le C_0 \quad \forall l \in \{0, \dots, n-1\}.
$$

Then

$$
|V^n| \le 4 \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} C_0. \qquad (14)
$$

Moreover, for all $l \in \{2, \dots, n\}$ such that $J(Z^l) \neq \emptyset$,

$$
(\nabla f_{\alpha}(Z^l), V^l + eV^{l-2}) \le \frac{L_f h}{2(1+e)} |V^l + eV^{l-2}|^2 \quad \forall \alpha \in J(Z^l).
$$
**Proof.** The proof is almost identical to the proof of Proposition 1 in [11]. More precisely, for all $l \in \{0, \dots, n\}$ we have

$$|U^l - U^0| \le \sum_{k=0}^{l-1} h|V^k| \le lhC_0 \le C_0\tau_h \le \delta \quad \forall l \in \{0, \dots, n\},$$

and thus $U^l \in B_0$. By definition of $Z^n$ we have

$$\|W^n - Z^n\|_{M(U^n)} \le \|W^n - Z^{n-1}\|_{M(U^n)}$$

since $Z^{n-1} \in K$, with the convention $Z^0 = U^0$. Recalling (10) we have

$$W^n - Z^n = \frac{h}{1+e}(V^{n-1} - V^n + hF^n),$$

and with formulae (5) and (6) we have also

$$W^n - Z^{n-1} = \begin{cases} \frac{h}{1+e}(V^{n-1} + eV^{n-2} + hF^n) & \text{if } n \ge 2, \\ \frac{h}{1+e}(2V^0 + hF^1) & \text{if } n = 1. \end{cases}$$

Hence

$$\|V^n\|_{M(U^n)} \le \begin{cases} 2\|V^{n-1}\|_{M(U^n)} + e\|V^{n-2}\|_{M(U^n)} + 2h\|F^n\|_{M(U^n)} & \text{if } n \ge 2, \\ 3\|V^0\|_{M(U^1)} + 2h\|F^1\|_{M(U^1)} & \text{if } n = 1, \end{cases} \quad (15)$$

and

$$|V^n| \le \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} (3C_0 + 2hC_F) \le 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} C_0.$$

It follows that $Z^l \in B_0$ for all $l \in \{1, \dots, n\}$. Indeed,

$$\begin{aligned} |Z^l - U^0| &= \left| \frac{U^{l+1} + eU^{l-1}}{1+e} - U^0 \right| \le \frac{1}{1+e} \left( |U^{l+1} - U^0| + e|U^{l-1} - U^0| \right) \\ &\le C_0 \tau_h \le \frac{\delta}{2} \quad \text{if } l \in \{1, \dots, n-1\}, \end{aligned}$$

and

$$\begin{aligned} |Z^l - U^0| &= \left| \frac{U^{l+1} + eU^{l-1}}{1+e} - U^0 \right| \le \frac{h}{1+e} |V^l| \\ &\quad + \frac{1}{1+e} \left( |U^l - U^0| + e|U^{l-1} - U^0| \right) \\ &\le 4h \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} C_0 + C_0 \tau_h \le \frac{\delta}{2} + C_0 \tau_h \le \delta \quad \text{if } l = n. \end{aligned}$$
Assume now that $n \ge 2$, and let $l \in \{2, \dots, n\}$ such that $J(Z^l) \ne \emptyset$. For all $\alpha \in J(Z^l)$, we have

$$0 \le f_{\alpha}(Z^{l-1}) - f_{\alpha}(Z^l) = \int_0^1 \left( \nabla f_{\alpha} \left(Z^l + t(Z^{l-1} - Z^l)\right), Z^{l-1} - Z^l \right) dt.$$

But

$$Z^{l-1} - Z^l = - \frac{h}{1+e} (V^l + eV^{l-2})$$

and thus

$$\begin{aligned} (\nabla f_{\alpha}(Z^l), V^l + eV^{l-2}) &\le -\int_0^1 (\nabla f_{\alpha}(Z^l + t(Z^{l-1} - Z^l)) \\ &\quad - \nabla f_{\alpha}(Z^l), V^l + eV^{l-2}) \, dt \\ &\le \int_0^1 \left| \nabla f_{\alpha}(Z^l + t(Z^{l-1} - Z^l)) - \nabla f_{\alpha}(Z^l) \right| \\ &\quad \times |V^l + eV^{l-2}| \, dt. \end{aligned}$$

It follows that

$$(\nabla f_{\alpha}(Z^l), V^l + eV^{l-2}) \le \frac{L_f h}{2(1+e)} |V^l + eV^{l-2}|^2.$$

□

Now we prove a more precise estimate for the discrete velocities.

**Proposition 2.** Let $C_0 > 0$ and assume that there exist $C_0^* > 0$ and $h_0^* \in (0, h^*]$ such that

$$h_0^* \le \min \left( \frac{C_0^*}{2C_F}, \frac{\delta}{8C_0} \sqrt{\frac{\lambda_{\min}}{\lambda_{\max}}}, \frac{\min(r_{B_0}, r'_{B_0})}{C_0 (v + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}})}, \frac{1}{72v^{3/2}C_e C_0 (v + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}})} \right)$$

and

$$|V^0| = \left| \frac{U^1 - U^0}{h} \right| \le C_0^* < \left( 4 \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} \right)^{-(v+1)} \quad C_0 \quad \forall h \in (0, h_0^*]$$

where $C_F$ is defined by (13), $\lambda_{\min}$ and $\lambda_{\max}$ are given by (12) and $C_e = \max(C_{B_0}, C'_{B_0})$ with $C'_{B_0}$ and $r'_{B_0}$ defined at Lemma 15 (see Appendix).

Then there exists $\tau_0 > 0$, depending only on $B$, $C_0$, $C_0^*$ and the data, such that

$$|V^n| = \left| \frac{U^{n+1} - U^n}{h} \right| \le C_0 \quad \forall nh \in [0, \min(\tau_0, T - t_{0h})], \quad \forall h \in (0, h_0^*].$$
**Proof.** As in Proposition 1, let us define $\tau_h = \min(\delta/(2C_0), T - t_{0h})$. We begin the proof with the study of the $(\nu + 1)$ first velocities. More precisely, we prove that they satisfy the announced estimate:

**Lemma 2.** For all $h \in (0, h_0^*]$ and for all $n \in \{0, \dots, \min(\nu, \lfloor \tau_h/h \rfloor)\}$ we have

$$|V^n| \le C_0^* \left( 4 \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} \right)^n < C_0.$$

**Proof.** The result is almost a direct consequence of Proposition 1. Indeed, let $h \in (0, h_0^*]$. By definition of $C_0^*$ we have

$$|V^0| = \left| \frac{U^1 - U^0}{h} \right| \le C_0^* \le C_0,$$

and with (15), we already know that $U^1 \in B_0$ and

$$\|V^1\|_{M(U^1)} \le 3\|V^0\|_{M(U^1)} + 2h\|F^1\|_{M(U^1)}.$$

Hence

$$|V^1| \le \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} (3|V^0| + 2h|F^1|) \le \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} (3C_0^* + 2hC_F) \le 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} C_0^*.$$

Now, let $n \in \{2, \dots, \nu - 1\}$ and assume that

$$|V^l| \le C_0^* \left( 4 \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} \right)^l \quad \forall l \in \{0, \dots, n-1\}.$$

Then $|V^l| \le C_0$ for all $l \in \{0, \dots, n-1\}$ and, once again using (15), we infer that $U^n \in B_0$ and

$$\|V^n\|_{M(U^n)} \le 2\|V^{n-1}\|_{M(U^{n-1})} + e\|V^{n-2}\|_{M(U^{n-2})} + 2h\|F^n\|_{M(U^n)}.$$

Hence

$$\begin{aligned} |V^n| & \le \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} \left( 2C_0^* \left( 4 \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} \right)^{n-1} + eC_0^* \left( 4 \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} \right)^{n-2} + 2hC_F \right) \\ & \le C_0^* \left( 4 \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} \right)^n. \end{aligned}$$

□
Let us assume now that $h \in (0, h_0^*]$, $v + 1 \le n \le \lfloor \tau_h / h \rfloor$ and $|V^l| \le C_0$ for all $l \in \{0, \dots, n-1\}$. We already know from Proposition 1 that

$$U^l \in B_0, \quad Z^l \in B_0 \quad \forall l \in \{0, \dots, n\},$$

and

$$|V^n| \le 4 \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} C_0.$$

But we may obtain a more precise estimate on $V^n$: using Proposition 3 (see below) there exists a constant $C_{B,C_0}$, depending only on $B$, $C_0$ and the data, such that

$$|\tilde{V}^n|^2 \le \max \left( |\tilde{V}^{n-1}|^2, \dots, |\tilde{V}^{n-v}|^2, |\tilde{V}^{n-v-1}|^2 \right) + C_{B,C_0}h$$

where we define

$$\tilde{V}^l = M^{1/2}(Z^l)V^l \quad \forall l \in \{0, \dots, \lfloor \frac{T-t_{0h}}{h} \rfloor\}.$$

By an immediate induction, we infer that

$$|V^n|^2 \le \frac{\lambda_{\max}}{\lambda_{\min}} \max \left( |V^0|^2, \dots, |V^\nu|^2 \right) + \frac{C_{B,C_0}}{\lambda_{\min}} (n-\nu)h,$$

and by defining

$$\tau_0 = \min \left( \frac{\lambda_{\min}}{C_{B,C_0}} \frac{15C_0^2}{16}, \frac{\delta}{2C_0} \right)$$

we obtain

$$|V^n| \le C_0 \quad \forall nh \in [0, \min(\tau_0, T - t_{0h})], \quad \forall h \in (0, h_0^*].$$

□

**Proposition 3.** Let us assume now that $h \in (0, h_0^*]$, $v + 1 \le n \le \lfloor \tau_h / h \rfloor$ and

$$|V^l| \le C_0 \quad \forall l \in \{0, \dots, n-1\}. \qquad (16)$$

There exists a constant $C_{B,C_0}$, depending only on $B$, $C_0$ and the data, such that

$$|\tilde{V}^n|^2 \le \max \left( |\tilde{V}^{n-1}|^2, \dots, |\tilde{V}^{n-v}|^2, |\tilde{V}^{n-v-1}|^2 \right) + C_{B,C_0}h \quad (17)$$

with

$$\tilde{V}^l = M^{1/2}(Z^l)V^l \quad \forall l \in \{0, \dots, \lfloor \frac{T-t_{0h}}{h} \rfloor\}. \qquad (18)$$
The proof of this result is rather technical and will be divided into several lemmas. Let us recall here some useful relations about the $Z^n$'s. By definition of the scheme, we have

$$Z^n = \frac{U^{n+1} + eU^{n-1}}{1+e} \quad \forall n \in \{1, \dots, \lfloor \frac{T-t_{0h}}{h} \rfloor\} \quad (19)$$

and by convention $Z^0 = U^0$. It follows that

$$U^n - Z^n = \frac{h}{1+e}(eV^{n-1} - V^n) \quad \forall n \in \{1, \dots, \lfloor \frac{T-t_{0h}}{h} \rfloor\} \quad (20)$$

and

$$Z^n - Z^{n-1} = \frac{h}{1+e}(V^n + eV^{n-2}) \quad \forall n \in \{2, \dots, \lfloor \frac{T-t_{0h}}{h} \rfloor\}. \quad (21)$$

**Proof.** First, we observe that (17) is obvious if $J(Z^n) = \emptyset$. Indeed, with Lemma 1, we know that

$$M(U^n)(V^{n-1} - V^n + hF^n) \in N_K(Z^n)$$

and thus

$$V^n = V^{n-1} + hF^n \quad \text{if } J(Z^n) = \emptyset.$$

With (21) we infer that

$$\begin{align}
|\tilde{V}^n|^2 &= |V^n|^2_{M(Z^n)} = |V^{n-1}|^2_{M(Z^n)} + 2h(V^{n-1}, F^n)_{M(Z^n)} + h^2|F^n|^2_{M(Z^n)} \nonumber \\
&\leq |\tilde{V}^{n-1}|^2 + L_M|Z^n - Z^{n-1}||V^{n-1}|^2 + 2h\lambda_{\max}|F^n||V^{n-1}| \nonumber \\
&\quad + h^2\lambda_{\max}|F^n|^2 \nonumber \\
&\leq |\tilde{V}^{n-1}|^2 + hL_M C_0^3 4 \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} + 3h\lambda_{\max}C_F C_0. \tag{22}
\end{align}$$

From now on, let us denote simply $J_l = J(Z^l)$ for all $l \in \{1, \dots, n\}$ and assume that $J_n \neq \emptyset$. We observe that there exists $k \in \{1, \dots, v\}$ such that

$$J_{n-k} \subset \bigcup_{p=0}^{k-1} J_{n-p}.$$

Indeed, for $k \geq 1$, $\text{Card}(\bigcup_{p=0}^k J_{n-p}) \geq \text{Card}(\bigcup_{p=0}^{k-1} J_{n-p}) + 1$ if $J_{n-k} \not\subset (\bigcup_{p=0}^{k-1} J_{n-p})$ and we have $J_n \neq \emptyset$, $(\bigcup_{p=0}^k J_{n-p}) \subset \{1, \dots, v\}$. We define $k \in \{1, \dots, v\}$ such that

$$J_{n-k} \subset \left( \bigcup_{p=0}^{k-1} J_{n-p} \right) \quad \text{and} \quad J_{n-m} \not\subset \left( \bigcup_{p=0}^{m-1} J_{n-p} \right), \quad \forall m \in \{1, \dots, k-1\}.$$
and with Proposition 1, we infer that

$$|\mu_{\beta}^{m}| \leq C_2 = \frac{\lambda_{\max}}{\sqrt{\lambda_{\min}}} \left(2 + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}}\right) C_0 \quad (24)$$

for all $\beta \in J_m$ and for all $m \in \{n-k, \dots, n\}$. It follows that

$$\tilde{V}^{m-1} - \tilde{V}^m + h\tilde{F}^m = (M^{1/2}(Z^{m-1}) - M^{1/2}(Z^m))V^{m-1} + \sum_{\alpha \in J_m} \mu_{\alpha}^m e_{\alpha}^n \\ + \sum_{\alpha \in J_m} \mu_{\alpha}^m (M^{1/2}(Z^m)M^{-1}(U^m)M^{1/2}(Z^m)e_{\alpha}^m - e_{\alpha}^n). \quad (25)$$

By summation for $m = n-k$ to $m = n$, we infer that

$$\tilde{V}^n - \tilde{V}^{n-k-1} = \sum_{m=n-k}^{n} \sum_{\alpha \in J_m} (-\mu_{\alpha}^m) M^{1/2}(Z^m) M^{-1}(U^m) M^{1/2}(Z^m) e_{\alpha}^m \\ + \sum_{m=n-k}^{n} (M^{1/2}(Z^m) - M^{1/2}(Z^{m-1})) V^{m-1} + h \tilde{F}^m. \quad (26)$$

But $\sum_{\alpha \in J_m} \mu_{\alpha}^m e_{\alpha}^n \in E_n$ since $J_m \subset \tilde{J}_n$, so

$$\operatorname{Proj} (E_n^{\perp}, \tilde{V}^n - \tilde{V}^{n-k-1}) \\ = \operatorname{Proj} \left( E_n^{\perp}, \sum_{m=n-k}^{n} \sum_{\alpha \in J_m} \mu_{\alpha}^{m} (e_{\alpha}^{n} - M^{1/2}(Z^{m}) M^{-1}(U^{m}) M^{1/2}(Z^{m}) e_{\alpha}^{m}) \right) \\ + \operatorname{Proj} \left( E_n^{\perp}, \sum_{m=n-k}^{n} (M^{1/2}(Z^{m}) - M^{1/2}(Z^{m-1})) V^{m-1} + h \tilde{F}^{m} \right)$$

which yields

$$|\operatorname{Proj}(E_n^{\perp}, \tilde{V}^n - \tilde{V}^{n-k-1})| \\ \leq \sum_{m=n-k}^{n} \sum_{\alpha \in J_m} |\mu_{\alpha}^m| \left( |e_{\alpha}^n - e_{\alpha}^m| + \|M^{1/2}(Z^m)\|^2 \|M^{-1}(Z^m) - M^{-1}(U^m)\| \right) \\ + \sum_{m=n-k}^{n} \|M^{1/2}(Z^m) - M^{1/2}(Z^{m-1})\| |V^{m-1}| + h \|M^{1/2}(Z^m)\| |F^m|.$$

Using the Lipschitz property of $M^{1/2}$ and $M^{-1}$ on $B_0$ we get

$$|\operatorname{Proj}(E_n^{\perp}, \tilde{V}^n - \tilde{V}^{n-k-1})| \leq \sum_{m=n-k}^{n} \sum_{\alpha \in J_m} |\mu_{\alpha}^m| (|e_{\alpha}^n - e_{\alpha}^m| + \lambda_{\max} L M^{-1} |U^m - Z^m|) \\ + \sum_{m=n-k}^{n} L M^{1/2} C_0 |Z^m - Z^{m-1}| + \sqrt{\lambda_{\max}} h C_F.$$
But $m \ge n-k \ge n-\nu \ge 1$, thus with (16), (14), (19) and (21)

$$
\begin{align}
|Z^m - Z^{m-1}| &= \frac{h}{1+e} |V^m + eV^{m-2}| \nonumber \\
&\le \begin{cases} hC_0 (1 + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}}) & \text{if } m = n, \\ hC_0 & \text{if } 2 \le m \le n-1, \end{cases} \tag{27} \\
|Z^1 - Z^0| &= \frac{h}{1+e} |V^1 + V^0| \le 2hC_0, \nonumber
\end{align}
$$

and with (20)

$$
\begin{align}
|U^m - Z^m| &= \frac{h}{1+e} |V^m - eV^{m-1}| \nonumber \\
&\le \begin{cases} hC_0 (1 + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}}) & \text{if } m = n, \\ hC_0 & \text{if } 1 \le m \le n-1. \end{cases} \tag{28}
\end{align}
$$

It follows that

$$
\begin{aligned}
|\operatorname{Proj}(E_n^\perp, \tilde{V}^n - \tilde{V}^{n-k-1})| \le & \sum_{m=n-k}^{n} h \left( L_{M^{1/2}} C_0^2 \left( 1 + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} \right) + \sqrt{\lambda_{\max}} C_F \right) \\
& + \sum_{m=n-k}^{n} \sum_{\alpha \in J_m} |\mu_\alpha^m| \left( |e_\alpha^n - e_\alpha^m| + \lambda_{\max} L_{M^{-1}} C_0 h \right) \\
& \times \left( 1 + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} \right).
\end{aligned}
$$

Since

$$
|Z^m - Z^n| \le (n-m)C_0h + \left(4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} - 1\right)C_0h \le \min(r_{B_0}, r'_{B_0})
$$

for all $m \in \{n-k, \dots, n\}$ (see (23)), we infer from Lemma 15 that

$$
|e_{\alpha}^{m} - e_{\alpha}^{n}| \le C'_{B_0} |Z^{m} - Z^{n}| \le h C'_{B_0} C_0 \left( 4 \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} + k - 1 \right).
$$

Finally, using (24) and recalling that $k \in \{1, \dots, \nu\}$, we may conclude with

$$
\begin{align*}
C_1 = (\nu + 1) & \left( L_{M^{1/2}} C_0^2 \left( 1 + 4 \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} \right) + \sqrt{\lambda_{\max}} C_F \right) \\
& + (\nu + 1) \nu C_2 C_0 \left( C'_{B_0} \left( 4 \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} + \nu - 1 \right) + \lambda_{\max} L_{M^{-1}} \left( 1 + 4 \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} \right) \right).
\end{align*}
$$
In order to compare $\tilde{V}_N^n$ and $\tilde{V}_N^{n-k-1}$, we first prove that

**Lemma 4.** There exists $C_3 > 0$ such that, for all $\beta \in \tilde{J}_n$:

$$ (\tilde{V}^{n-k-1}, e_{\beta}^n) - C_3h \le (\tilde{V}^n, e_{\beta}^n) \le -e (\tilde{V}^{n-k-1}, e_{\beta}^n) + C_3h. $$

**Proof.** With (26)

$$ \begin{aligned} & \left( e_{\beta}^{n}, \tilde{V}^{n} - \tilde{V}^{n-k-1} - h \sum_{m=n-k}^{n} \tilde{F}^{m} \right) \\ &= \sum_{m=n-k}^{n} \sum_{\alpha \in J_m} (-\mu_{\alpha}^{m}) (e_{\beta}^{n}, M^{1/2}(Z^{m})M^{-1}(U^{m})M^{1/2}(Z^{m})e_{\alpha}^{m}) \\ &\quad + \sum_{m=n-k}^{n} \left( e_{\beta}^{n}, (M^{1/2}(Z^{m}) - M^{1/2}(Z^{m-1}))V^{m-1} \right) \quad \forall \beta \in \tilde{J}_n. \end{aligned} $$

But, for all $\beta \in \tilde{J}_n$, there exists $l \in \{n-k+1, \dots, n\}$ such that $\beta \in J_l$. Thus, if $\alpha \in J_m$ and $\alpha \ne \beta$, we get

$$ \begin{aligned} |(e_{\alpha}^m, e_{\beta}^n)| &\le |(e_{\alpha}^m, e_{\beta}^l)| + |(e_{\alpha}^m, e_{\beta}^n - e_{\beta}^l)| \\ &\le C_{B_0} |Z^m - Z^l| + C'_{B_0} |Z^l - Z^n| \\ &\le C_{B_0} \left( |l-m| + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} - 1 \right) hC_0 \\ &\quad + C'_{B_0} \left( (n-l) + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} - 1 \right) hC_0 \\ &\le 2C_e \left( k + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} - 1 \right) hC_0, \quad \text{with } C_e = \max(C_{B_0}, C'_{B_0}). \end{aligned} $$

If $\beta = \alpha$, we have

$$ \begin{aligned} (e_{\alpha}^m, e_{\alpha}^n) &= |e_{\alpha}^n|^2 + (e_{\alpha}^m - e_{\alpha}^n, e_{\alpha}^n) \\ &\ge 1 - C'_{B_0} |Z^m - Z^n| \ge 1 - C_e \left( k + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} - 1 \right) hC_0. \end{aligned} $$
It follows that

$$
\begin{aligned}
& \left( e_{\beta}^{n}, \tilde{V}^{n} - \tilde{V}^{n-k-1} - h \sum_{m=n-k}^{n} \tilde{F}^{m} \right) \\
\ge & - \sum_{m=n-k}^{n} \sum_{\alpha \in J_{m}} |\mu_{\alpha}^{m}| 2C_{e} \left( k + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} - 1 \right) hC_{0} \\
& - \sum_{m=n-k}^{n} \sum_{\alpha \in J_{m}} |\mu_{\alpha}^{m}| \|M^{1/2}(Z^{m})\|^{2} \|M^{-1}(Z^{m}) - M^{-1}(U^{m})\| \\
& - \sum_{m=n-k}^{n} \left\| M^{1/2}(Z^{m}) - M^{1/2}(Z^{m-1}) \right\| |V^{m-1}|
\end{aligned}
$$

and using estimates (24), (27), (28) and the Lipschitz property of $M^{-1}$ and $M^{-1/2}$
we get

$$
\begin{aligned}
(e_{\beta}^{n}, \tilde{V}^{n}) \geq (e_{\beta}^{n}, \tilde{V}^{n-k-1}) & - (\nu+1)L_{M^{1/2}} C_{0}^{2} h \left(1 + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}}\right) \\
& - (\nu+1)\sqrt{\lambda_{\max}} C_{F} h - \hat{C}_{3} h
\end{aligned}
$$

with

$$
\hat{C}_3 = (\nu+1)\nu C_2 C_0 h \left( 2C_e \left( k + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} - 1 \right) + \lambda_{\max} L M^{-1} \left( 1 + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} \right) \right).
$$

This is the left side of Lemma 4. For the right-hand side we consider $l \in$
$\{n-k+1, \dots, n\}$ such that $\beta \in J_l$ and $\beta \notin \bigcup_{m=l+1}^n J_m$. Then,

$$
\begin{aligned}
& \left( e_{\beta}^{n}, \tilde{V}^{n} - \tilde{V}^{l} - h \sum_{m=l+1}^{n} \tilde{F}^{m} \right) \\
&= \sum_{m=l+1}^{n} \sum_{\alpha \in J_m} (-\mu_{\alpha}^{m}) (e_{\beta}^{n}, M^{1/2}(Z^{m}) M^{-1}(U^{m}) M^{1/2}(Z^{m}) e_{\alpha}^{m}) \\
&\quad + \sum_{m=l+1}^{n} (e_{\beta}^{n}, (M^{1/2}(Z^{m}) - M^{1/2}(Z^{m-1})) V^{m-1}).
\end{aligned}
$$

Since $\beta \notin \bigcup_{m=l+1}^{n} J_m$, we have

$$
\begin{align}
| (e_{\beta}^{n}, e_{\alpha}^{m}) | &\le | (e_{\beta}^{n} - e_{\beta}^{l}, e_{\alpha}^{m}) | + | (e_{\beta}^{l}, e_{\alpha}^{m}) | \nonumber \\
&\le C_e \left( (n-l) + (m-l) + 8 \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} - 2 \right) C_0 h \nonumber \\
&\le 2C_e \left( (n-l) + 4 \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} - 1 \right) C_0 h \tag{29}
\end{align}
$$
for all $m \in \{l+1, \dots, n\}$, for all $\alpha \in J_m$ and thus

$$ (\tilde{V}^n, e_\beta^n) \le (\tilde{V}^l, e_\beta^n) + h \sum_{m=l+1}^n (\tilde{F}^m, e_\beta^n) \\ + (n-l)\nu C_2 \left( 2C_e \left( (n-l) + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} - 1 \right) \right. \\ \left. + L_{M^{-1}}\lambda_{\max} \left( 1 + \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} \right) \right) C_0 h \\ + (n-l)L_{M^{1/2}} C_0^2 h \left( 1 + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} \right). $$

Moreover, $\beta \in J_l$ and $l \ge n-k+1 \ge 2$ thus, with Proposition 1,

$$ (\nabla f_\beta(Z^l), V^l + eV^{l-2}) \le \frac{L_f h}{2(1+e)} |V^l + eV^{l-2}|^2 $$

and

$$ (e_\beta^l, M^{1/2}(Z^l)(V^l + eV^{l-2})) \le \frac{L_f h}{2m_{B_0}} |V^l + eV^{l-2}|^2. $$

With (16), (14) and (21) it follows that

$$ (\tilde{V}^l + e \tilde{V}^{l-2}, e_\beta^l) \le \frac{L_f h}{2m_{B_0}} |V^l + eV^{l-2}|^2 \\ + e \left( (M^{1/2}(Z^{l-2}) - M^{1/2}(Z^l)) V^{l-2}, e_\beta^l \right) \le C_4 h $$

with

$$ C_4 = \frac{L_f}{2m_{B_0}} C_0^2 \left( 4 \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} + e \right)^2 + e L_{M^{1/2}} C_0^2 \left( 4 \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} + 3 \right). $$

Recalling that $|e_\beta^l - e_\beta^n| \le C_e |Z^l - Z^n|$, we infer that

$$ (\tilde{V}^l + e \tilde{V}^{l-2}, e_\beta^n) \le C_5 h $$

with

$$ C_5 = C_4 + C_e \left( (n-l) + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} - 1 \right) C_0^2 \left( 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} + e \right) \sqrt{\lambda_{\max}}. $$

Finally, again using (25), we get

$$ \tilde{V}^{l-2} - \tilde{V}^{n-k-1} = \sum_{m=n-k}^{l-2} h \tilde{F}^m - (M^{1/2}(Z^{m-1}) - M^{1/2}(Z^m)) V^{m-1} \\ - \sum_{m=n-k}^{l-2} \sum_{\alpha \in J_m} \mu_\alpha^m (M^{1/2}(Z^m) M^{-1}(U^m) M^{1/2}(Z^m) e_\alpha^m) $$
and with (H6) (see (8)), we obtain

$$ \begin{aligned} & (\tilde{V}^{l-2} - \tilde{V}^{n-k-1}, e_{\beta}^{l-2}) \\ & \geq -(k' + 1)\sqrt{\lambda_{\max}} C_F h - (k' + 1)\nu C_2 \\ & \times \left( 2C_e \left( k' + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} - 1 \right) + \lambda_{\max} L_{M^{-1}} \right) C_0 h \\ & - 2(k' + 1)L_{M^{1/2}} C_0^2 h \end{aligned} $$

with $k' = l - (n - k) - 2$. Hence, with (29)

$$ \begin{aligned} (\tilde{V}^{l-2} - \tilde{V}^{n-k-1}, e_{\beta}^{n}) &\geq -(k' + 1)h \left( \nu C_2 C_0 \left( 2C_e \left( k' + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} - 1 \right) + \lambda_{\max} L_{M^{-1}} \right) + 2L_{M^{1/2}} C_0^2 + \sqrt{\lambda_{\max}} C_F \right) \\ &\quad - 2C_0^2 C_e h \left( n - l + 1 + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} \right) \sqrt{\lambda_{\max}}. \end{aligned} $$

Observing that $n - \nu \leq n - k < l \leq n$, the conclusion follows with

$$ C_3 = \max(\hat{C}_3, \tilde{C}_3) + (\nu + 1)\sqrt{\lambda_{\max}} C_F + (\nu + 1)L_{M_{1/2}} C_0^2 \left( 1 + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} \right) $$

and

$$ \tilde{C}_3 = (\nu - 1)\nu C_2 C_0 \left( 2C_e \left( \nu + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} - 2 \right) + \lambda_{\max} L_{M^{-1}} \left( 1 + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} \right) \right) \\ + C_4 + 2C_0^2 C_e \left( \nu + 1 + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} \right) \left( 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} + 3e \right) \sqrt{\lambda_{\max}}. $$

□

Now, we may apply the following lemma.

**Lemma 5.** There exists $C_5 > 0$ such that, for all $w \in E_n$, we have

$$ \left| |w|^2 - \sum_{\alpha \in \tilde{J}_n} (w, e_{\alpha}^n)^2 \right| \leq C_5 h \sum_{\alpha \in \tilde{J}_n} (w, e_{\alpha}^n)^2. $$

**Proof.** Let $w \in E_n = \text{span}\{e_{\alpha}^n, \alpha \in \tilde{J}_n\}$. There exists $(\mu_{\beta})_{\beta \in \tilde{J}_n} \in \mathbb{R}^{\text{Card}\tilde{J}_n}$ such that

$$ w = \sum_{\beta \in \tilde{J}_n} \mu_{\beta} e_{\beta}^{n}, $$
and for all $\alpha \in \tilde{J}_n$

$$ (w, e_{\alpha}^{n}) = \sum_{\beta \in \tilde{J}_n} \mu_{\beta}(e_{\alpha}^{n}, e_{\beta}^{n}) = \mu_{\alpha} + \sum_{\beta \in \tilde{J}_n \setminus \{\alpha\}} \mu_{\beta}(e_{\alpha}^{n}, e_{\beta}^{n}). $$

But $\tilde{J}_n = \bigcup_{p=0}^{k-1} J_{n-p} = \bigcup_{p=0}^{k} J_{n-p}$, and there exists $(p, q) \in \{0, \dots, k-1\}^2$ such that $\alpha \in J_{n-p}$ and $\beta \in J_{n-q}$. Hence,

$$ \begin{aligned} |(e_{\alpha}^{n}, e_{\beta}^{n})| &\le |(e_{\alpha}^{n} - e_{\alpha}^{n-p}, e_{\beta}^{n})| + |(e_{\alpha}^{n-p}, e_{\beta}^{n-q})| + |(e_{\alpha}^{n-p}, e_{\beta}^{n} - e_{\beta}^{n-q})| \\ &\le |e_{\alpha}^{n} - e_{\alpha}^{n-p}| + |(e_{\alpha}^{n-p}, e_{\beta}^{n-q})| + |e_{\beta}^{n} - e_{\beta}^{n-q}|. \end{aligned} $$

We recall that, for all $(m, l) \in \{0, \dots, k-1\}^2$ we have

$$ |Z^{n-m} - Z^{n-l}| \le \left( \nu + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} - 2 \right) C_{0h} \le \min(r_{B_0}, r'_{B_0}). $$

Thus, with (8) and Lemma 15 we get for all $\beta \in \tilde{J}_n \setminus \{\alpha\}$

$$ \begin{aligned} |e_{\alpha}^{n} - e_{\alpha}^{n-p}| &\le \left( p + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} - 1 \right) C'_{B_0} C_{0h}, \\ |e_{\beta}^{n} - e_{\beta}^{n-q}| &\le \left( q + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} - 1 \right) C'_{B_0} C_{0h}, \\ |(e_{\alpha}^{n-p}, e_{\beta}^{n-q})| &\le \left( |p-q| + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} - 1 \right) C_{B_0} C_{0h}, \end{aligned} $$

and

$$ |(e_{\alpha}^{n}, e_{\beta}^{n})| \le 3 \left( k + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} - 2 \right) C_e C_{0h}. $$

Let us denote

$$ C_6 = 3C_eC_0 \left( \nu + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} - 2 \right). $$

We infer that

$$ |(w, e_{\alpha}^{n}) - \mu_{\alpha}| \le \sum_{\beta \in \tilde{J}_n \setminus \{\alpha\}} |\mu_{\beta}||(e_{\alpha}^{n}, e_{\beta}^{n})| \le C_6 h \sum_{\beta \in \tilde{J}_n \setminus \{\alpha\}} |\mu_{\beta}| $$

and since $\text{Card}(\tilde{J}_n) \le \nu$ and $k \le \nu$ we get

$$ |(w, e_{\alpha}^{n}) - \mu_{\alpha}| \le C_6 h \sqrt{\nu - 1} \left( \sum_{\beta \in \tilde{J}_n \setminus \{\alpha\}} |\mu_{\beta}|^2 \right)^{1/2}. $$
On the other hand, we also have

$$|w|^2 = \sum_{(\alpha, \beta) \in \tilde{J}_n^2} \mu_\alpha \mu_\beta (e_\alpha^n, e_\beta^n) = \sum_{\alpha \in \tilde{J}_n} \mu_\alpha^2 + \sum_{(\alpha, \beta) \in \tilde{J}_n^2, \alpha \neq \beta} \mu_\alpha \mu_\beta (e_\alpha^n, e_\beta^n)$$

and, with the same arguments as above, we get

$$ \begin{aligned} \left| |w|^2 - \sum_{\alpha \in \tilde{J}_n} \mu_\alpha^2 \right| &\le \sum_{(\alpha, \beta) \in \tilde{J}_n^2, \alpha \ne \beta} \left| \mu_\alpha \mu_\beta \right| \left| (e_\alpha^n, e_\beta^n) \right| \le C_6 h \sum_{(\alpha, \beta) \in \tilde{J}_n^2, \alpha \ne \beta} \left| \mu_\alpha \mu_\beta \right| \\ &\le C_6 h (\nu - 1) \sum_{\alpha \in \tilde{J}_n} \mu_\alpha^2. \end{aligned} $$

It follows that

$$ \begin{aligned} \left| |w|^2 - \sum_{\alpha \in \tilde{J}_n} (w, e_\alpha^n)^2 \right| &\le \left| |w|^2 - \sum_{\alpha \in \tilde{J}_n} \mu_\alpha^2 \right| + \left| \sum_{\alpha \in \tilde{J}_n} \mu_\alpha^2 - (w, e_\alpha^n)^2 \right| \\ &\le C_6 h (\nu - 1) \sum_{\alpha \in \tilde{J}_n} \mu_\alpha^2 + \sum_{\alpha \in \tilde{J}_n} \left| \mu_\alpha^2 - (w, e_\alpha^n)^2 \right| \end{aligned} $$

and for all $\alpha \in \tilde{J}_n$

$$ \begin{aligned} \left| \mu_{\alpha}^{2} - (w, e_{\alpha}^{n})^{2} \right| &= \left| \mu_{\alpha} - (w, e_{\alpha}^{n}) \right| \left| \mu_{\alpha} + (w, e_{\alpha}^{n}) \right| \le \left| \mu_{\alpha} - (w, e_{\alpha}^{n}) \right| \\ & \quad \times (2\left|\mu_{\alpha}\right| + \left|\mu_{\alpha} - (w, e_{\alpha}^{n})\right|) \\ &\le \left| \mu_{\alpha} - (w, e_{\alpha}^{n}) \right| \left( 2 \left( \sum_{\beta \in \tilde{J}_n} |\mu_{\beta}|^2 \right)^{1/2} + \left| \mu_{\alpha} - (w, e_{\alpha}^{n}) \right| \right) \\ &\le C_6 h \sqrt{\nu - 1} \left( 2 + C_6 h \sqrt{\nu - 1} \right) \sum_{\beta \in \tilde{J}_n} \mu_{\beta}^2. \end{aligned} $$

Since

$$h_0^* \le \frac{1}{72\nu^{3/2}C_eC_0(\nu + 4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}})}$$

we get

$$\left| \mu_{\alpha}^{2} - (w, e_{\alpha}^{n})^{2} \right| \le 3C_{6}h\sqrt{\nu-1} \sum_{\beta \in \tilde{J}_{n}} \mu_{\beta}^{2}$$

for all $\alpha \in \tilde{J}_n$. Thus

$$\sum_{\alpha \in \tilde{J}_n} (w, e_\alpha^n)^2 \ge (1 - 3C_6 h \nu \sqrt{\nu-1}) \sum_{\alpha \in \tilde{J}_n} \mu_\alpha^2 \ge \frac{1}{2} \sum_{\alpha \in \tilde{J}_n} \mu_\alpha^2$$
which yields finally

$$
\begin{aligned}
\left| w \right|^2 - \sum_{\alpha \in \tilde{J}_n} (w, e_\alpha^n)^2 &\le C_6 h \left( (\nu - 1) + 3\nu\sqrt{\nu - 1} \right) \sum_{\alpha \in \tilde{J}_n} \mu_\alpha^2 \\
&\le 4\nu^{3/2} C_6 h \sum_{\alpha \in \tilde{J}_n} (w, e_\alpha^n)^2.
\end{aligned}
$$

The conclusion follows with $C_7 = 4\nu^{3/2}C_6$. $\square$

Now we can prove that

$$
|\tilde{V}^n|^2 \le \max \left( |\tilde{V}^{n-1}|^2, \dots, |\tilde{V}^{n-\nu}|^2, |\tilde{V}^{n-\nu-1}|^2 \right) + C_{B,C_0} h
$$

where $C_{B,C_0}$ is a constant which depends only on $\mathcal{B}$ and $C_0$.

Indeed, if $J_n = \emptyset$, we have (see (22))

$$
|\tilde{V}^n|^2 \le |\tilde{V}^{n-1}|^2 + hL_M C_0^3 4 \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} + 3h\lambda_{\max}C_F C_0.
$$

If $J_n \neq \emptyset$, we have with Lemma 3 and Lemma 4

$$
|\tilde{V}_T^n| \le |\tilde{V}_T^{n-k-1}| + |\mathrm{Proj}(E_n^\perp, \tilde{V}^n - \tilde{V}^{n-k-1})| \le |\tilde{V}_T^{n-k-1}| + C_1 h
$$

and, for all $\beta \in \tilde{J}_n$

$$
| (\tilde{V}^n, e_\beta^n) | = | (\tilde{V}_N^n, e_\beta^n) | \le | (\tilde{V}^{n-k-1}, e_\beta^n) | + C_3 h = | (\tilde{V}_N^{n-k-1}, e_\beta^n) | + C_3 h.
$$

With Lemma 5 we get

$$
\begin{aligned}
|\tilde{V}^n|^2 &= |\tilde{V}_T^n|^2 + |\tilde{V}_N^n|^2 \\
&\le |\tilde{V}_T^{n-k-1}|^2 + 2C_1h|\tilde{V}_T^{n-k-1}| + (C_1h)^2 + (1+C_7h) \sum_{\beta \in \tilde{J}_n} (\tilde{V}_N^n, e_\beta^n)^2
\end{aligned}
$$

But

$$
|\tilde{V}_T^{n-k-1}| \le |\tilde{V}^{n-k-1}| \le C_0 \sqrt{\lambda_{\max}}
$$

and

$$
| (\tilde{V}_N^{n-k-1}, e_\beta^n) | \le | \tilde{V}^{n-k-1} | \le C_0 \sqrt{\lambda_{\max}}.
$$

Thus

$$
\begin{aligned}
|\tilde{V}^n|^2 &\le |\tilde{V}_T^{n-k-1}|^2 + 2C_1hC_0\sqrt{\lambda_{\max}} + (C_1h)^2 \\
&\quad + (1+C_7h) \sum_{\beta \in \tilde{J}_n} \left( (\tilde{V}_N^{n-k-1}, e_\beta^n)^2 + 2C_3C_0\sqrt{\lambda_{\max}}h + (C_3h)^2 \right).
\end{aligned}
$$
Since $h_0^* \le \frac{1}{3C_7}$, we have

$$1 - C_7h > 0 \quad \text{and} \quad \frac{1 + C_7h}{1 - C_7h} \le 1 + 3C_7h.$$

Thus

$$|\tilde{V}^n|^2 \le |\tilde{V}_T^{n-k-1}|^2 + \frac{1+C_7h}{1-C_7h} |\tilde{V}_N^{n-k-1}|^2 + C_0 \sqrt{\lambda_{\max}} h (2C_1 + 2\nu C_3(1+C_7h)) \\ + (C_1h)^2 + (C_3h)^2(1+C_7h)\nu.$$

Finally, in both cases ($J_n = \emptyset$ or $J_n \ne \emptyset$), we have

$$|\tilde{V}^n|^2 \le \max \left( |\tilde{V}^{n-1}|^2, \dots, |\tilde{V}^{n-\nu-1}|^2 \right) + C_{B,C_0} h$$

with

$$C_{B,C_0} = \max \left( L_M C_0^3 4 \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} + 3\lambda_{\max} C_F C_0, C_0 \sqrt{\lambda_{\max}} (2C_1 + 2\nu C_3(1+C_7h^*)) \right. \\ \left. + C_1^2 h^* + C_3^2 h^* (1+C_7h^*)\nu + 3C_7\lambda_{\max} C_0^2 \right)$$

which allows us to conclude the proof. □

Let us now consider the initialization procedure given by formula (4), that is, let $t_{0h} = 0$ and

$$U^0 = u_0, \quad U^1 \in \operatorname{Argmin}_{Z \in K} \|u_0 + h v_0 + h z(h) - Z\|_{M(u_0)}, \quad \lim_{h \to 0} z(h) = 0,$$

for all $h \in (0, h^*]$. We choose $B = \bar{B}(u_0, C+1)$ with $C \ge 0$. By applying the preceding results, we get a uniform estimate of the discrete velocities on a non-trivial time interval:

**Theorem 1.** For all $C_0^* \ge 2\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} (|v_0| + 1)$ and for all $C_0 > (4\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}})^{\nu+1} C_0^*$, there exist $h_0^* \in (0, h^*]$ and $\tau_0 > 0$, depending only on $B$, $C_0$ and $C_0^*$ and the data, such that

$$|V^n| = \left| \frac{U^{n+1} - U^n}{h} \right| \le C_0 \quad \forall nh \in [0, \min(\tau_0, T)], \quad \forall h \in (0, h_0^*].$$

**Proof.** Let $C_0^* \ge 2\sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} (|v_0|+1)$, and let $C_0$ and $h_0^*$ be defined as in Proposition 2. With (9) we have

$$|z(h)| \le 1 \quad \forall h \in (0, h^*]$$

and by definition of $U^1$, we have

$$\|u_0 + h v_0 + h z(h) - U^1\|_{M(u_0)} \le \|u_0 + h v_0 + h z(h) - Z\|_{M(u_0)} \quad \forall Z \in K.$$
By choosing $Z = U^0 = u_0$, we get

$$ \|V^0\|_{M(u_0)} \le 2 \left( \|v^0\|_{M(u_0)} + \|z(h)\|_{M(u_0)} \right). $$

Since $u_0 \in \mathcal{B} \subset B_0$, we infer that

$$ |V^0| \le 2 \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} (|v_0| + 1) \le C_0^* \quad \forall h \in (0, h^*]. $$

It follows that

$$ |V^0| \le C_0^* < \left( 4 \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} \right)^{-(v+1)} C_0 \quad \forall h \in (0, h_0^*] $$

and we may apply Proposition 2, which yields the announced result. □

### 3. Convergence of the approximate solutions $(u_h)_{h^* \ge h > 0}$

Before passing to the limit as $h$ tends to zero in the sequence $(u_h)_{h^* \ge h > 0}$, we prove an estimate for the discrete accelerations.

**Proposition 4.** Let us assume that there exist $C_0 > 0$, $\tau_0 > 0$, $h_0^* \in (0, h^*]$ and a subsequence $(h_i)_{i \in \mathbb{N}}$, decreasing to zero, such that

$$ |V^n| \le C_0 \quad \forall nh_i \in [0, \min(\tau_0, T)], \quad \forall h_i \in (0, h_0^*]. \quad (30) $$

Then there exist $h_1^* \in (0, h_0^*]$ and $C_0' > 0$ such that, for all $h_i \in (0, h_1^*]$

$$ \sum_{n=1}^{N} |V^n - V^{n-1}| \le C_0', \quad \text{with } N = \left\lfloor \frac{\min(\tau_0, T)}{h_i} \right\rfloor. $$

**Proof.** We begin the proof as in [11] Proposition 3. More precisely, let $\mathcal{B} = \bar{B}(u_0, C+1)$ with $C \ge 0$, $B_0$ be defined by (11) and $C_F$ be defined by (13). Without loss of generality, possibly decreasing $h_0^*$, we assume that $C_0 h_0^* \le 1$ and $C_F h_0^* \le C_0$. We denote $K_1 = K \cap B_1 = K \cap \bar{B}(u_0, C_0 T + 1)$ and

$$ \lambda_{\min, B_1} = \inf_{u \in B_1} \lambda_{\min}(u) = \frac{1}{\sup_{u \in B_1} \|M^{-1}(u)\|}, $$

$$ \lambda_{\max, B_1} = \sup_{u \in B_1} \lambda_{\max}(u) = \sup_{u \in B_1} \|M(u)\|. $$

Let $h_i \in (0, h_0^*]$. By definition of the scheme, we have $Z^n \in K$ for all $n \in \{0, \dots, \lfloor T/h_i \rfloor\}$ and assumption (30) implies that

$$ |U^n - U^0| \le h_i \sum_{k=0}^{n-1} |V^k| \le nh_i C_0 \le C_0 T \quad \forall n \in \{0, \dots, N+1\} $$
and with (20)

$$
\begin{aligned}
\left|Z^n - U^0\right| &\le \left|U^n - U^0\right| + \frac{h_i}{1+e} \left|V^n - eV^{n-1}\right| \\
&\le C_0 T + C_0 h_i \le C_0 T + 1 \quad \forall n \in \{1, \dots, N\},
\end{aligned}
$$

thus $Z^n \in K_1$ for all $n \in \{1, \dots, N\}$.

The continuity of the mappings $f_\alpha, \alpha \in \{1, \dots, \nu\}$, implies that, for all $q \in \mathbb{R}^d$, there exists $r_q > 0$ such that

$$
f_{\alpha}(q') \ge \frac{f_{\alpha}(q)}{2} > 0 \quad \forall q' \in \bar{B}(q, r_q), \quad \forall \alpha \notin J(q). \quad (31)
$$

Without loss of generality, we may assume that $r_q \le \min(r_{B_1}, r'_{B_1}, 1/(4\nu^2 C_{e,B_1}))$
where $C_{e,B_1} = \max(C_{B_1}, C'_{B_1})$ and $C_{B_1}, r_{B_1}, C'_{B_1}$ and $r'_{B_1}$ are the constants defined
at assumption (H6) and Lemma 15 for the compact set $B_1$. Then, the compactness
of $K_1$ implies that there exists $(q_j)_{1 \le j \le l}$ such that $q_j \in K_1$ for all $j \in \{1, \dots, l\}$
and

$$
K_1 \subset \bigcup_{i=1}^{l} B \left( q_i, \frac{r_{q_i}}{2} \right).
$$

We define

$$
r = \min_{1 \le i \le l} \frac{r_{q_i}}{2}.
$$

Let $h_1^* \in (0, \min(h_0^*, r/(2C_0))]$ and $h_i \in (0, h_1^*]$. Using (21), we infer that for all $(n, m) \in \{1, \dots, N\}^2$ such that $n \le m-1$ we have

$$
|Z^m - Z^n| \le \sum_{k=n+1}^{m} \frac{h_i}{1+e} \left|V^k + eV^{k-2}\right| \le (m-n)h_i C_0.
$$

Let $p = \lfloor r/(C_0h_i) \rfloor$ and let $n \in \{1, \dots, N-1\}$. Then, there exists $i \in \{1, \dots, l\}$ such that $Z^n \in B(q_i, r_{q_i}/2)$ and $Z^m \in B(q_i, r_{q_i})$ for all $m \in \{n, \dots, \min(N, n+p)\}$. With (31) we infer that

$$
f_{\alpha}(Z^m) > 0 \quad \forall \alpha \notin J(q_i) \quad \forall m \in \{n, \dots, \min(N, n+p)\}
$$

that is, $J(Z^m) \subset J(q_i)$ for all $m \in \{n, \dots, \min(N, n+p)\}$.

Let $m \in \{n+1, \dots, \min(N, n+p)\}$. Since $|Z^m - q_i| \le r_{q_i} \le \min(r_{B_1}, r'_{B_1})$,
we may define

$$
e_{\alpha}^{m} = e_{\alpha}(Z^{m}) = \frac{M^{-1/2}(Z^{m}) \nabla f_{\alpha}(Z^{m})}{|M^{-1/2}(Z^{m}) \nabla f_{\alpha}(Z^{m})|} \quad \forall \alpha \in J(q_{i})
$$

and we let $\mu_{\alpha}^{m} = 0$ if $\alpha \in J(q_i) \setminus J(Z^m)$. It follows (with Lemma 1) that

$$
M(U^m)(V^{m-1} - V^m + h_i F^m) = \sum_{\beta \in J(q_i)} \mu_{\beta}^m M^{1/2}(Z^m) e_{\beta}^m,
$$
with $\mu_{\beta}^m \le 0$ for all $\beta \in J(q_i)$. From assumption (H6) we know that, for all $(\alpha, \beta) \in J(Z^m)^2$

$$ (e_{\alpha}^m, e_{\beta}^m) = \begin{cases} 1 & \text{if } \alpha = \beta, \\ 0 & \text{if } \alpha \neq \beta. \end{cases} $$

Thus for all $\beta \in J(Z^m)$

$$ (M^{-1/2}(Z^m)M(U^m)(V^{m-1} - V^m + h_i F^m), e_{\beta}^m) = \mu_{\beta}^m \le 0 $$

and

$$ \begin{aligned} |\mu_{\beta}^m| &= (M^{-1/2}(Z^m)M(U^m)(V^m - V^{m-1} - h_i F^m), e_{\beta}^m) \\ &\le |M^{-1/2}(Z^m)M(U^m)(V^m - V^{m-1} - h_i F^m)| \\ &\le \|M^{-1/2}(Z^m)\| \|M(U^m)\| |V^m - V^{m-1} - h_i F^m|. \end{aligned} $$

On the other hand, for all $\alpha \in J(q_i) \setminus J(Z^m)$

$$ (M^{-1/2}(Z^m)M(U^m)(V^{m-1} - V^m + h_i F^m), e_{\alpha}^m) = \sum_{\beta \in J(Z^m)} \mu_{\beta}^m (e_{\beta}^m, e_{\alpha}^m) $$

and since $(\alpha, \beta) \in J(q_i)^2$ with $\alpha \ne \beta$, we get with assumption (H6) (see (8)) and Lemma 15

$$ \begin{aligned} |(e_{\beta}^m, e_{\alpha}^m)| &\le |(e_{\beta}^m, e_{\alpha}^m - e_{\alpha}(q_i))| + |(e_{\alpha}(q_i), e_{\beta}^m)| \\ &\le (C'_{B_1} + C_{B_1})|Z^m - q_i| \le 2C_{e, B_1}r_{q_i}. \end{aligned} $$

It follows that, for all $\alpha \in J(q_i) \setminus J(Z^m)$

$$ \begin{aligned} & |(M^{-1/2}(Z^m)M(U^m)(V^{m-1} - V^m + h_i F^m), e_{\alpha}^m)| \\ &\le \sum_{\beta \in J(Z^m)} |\mu_{\beta}^m| |(e_{\beta}^m, e_{\alpha}^m)| \\ &\le 2\nu C_{e, B_1} r_{q_i} |M^{-1/2}(Z^m)M(U^m)(V^m - V^{m-1} - h_i F^m)|. \end{aligned} $$

Then we infer that

$$ \begin{aligned} & |M^{-1/2}(Z^m)M(U^m)(V^m - V^{m-1} - h_i F^m)| \\ &= \left| \sum_{\beta \in J(Z^m)} -\mu_{\beta}^m e_{\beta}^m \right| \le \sum_{\beta \in J(Z^m)} |\mu_{\beta}^m| \\ &= \sum_{\beta \in J(Z^m)} (M^{-1/2}(Z^m)M(U^m)(V^m - V^{m-1} - h_i F^m), e_{\beta}^m) \\ &\le \sum_{\beta \in J(q_i)} (M^{-1/2}(Z^m)M(U^m)(V^m - V^{m-1} - h_i F^m), e_{\beta}^m) \\ &\quad + 2\nu^2 C_{e, B_1} r_{q_i} |M^{-1/2}(Z^m)M(U^m)(V^m - V^{m-1} - h_i F^m)|. \end{aligned} $$
Thus, for all $m \in \{n+1, \dots, \min(N, n+p)\}$

$$ (1 - 2\nu^2 C_{e,B_1} r_{q_i}) \left| M^{-1/2}(Z^m) M(U^m) (V^m - V^{m-1} - h_i F^m) \right| \\ \le \sum_{\beta \in J(q_i)} \left( M^{-1/2}(Z^m) M(U^m) (V^m - V^{m-1} - h_i F^m), e_{\beta}^m \right). $$

But $r_{q_i} \le 1/(4\nu^2 C_{e,B_1})$, so we get

$$ \frac{1}{2} \left| M^{-1/2}(Z^m) M(U^m) (V^m - V^{m-1} - h_i F^m) \right| \\ \le \sum_{\beta \in J(q_i)} \left( M^{-1/2}(Z^m) M(U^m) (V^m - V^{m-1} - h_i F^m), e_{\beta}^m \right) $$

for all $m \in \{n+1, \dots, \min(N, n+p)\}$.

Now, we rewrite the right-hand side in order to obtain a telescopic sum. More precisely

$$ \begin{aligned} & \left( M^{-1/2}(Z^m) M(U^m) (V^m - V^{m-1} - h_i F^m), e_{\beta}^m \right) \\ & \le - \left( M^{-1/2}(Z^{m-1}) M(U^{m-1}) V^{m-1}, e_{\beta}^{m-1} \right) \\ & \quad + \left( M^{-1/2}(Z^m) M(U^m) V^m, e_{\beta}^m \right) \\ & \quad - \left( (M^{-1/2}(Z^m) M(U^m) - M^{-1/2}(Z^{m-1}) M(U^{m-1})) V^{m-1}, e_{\beta}^{m-1} \right) \\ & \quad - \left( M^{-1/2}(Z^m) M(U^m) V^{m-1}, e_{\beta}^m - e_{\beta}^{m-1} \right) \\ & \quad - h_i \left( M^{-1/2}(Z^m) M(U^m) F^m, e_{\beta}^m \right). \end{aligned} $$

Using the Lipschitz properties of $M^{-1/2}$ and $M$ on $B_1$ we get

$$ \begin{aligned} & \left| \left( (M^{-1/2}(Z^m) M(U^m) - M^{-1/2}(Z^{m-1}) M(U^{m-1})) V^{m-1}, e_{\beta}^{m-1} \right) \right| \\ & \le L_{M^{-1/2}} |Z^m - Z^{m-1}| \|M(U^m)\| |V^{m-1}| \\ & \quad + L_M |U^m - U^{m-1}| \|M^{-1/2}(Z^{m-1})\| |V^{m-1}| \\ & \le \left( L_{M^{-1/2}} \lambda_{\max, B_1} + \frac{L_M}{\sqrt{\lambda_{\min, B_1}}} \right) C_0^2 h_i \end{aligned} $$

and with Lemma 15

$$ \begin{aligned} & \left| \left( M^{-1/2}(Z^m) M(U^m) V^{m-1}, e_{\beta}^m - e_{\beta}^{m-1} \right) \right| \\ & \le \left\| M^{-1/2}(Z^m) \right\| \|M(U^m)\| |V^{m-1}| C_{e,B_1} |Z^m - Z^{m-1}| \le \frac{\lambda_{\max, B_1}}{\sqrt{\lambda_{\min, B_1}}} C_{e,B_1} C_0^2 h_i. \end{aligned} $$
Finally we obtain

$$
\begin{aligned}
& \frac{1}{2} |M^{-1/2}(Z^m)M(U^m)(V^m - V^{m-1} - h_i F^m)| \\
\le & \sum_{\beta \in J(q_i)} \left( -(M^{-1/2}(Z^{m-1})M(U^{m-1})V^{m-1}, e_{\beta}^{m-1}) \right. \\
& \qquad \left. + (M^{-1/2}(Z^m)M(U^m)V^m, e_{\beta}^m) \right) \\
& + h_i \nu \left( C_0^2 L M^{-1/2} \lambda_{\max, B_1} + \frac{L_M C_0^2}{\sqrt{\lambda_{\min, B_1}}} + (C_{e, B_1} C_0^2 + C_F) \frac{\lambda_{\max, B_1}}{\sqrt{\lambda_{\min, B_1}}} \right)
\end{aligned}
$$

for all $m \in \{n+1, \dots, \min(N, n+p)\}$. It follows that

$$
\sum_{m=n+1}^{\min(N,n+p)} |M^{-1/2}(Z^m)M(U^m)(V^m - V^{m-1} - h_i F^m)| \le 4 \frac{\lambda_{\max, B_1}}{\sqrt{\lambda_{\min, B_1}}} C_0 \\
+ 2p'\nu h_i \left( C_0^2 L M^{-1/2} \lambda_{\max, B_1} + \frac{L_M C_0^2}{\sqrt{\lambda_{\min, B_1}}} + (C_{e, B_1} C_0^2 + C_F) \frac{\lambda_{\max, B_1}}{\sqrt{\lambda_{\min, B_1}}} \right)
$$

with $p' = \min(N, n + p) - n$. Let

$$
\tilde{C}_0 = 2\nu \frac{\sqrt{\lambda_{\max, B_1}}}{\lambda_{\min, B_1}} \left( C_0^2 L M^{-1/2} \lambda_{\max, B_1} + \frac{L_M C_0^2}{\sqrt{\lambda_{\min, B_1}}} \right. \\
\left. \qquad + (C_{e, B_1} C_0^2 + C_F) \frac{\lambda_{\max, B_1}}{\sqrt{\lambda_{\min, B_1}}} \right).
$$

We have

$$
\begin{aligned}
\sum_{m=n+1}^{\min(N,n+p)} |V^m - V^{m-1}| &\le p' h_i C_F + \sum_{m=n+1}^{\min(N,n+p)} \|M^{-1}(U^m)\| \|M^{1/2}(Z^m)\| \\
&\quad \times |M^{-1/2}(Z^m)M(U^m)(V^m - V^{m-1} - h_i F^m)| \\
&\le 4 \left(\frac{\lambda_{\max, B_1}}{\lambda_{\min, B_1}}\right)^{3/2} C_0 + p' h_i (C_F + \tilde{C}_0).
\end{aligned}
$$

Finally,

$$
\begin{aligned}
\sum_{m=1}^{N} |V^m - V^{m-1}| &\le \sum_{k=0}^{\lfloor N/p \rfloor - 1} \sum_{m=kp+1}^{(k+1)p} |V^m - V^{m-1}| + \sum_{m=\lfloor N/p \rfloor p+1}^{N} |V^m - V^{m-1}| \\
&\le \left(\lfloor \frac{N}{p} \rfloor + 1\right) 4 \left(\frac{\lambda_{\max, B_1}}{\lambda_{\min, B_1}}\right)^{3/2} C_0 + N h_i (C_F + \tilde{C}_0).
\end{aligned}
$$
Since $p = \lfloor \frac{r}{C_0h_i} \rfloor$ and $N = \lfloor \frac{\min(\tau_0, T)}{h_i} \rfloor$ we get $\lfloor \frac{N}{p} \rfloor \le \frac{C_0T}{r-C_0h_i^*}$ for all $h_i \in (0, h_1^*]$ and the conclusion follows with

$$C_0' = \left( \frac{C_0 T}{r - C_0 h_1^*} + 1 \right) 4 \left( \frac{\lambda_{\max, B_1}}{\lambda_{\min, B_1}} \right)^{3/2} C_0 + T(C_F + \tilde{C}_0). \quad \square$$

Now we can pass to the limit as $h$ tends to zero. The steps of the proof are quite similar to [11]. For the sake of self-sufficiency, we state the main results here.

Let us recall the definition of the approximate solutions $(u_h)_{h^* \ge h > 0}$

$$u_h(t) = U^n + (t - nh) \frac{U^{n+1} - U^n}{h} \quad \forall t \in [nh, (n+1)h] \cap [0, T] \quad (32)$$

and let us define

$$v_h(t) = V^n = \frac{U^{n+1} - U^n}{h} \quad \forall t \in [nh, (n+1)h] \cap [0, T] \quad (33)$$

for all $n \in \{0, \dots, \lfloor T/h \rfloor\}$ and $h \in (0, h^*]$.

Let us assume from now on that

(H7) there exist $C_0 > 0$, $\tau_0 > 0$, $h_0^* \in (0, h^*]$ and a subsequence $(h_i)_{i \in \mathbb{N}}$, decreasing to zero, such that

$$|V^n| \le C_0 \quad \forall nh_i \in [0, \min(\tau_0, T)], \quad \forall h_i \in (0, h_0^*].$$

We define $B = \bar{B}(u_0, C+1)$ with $C \ge 0$. Let $B_0$ and $C_F$ be defined by (11) and (13), respectively. We assume (without loss of generality) that $C_0h_0^* \le 1$ and $C_Fh_0^* \le C_0$. Let us denote $\tau = \min(\tau_0, T)$.

From assumption (H7) and Proposition 4 we know that $(u_{h_i})_{h_i^* \ge h_i > 0}$ is uniformly $C_0$-Lipschitz continuous on $[0, \tau]$ and $(v_{h_i})_{h_i^* \ge h_i > 0}$ is uniformly bounded in $L^\infty(0, \tau; \mathbb{R}^d) \cap BV(0, \tau; \mathbb{R}^d)$. Hence, $(u_{h_i})_{h_i^* \ge h_i > 0}$ is equicontinuous and, using Ascoli's and Helly's theorems, we infer that there exist a subsequence, still denoted $(h_i)_{i \in \mathbb{N}}$, $u \in C^0([0, \tau]; \mathbb{R}^d)$ and $v \in BV(0, \tau; \mathbb{R}^d)$ such that

$$u_{h_i} \to u \text{ strongly in } C^0([0, \tau]; \mathbb{R}^d), \quad (34)$$

and

$$v_{h_i} \to v \text{ pointwise in } [0, \tau]. \quad (35)$$

Moreover we have

$$u_{h_i}(t) = u_0 + \int_0^t v_{h_i}(s) \, ds \quad \forall t \in [0, T], \quad \forall h_i \in (0, h^*].$$

Thus, with Lebesgue's theorem, we get

$$u(t) = \lim_{h_i \to 0} \left( u_0 + \int_0^t v_{h_i}(s) \, ds \right) = u_0 + \int_0^t v(s) \, ds \quad \forall t \in [0, \tau]. \quad (36)$$
We infer that $u$ is $C_0$-Lipschitz continuous and

$$u_{h_i}(t), u(t) \in \bar{B}(u_0, C_0\tau) \subset B_1 = \bar{B}(u_0, C_0T + 1) \quad \forall t \in [0, \tau], \forall h_i \in (0, h_1^*].$$

Moreover, $u$ is absolutely continuous on $[0, \tau]$, thus $u$ admits a derivative (in the classical sense) almost everywhere on $[0, \tau]$ and $\dot{u} \in L^1(0, \tau; \mathbb{R}^d)$. From (36) we infer that $\dot{u}(t) = v(t)$ for all $t \in [0, \tau]$ such that $v$ is continuous at $t$. Possibly modifying $\dot{u}$ on a countable subset of $[0, \tau]$, we may assume without loss of generality that $\dot{u} = v$.

As usual, we adopt the convention

$$\begin{aligned} \dot{u}(0-0) &= v(0-0) = v(0) = \dot{u}(0), \\ \dot{u}(\tau+0) &= v(\tau+0) = v(\tau) = \dot{u}(\tau). \end{aligned} \quad (37)$$

Then we observe that

**Lemma 6.** For all $t \in [0, \tau]$, $u(t) \in K$.

**Proof.** Let $t \in (0, \tau)$. For all $h_i \in (0, \min(t, \tau - t))$ there exists $n \in \{1, \dots, \lfloor \frac{\tau}{h_i} \rfloor\}$ such that $t \in [nh_i, (n+1)h_i) \subset (0, \tau)$. Then, observing that $Z^n \in K$ we get

$$\mathrm{dist}(u(t), K) \le |u(t) - Z^n| \le |u(t) - u_{h_i}(t)| + |u_{h_i}(t) - U^n| + |U^n - Z^n|.$$

But

$$\begin{aligned} |u_{h_i}(t) - U^n| &= |u_{h_i}(t) - u_{h_i}(nh_i)| \le |t - nh_i||V^n| \le C_0h_i, \\ |U^n - Z^n| &= \frac{h_i}{1+e} |V^n - eV^{n-1}| \le C_0h_i, \end{aligned}$$

thus

$$\mathrm{dist}(u(t), K) \le \|u - u_{h_i}\|_{C^0([0, \tau]; \mathbb{R}^d)} + 2C_0h_i.$$

By passing to the limit as $h_i$ tends to zero, we obtain $\mathrm{dist}(u(t), K) \le 0$, that is, $u(t) \in K$ for all $t \in (0, \tau)$.

Since $K$ is a closed subset of $\mathbb{R}^d$ and $u$ is continuous at $t = 0$ and $t = \tau$, the same property holds on $[0, \tau]$. □

### 3.1. Study of property (P2)

Now, let us prove that $u$ satisfies property (P2), that is, the differential inclusion (3). First, we observe (as in [11]) that there exists at least one non-negative measure $\mu$ such that both the Stieltjes measure $\ddot{u} = d\dot{u} = dv$ and the Lebesgue measure $dt$ admit densities with respect to $\mu$. Indeed, let $\mu$ be defined by $d\mu = |d\dot{u}| + dt$: $\mu$ is non-negative and the measures $\ddot{u} = d\dot{u}$ and $dt$ are absolutely continuous with respect to $\mu$.

Let now $\mu = |d\dot{u}| + dt$. We denote by $v'_\mu$ and $t'_\mu$ the densities of $d\dot{u} = dv$ and $dt$ with respect to $d\mu$. We have to prove that

$$M(u(t)) v'_{\mu}(t) - g(t, u(t), \dot{u}(t)) t'_{\mu}(t) \in -N_K(u(t)) \quad d\mu \text{ almost everywhere.}$$
By Jeffery's theorem (see [4] or [6]) we know that there exists a $d\mu$-negligible set $N \subset [0, \tau]$ such that, for all $t \in [0, \tau] \setminus N$:

$$v'_{\mu}(t) = \lim_{\epsilon \to 0^+} \frac{d\dot{u}(I_{\epsilon})}{d\mu(I_{\epsilon})}, \quad t'_{\mu}(t) = \lim_{\epsilon \to 0^+} \frac{dt(I_{\epsilon})}{d\mu(I_{\epsilon})}$$

with $I_{\epsilon} = [t, t + \epsilon] \cap [0, \tau]$.

We define

$$N' = \{t \in [0, \tau]; \dot{u}(t+0) = \dot{u}(t-0) \neq \dot{u}(t)\}$$

(we may observe that the convention (37) implies that $0 \notin N'$ and $\tau \notin N'$). Since $\dot{u} = v$ belongs to $BV(0, \tau; \mathbb{R}^d)$, $N'$ is at most a countable subset of $[0, \tau]$ and we infer that $N'$ is negligible with respect to $|d\dot{u}|$.

Finally, let $N_0 = \{t \in \{0\} \cup \{\tau\}; \dot{u}$ is continuous at $t\}$). The set $N_0$ is finite (it contains at most the two points $t=0$ and $t=\tau$), so it is negligible with respect to $|d\dot{u}|$, and it follows that $N \cup N' \cup N_0$ is also negligible with respect to $d\mu$. We have:

**Proposition 5.** Let $t \in [0, \tau] \setminus (N \cup N' \cup N_0)$ such that $\dot{u}$ is continuous at $t$. Then

$$M(u(t)) v'_{\mu}(t) - g(t, u(t), \dot{u}(t)) t'_{\mu}(t) \in -N_K(u(t)). \quad (38)$$

**Proof.** We begin the proof in the same way as in the proof of Proposition 4 in [11]. More precisely, let $t \in [0, \tau] \setminus (N \cup N' \cup N_0)$ such that $\dot{u}$ is continuous at $t$. Then $t \in (0, \tau)$ and, for simplicity, we will denote $\bar{u} = u(t)$ in the remainder of the proof. By definition of $N_K(\bar{u})$, (38) is equivalent to

$$(g(t, \bar{u}, \dot{u}(t)) t'_{\mu}(t) - M(\bar{u}) v'_{\mu}(t), w) \le 0 \quad (39)$$

for all $w \in T_K(\bar{u}) = \{w \in \mathbb{R}^d; (\nabla f_{\alpha}(\bar{u}), w) \ge 0, \forall \alpha \in J(\bar{u})\}$.

Let us consider $\tilde{T}_K(\bar{u})$ defined by

$$\tilde{T}_K(\bar{u}) = \begin{cases} \{w \in \mathbb{R}^d; (\nabla f_{\alpha}(\bar{u}), w) > 0 \ \forall \alpha \in J(\bar{u})\} & \text{if } J(\bar{u}) \neq \emptyset, \\ \mathbb{R}^d & \text{otherwise.} \end{cases}$$

Since $\tilde{T}_K(\bar{u})$ is dense in $T_K(\bar{u})$, we only need to prove that (39) holds for all $w \in \tilde{T}_K(\bar{u})$. As in the proof of Proposition 4 in [11], we define $r_{\bar{u}} > 0$ such that

$$J(q) \subset J(\bar{u}) \quad \forall q \in \bar{B}(\bar{u}, r_{\bar{u}}),$$

and, for all $w \in \tilde{T}_K(\bar{u})$, we define $r_w \in (0, r_{\bar{u}}]$ such that $w \in T_K(q)$ for all $q \in \bar{B}(\bar{u}, r_w)$.

Using the continuity of $u$ and the uniform convergence of $(u_{h_i})_{i \in \mathbb{N}}$ to $u$ on $[0, \tau]$, we also infer that, for all $w \in \tilde{T}_K(\bar{u})$, there exists $\tilde{\epsilon}_w \in (0, \min(t, \frac{\tau-t}{2}))$, such that for all $\epsilon \in (0, \tilde{\epsilon}_w]$ there exists $h_{\epsilon} \in (0, h_1^*)$, such that

$$u(s) \in \bar{B}(\bar{u}, \frac{r_w}{3}) \quad \forall s \in [t, t + \epsilon],$$

$$h_{\epsilon} \le \min \left( \frac{r_w}{3C_0}, \frac{\epsilon}{3} \right), \quad \|u - u_{h_i}\|_{C^0([0, \tau]; \mathbb{R}^d)} \le \frac{r_w}{3} \quad \forall h_i \in (0, h_{\epsilon}].$$
It follows that for all $\varepsilon \in (0, \tilde{\varepsilon}_w]$ and for all $h_i \in (0, h_\varepsilon]$

$$u_{h_i}(s) \in \bar{B}\left(\bar{u}, \frac{2r_w}{3}\right) \quad \forall s \in [t, t + \varepsilon],$$

and

$$Z^n \in \bar{B}(\bar{u}, r_w) \quad \forall nh_i \in [t, t + \varepsilon],$$

since

$$|Z^n - u_{h_i}(nh_i)| = |Z^n - U^n| \le C_0h_i \le C_0h_\varepsilon \le \frac{r_w}{3}.$$

Now let $w \in \tilde{T}_K(\bar{u})$, $\varepsilon \in (0, \tilde{\varepsilon}_w]$ and $h_i \in (0, h_\varepsilon]$. We define $j$ and $k$ by

$$j = \left\lfloor \frac{t}{h_i} \right\rfloor, \quad k = \left\lfloor \frac{t + \varepsilon}{h_i} \right\rfloor.$$

We have

$$0 < t_j = jh_i \le t < t_{j+1} < \dots < t_k = kh_i \le t + \varepsilon < t_{k+1} < \tau.$$

From Lemma 1 we know that, for all $n \in \{j+1, \dots, k\}$, we have

$$(V^{n-1} - V^n + h_i F^n, w)_{M(U^n)} \le 0$$

since $w \in T_K(Z^n)$. Next we continue the proof exactly as in [11]. $\square$

Let us now consider the case of discontinuities of the velocity. Let $t \in [0, \tau] \setminus (N \cup N' \cup N_0)$ such that $\dot{u}$ is discontinuous at $t$. Then, $\dot{u}(t+0) \ne \dot{u}(t-0)$ and $d\dot{u}$ possesses a Dirac mass at $\{t\}$. Thus $\{t\}$ is not negligible anymore with respect to $d\mu$ and (3) is equivalent to

$$M(u(t)) (\dot{u}(t+0) - \dot{u}(t-0)) \in -N_K(u(t)).$$

This property is a direct consequence of the following result:

**Proposition 6.** For all $t \in [0, \tau]$ we have

$$M(u(t)) (\dot{u}(t+0) - \dot{u}(t-0)) \in -N_K(u(t)).$$

**Proof.** Once again, the proof is similar to the proof of the analogous result in [11]. More precisely, let $t \in [0, \tau]$ and for simplicity denote $\bar{u} = u(t)$. Thanks to the density of $\tilde{T}_K(\bar{u})$ in $T_K(\bar{u})$, we only need to prove that

$$(M(\bar{u}) (\dot{u}(t-0) - \dot{u}(t+0)), w) \le 0 \quad \forall w \in \tilde{T}_K(\bar{u}).$$

Let $w \in \tilde{T}_K(\bar{u})$. As in the proof of the previous proposition, we define $r_w > 0$ such that

$$J(q) \subset J(\bar{u}) \quad \text{and} \quad w \in T_K(q) \text{ for all } q \in \bar{B}(\bar{u}, r_w).$$
We also define $\tilde{\epsilon}_w \in (0, \tau/2)$ such that for all $\epsilon \in (0, \tilde{\epsilon}_w]$ we have

$$u(s) \in \bar{B} \left( \bar{u}, \frac{r_w}{3} \right) \quad \forall s \in [t - \epsilon, t + \epsilon] \cap [0, \tau],$$

and there exists $h_{\epsilon} \in (0, \min(h_1^*, r_w/3C_0, \epsilon/3)]$ such that

$$u_{h_i}(s) \in \bar{B} \left( \bar{u}, \frac{2r_w}{3} \right) \quad \forall s \in [t - \epsilon, t + \epsilon] \cap [0, \tau], \quad \forall h_i \in (0, h_{\epsilon}],$$

and

$$Z^n \in \bar{B}(\bar{u}, r_w) \quad \forall nh_i \in [t - \epsilon, t + \epsilon] \cap [0, \tau], \quad \forall h_i \in (0, h_{\epsilon}],$$

(we recall that $Z^0 = U^0$ by convention). Now we consider $\epsilon \in (0, \tilde{\epsilon}_w]$ and $h_i \in (0, h_{\epsilon}]$. We define $t_{\epsilon}^{-} = \max(t - \epsilon, 0)$, $t_{\epsilon}^{+} = \min(t + \epsilon, \tau)$ and

$$j = \left\lfloor \frac{t_{\epsilon}^{-}}{h_i} \right\rfloor, \quad k = \left\lfloor \frac{t_{\epsilon}^{+}}{h_i} \right\rfloor$$

that is, we have

$$0 \le t_j = jh_i \le t_{\epsilon}^{-} < t_{j+1} < \dots < t_k = kh_i \le t_{\epsilon}^{+} \le \tau.$$

It follows that

$$v_{h_i}(t_{\epsilon}^{-}) = V^j, \quad v_{h_i}(t_{\epsilon}^{+}) = V^k$$

and we have $w \in T_K(Z^n)$ for all $n \in \{j+1, \dots, k\}$. Then, we continue the proof exactly as in Proposition 5 of [11]. $\square$

## 3.2. Transmission of the velocity at impacts

With the previous proposition, we observe that $\dot{u}(t-0) = \dot{u}(t+0)$ for all $t \in [0, \tau]$ such that $J(u(t)) = \emptyset$. That is, $\dot{u}$ is continuous at $t$ if $u(t) \in \text{Int}(K)$ and in this case the impact law (2) is satisfied. Thus it remains only to prove that

$$\dot{u}(\bar{t} + 0) = -e\dot{u}(\bar{t} - 0) + (1 + e)\text{Proj}_{M(u(\bar{t}))}(T_K(u(\bar{t})), \dot{u}(\bar{t} - 0)) \quad (40)$$

for all $\bar{t} \in (0, \tau)$ such that $J(u(\bar{t})) \neq \emptyset$.

Let $\bar{t} \in (0, \tau)$ be such that $J(u(\bar{t})) \neq \emptyset$. For simplicity, let us denote $\bar{u} = u(\bar{t})$ and $\dot{u}^{+} = \dot{u}(\bar{t} + 0)$, $\dot{u}^{-} = \dot{u}(\bar{t} - 0)$. With Proposition 6 we already know that $M(\bar{u})(\dot{u}^{-} - \dot{u}^{+}) \in N_K(\bar{u})$, that is, there exist non-positive real numbers $(\mu_{\alpha})_{\alpha \in J(\bar{u})}$ such that

$$M^{1/2}(\bar{u}) (\dot{u}^{-} - \dot{u}^{+}) = \sum_{\alpha \in J(\bar{u})} \mu_{\alpha} e_{\alpha}(\bar{u}), \quad (41)$$

where we recall that

$$e_{\alpha}(\bar{u}) = \frac{M^{-1/2}(\bar{u}) \nabla f_{\alpha}(\bar{u})}{|M^{-1/2}(\bar{u}) \nabla f_{\alpha}(\bar{u})|} \quad \forall \alpha \in J(\bar{u}).$$
Then (40) reduces to

$$ \dot{u}^{+} + e\dot{u}^{-} \in T_K(\bar{u}), \quad (\dot{u}^{-} - \dot{u}^{+}, \dot{u}^{+} + e\dot{u}^{-})_{M(\bar{u})} = 0, $$

that is,

$$ (\nabla f_{\alpha}(\bar{u}), \dot{u}^{+} + e\dot{u}^{-}) \geq 0 \quad \forall \alpha \in J(\bar{u}) $$

and

$$ (M^{1/2}(\bar{u})(\dot{u}^{-} - \dot{u}^{+}), M^{1/2}(\bar{u})(\dot{u}^{+} + e\dot{u}^{-})) = 0. $$

From (41) we infer that (40) is equivalent to

$$ (e_{\alpha}(\bar{u}), M^{1/2}(\bar{u})(\dot{u}^{+} + e\dot{u}^{-})) \geq 0 \quad \forall \alpha \in J(\bar{u}) $$

and

$$ \sum_{\alpha \in J(\bar{u})} \mu_{\alpha} (e_{\alpha}(\bar{u}), M^{1/2}(\bar{u})(\dot{u}^{+} + e\dot{u}^{-})) = 0. $$

But $\mu_{\alpha} \leq 0$ for all $\alpha \in J(\bar{u})$, so (40) is satisfied if and only if

$$ (e_{\alpha}(\bar{u}), M^{1/2}(\bar{u})(\dot{u}^{+} + e\dot{u}^{-})) \geq 0 \quad \text{for all } \alpha \in J(\bar{u}) \text{ such that } \mu_{\alpha} = 0, $$

$$ (e_{\alpha}(\bar{u}), M^{1/2}(\bar{u})(\dot{u}^{+} + e\dot{u}^{-})) = 0 \quad \text{for all } \alpha \in J(\bar{u}) \text{ such that } \mu_{\alpha} \neq 0. $$

First we observe that, for all $\alpha \in J(\bar{u})$, we have with (41) and assumption (H6)

$$ (e_{\alpha}(\bar{u}), M^{1/2}(\bar{u})\dot{u}^{-}) = (e_{\alpha}(\bar{u}), M^{1/2}(\bar{u})\dot{u}^{+}) + \mu_{\alpha}, $$

and, since $\dot{u}^{+} \in T_K(\bar{u})$ and $\dot{u}^{-} \in -T_K(\bar{u})$,

$$ (e_{\alpha}(\bar{u}), M^{1/2}(\bar{u})\dot{u}^{+}) \geq 0, \quad (e_{\alpha}(\bar{u}), M^{1/2}(\bar{u})\dot{u}^{-}) \leq 0. $$

It follows that

$$ (e_{\alpha}(\bar{u}), M^{1/2}(\bar{u})\dot{u}^{+}) = (e_{\alpha}(\bar{u}), M^{1/2}(\bar{u})\dot{u}^{-}) = 0 $$

if $\alpha \in J(\bar{u})$ such that $\mu_{\alpha} = 0$ and it remains only to prove that $(e_{\alpha}(\bar{u}), M^{1/2}(\bar{u})(\dot{u}^{+} + e\dot{u}^{-})) = 0$ for all $\alpha \in J(\bar{u})$ such that $\mu_{\alpha} \neq 0$.

As in [11], we construct a neighbourhood of $\bar{u}$, $\bar{B}(\bar{u}, r_{\bar{u}})$ with $r_{\bar{u}} \in (0, \min(r_{B_1}, r'_{B_1})/2)$, such that $J(q) \subset J(\bar{u})$ for all $q \in \bar{B}(\bar{u}, r_{\bar{u}})$, and $d$ Lipschitz continuous mappings $v_{\alpha}$, $\alpha \in \{1, \dots, d\}$, such that, for all $q \in \bar{B}(\bar{u}, r_{\bar{u}})$ $(v_{\alpha}(q))_{\alpha \in \{1, \dots, d\}}$ is a basis of $\mathbb{R}^d$, $|v_{\alpha}(q)| = 1$ for all $\alpha \in \{1, \dots, d\}$ and $v_{\alpha}(q) = e_{\alpha}(q)$ for all $\alpha \in J(q)$, for all $q \in \bar{B}(\bar{u}, r_{\bar{u}}) \cap K$.

We define the dual basis $(w_{\alpha}(q))_{1 \leq \alpha \leq d}$ for all $q \in \bar{B}(\bar{u}, r_{\bar{u}})$. The mappings $w_{\alpha}$ $(1 \leq \alpha \leq d)$ are also Lipschitz continuous on $\bar{B}(\bar{u}, r_{\bar{u}})$: we let

$$ C_{\bar{u}} = \sup \{|v_{\alpha}(q)|, |w_{\alpha}(q)|; q \in \bar{B}(\bar{u}, r_{\bar{u}}), \alpha \in \{1, \dots, d\}\} $$
and $L_{\bar{u}} \in \mathbb{R}_*^+$ be such that, for all $\alpha \in \{1, \dots, d\}$ and for all $(q, q') \in \overline{B}(\bar{u}, r_{\bar{u}})^2$

$$|v_{\alpha}(q) - v_{\alpha}(q')| \leq L_{\bar{u}}|q - q'|, \quad |w_{\alpha}(q) - w_{\alpha}(q')| \leq L_{\bar{u}}|q - q'|.$$

Since the mappings $\nabla f_{\alpha}$ ($\alpha \in \{1, \dots, \nu\}$) are locally Lipschitz continuous, possibly modifying $C_{\bar{u}}$, we may assume without loss of generality that

$$|\nabla f_{\alpha}(q)| \leq C_{\bar{u}},$$

for all $\alpha \in \{1, \dots, \nu\}$ and for all $(q, q') \in \overline{B}(\bar{u}, r_{\bar{u}})^2$.

From the continuity of $u$ and the uniform convergence of $(u_{h_i})_{i \in \mathbb{N}}$ to $u$ on $[0, \tau]$,
we infer that there exist $\bar{\epsilon} \in (0, \min(\bar{t}, \tau - \bar{t})/2)$ and $h_2^* \in (0, \min(h_1^*, \frac{\bar{\epsilon}}{3}, \frac{r_{\bar{u}}}{6C_0})]$,
where $h_1^*$ is given by Proposition 4, such that

$$u(t) \in \bar{B}\left(\bar{u}, \frac{r_{\bar{u}}}{3}\right) \quad \forall t \in [\bar{t} - \bar{\epsilon}, \bar{t} + \bar{\epsilon}],$$

$$\|u - u_{h_i}\|_{C^0([0, \tau]; \mathbb{R}^d)} \leq \frac{r_{\bar{u}}}{3} \quad \forall h_i \in (0, h_2^*].$$

It follows that

$$U^{n+1}, U^n, Z^n, Z^{n\pm 1} \in \overline{B}(\bar{u}, r_{\bar{u}}) \quad \forall nh_i \in [\bar{t} - \bar{\epsilon}, \bar{t} + \bar{\epsilon}], \quad \forall h_i \in (0, h_2^*]. \quad (42)$$

Furthermore, we infer from Lemma 1 that, for all $nh_i \in [\bar{t} - \bar{\epsilon}, \bar{t} + \bar{\epsilon}]$, for all $h_i \in (0, h_2^*]$

$$M(U^n)(V^{n-1} - V^n + h_i F^n) \in N_K(Z^n).$$

If $J(Z^n) \neq \emptyset$, there exist non-positive real numbers $(\mu_{\beta}^n)_{\beta \in J(Z^n)}$ such that

$$M(U^n) \left(V^{n-1} - V^n + h_i F^n\right) = \sum_{\beta \in J(Z^n)} \mu_{\beta}^n M^{1/2}(Z^n) e_{\beta}(Z^n) \quad (43)$$

and observing that $e_{\beta}(Z^n) = v_{\beta}(Z^n)$ for all $\beta \in J(Z^n)$, we get with assumption (H6) (see (8))

$$\begin{aligned} |\mu_{\beta}^n| & \leq \left| \left(M(U^n)(V^{n-1} - V^n + h_i F^n), M^{-1/2}(Z^n)v_{\beta}(Z^n)\right) \right| \\ & \leq \frac{\lambda_{\max, B_1}}{\sqrt{\lambda_{\min, B_1}}} \left( |V^{n-1} - V^n| + h_i C_F \right) \leq 3C_0 \frac{\lambda_{\max, B_1}}{\sqrt{\lambda_{\min, B_1}}}. \end{aligned} \quad (44)$$

From now on, let us denote

$$C_2' = 3C_0 \frac{\lambda_{\max, B_1}}{\sqrt{\lambda_{\min, B_1}}}. $$

We begin with the following lemma.

**Lemma 7.** Let $\alpha \in J(\bar{u})$ such that $\mu_{\alpha} \neq 0$. Then, for all $\varepsilon_1 \in (0, \bar{\epsilon}]$ there exists $h_{\varepsilon_1} \in (0, \min(h_2^*, \varepsilon_1/3)]$ such that for all $h_i \in (0, h_{\varepsilon_1}]$, there exists $nh_i \in [\bar{t} - \varepsilon_1, \bar{t} + \varepsilon_1]$ such that $f_{\alpha}(Z^n) \leq 0$.
This result is analogous to Lemma 3 in [11]: it means that on any neighbourhood $[\tilde{t} - \varepsilon_1, \tilde{t} + \varepsilon_1]$ of $\tilde{t}$ (with $\varepsilon_1 \in (0, \bar{\varepsilon}]$), the constraint numbered $\alpha$ is saturated by at least one discrete average position $Z^n$, if $h_i$ is small enough.

**Proof.** As in the proof of Lemma 3 in [11], we assume that the announced result does not hold, that is, assume that there exists $\varepsilon_1 \in (0, \bar{\varepsilon}]$ such that, for all $h_{\varepsilon_1} \in (0, \min(h_2^*, \varepsilon_1/3)]$ there exists $h_i \in (0, h_{\varepsilon_1}]$ such that $f_\alpha(Z^n) > 0$ for all $nh_i \in [\tilde{t} - \varepsilon_1, \tilde{t} + \varepsilon_1]$.

Hence we can extract from $(h_i)_{i \in \mathbb{N}}$ a subsequence denoted $(h_{\varphi(i)})_{i \in \mathbb{N}}$ decreasing to zero, such that $h_{\varphi(i)} \in (0, \min(h_2^*, \varepsilon_1/3)]$ and

$$f_{\alpha}(Z^n) > 0 \quad \forall nh_{\varphi(i)} \in [\tilde{t} - \varepsilon_1, \tilde{t} + \varepsilon_1] \quad (45)$$

for all $i \in \mathbb{N}$.

For all $\varepsilon \in (0, \varepsilon_1]$, let us establish the following estimate:

$$\left| \left( M^{1/2} \left( u_{h_{\varphi(i)}}(t_{\varepsilon}^{-}) \right) v_{h_{\varphi(i)}}(t_{\varepsilon}^{-}) - M^{1/2} \left( u_{h_{\varphi(i)}}(t_{\varepsilon}^{+}) \right) v_{h_{\varphi(i)}}(t_{\varepsilon}^{+}) \right) w_{\alpha}(\bar{u}) \right| \\ \le \mathcal{O} \left( \varepsilon + h_{\varphi(i)} + \|u - u_{h_{\varphi(i)}}\|_{C^0([0, \tau]; \mathbb{R}^d)} \right)$$

where $t_{\varepsilon}^{-} = \max(\tilde{t} - \varepsilon, 0) = \tilde{t} - \varepsilon$ and $t_{\varepsilon}^{+} = \min(\tilde{t} + \varepsilon, \tau) = \tilde{t} + \varepsilon$. Then, by passing to the limit when $i$ tends to $+\infty$, we will infer with (34)–(35) that

$$\left| \left( M^{1/2} \left( u(t_{\varepsilon}^{-}) \right) v(t_{\varepsilon}^{-}) - M^{1/2} \left( u(t_{\varepsilon}^{+}) \right) v(t_{\varepsilon}^{+}) \right) w_{\alpha}(\bar{u}) \right| \le \mathcal{O}(\varepsilon)$$

and, when $\varepsilon$ tends to zero, we will obtain

$$\left| \left( M^{1/2}(\bar{u}) \left( \dot{u}(\tilde{t} - 0) - \dot{u}(\tilde{t} + 0) \right) , w_{\alpha}(\bar{u}) \right) \right| = |\mu_{\alpha}| \le 0$$

which gives a contradiction.

Let $\varepsilon \in (0, \varepsilon_1]$. There exists $i_\varepsilon \in \mathbb{N}$ such that, for all $i \ge i_\varepsilon$ we have $h_{\varphi(i)} \in (0, \varepsilon/2)$ and we define

$$n_i = \left\lfloor \frac{t_{\varepsilon}^{-}}{h_{\varphi(i)}} \right\rfloor, \quad p_i = \left\lfloor \frac{t_{\varepsilon}^{+}}{h_{\varphi(i)}} \right\rfloor.$$

Then, $n_i + 1 < p_i$ and for all $n \in \{n_i + 1, \dots, p_i\}$ we have $nh_{\varphi(i)} \in [t_{\varepsilon}^{-}, t_{\varepsilon}^{+}]$. With (20) we infer that

$$|U^n - Z^n| \le C_0 h_{\varphi(i)} \quad (46)$$

and with (45) we know that $\alpha \notin J(Z^n)$, thus with (43) and (44)

$$\left| \left( M^{1/2}(Z^n) \left( V^{n-1} - V^n + h_{\varphi(i)} F^n \right), w_{\alpha}(Z^n) \right) \right| \\ = \left| \sum_{\beta \in J(Z^n)} \mu_{\beta}^n \left( M^{1/2}(Z^n) M^{-1}(U^n) M^{1/2}(Z^n) v_{\beta}(Z^n), w_{\alpha}(Z^n) \right) \right| \\ \le \sum_{\beta \in J(Z^n)} \left| \mu_{\beta}^n \right| \left\| M^{1/2}(Z^n) \right\|^2 \left\| M^{-1}(Z^n) - M^{-1}(U^n) \right\| \left| v_{\beta}(Z^n) \right| \left| w_{\alpha}(Z^n) \right| \\ \le v C'_2 C_{\bar{u}} \lambda_{\max, B_1} L_{M^{-1}} |Z^n - U^n| \le v C'_2 C_{\bar{u}} C_0 \lambda_{\max, B_1} L_{M^{-1}} h_{\varphi(i)}$$
for all $n \in \{n_i + 1, \dots, p_i\}$, if $J(Z^n) \neq \emptyset$. If $J(Z^n) = \emptyset$, this inequality remains true, since $V^{n-1} - V^n + h_{\varphi(i)}F^n = 0$ in this case.

It follows that, for all $n \in \{n_i + 1, \dots, p_i\}$

$$ 
\begin{aligned}
& \left(M^{1/2}(Z^{n_i+1})V^{n_i} - M^{1/2}(Z^{p_i+1})V^{p_i}, w_\alpha(\bar{u})\right) \\
&= \sum_{n=n_i+1}^{p_i} \left(M^{1/2}(Z^n)V^{n-1} - M^{1/2}(Z^{n+1})V^n, w_\alpha(\bar{u})\right) \\
&= \sum_{n=n_i+1}^{p_i} \left(M^{1/2}(Z^n)(V^{n-1} - V^n + h_{\varphi(i)}F^n), w_\alpha(Z^n)\right) \\
&\quad + \sum_{n=n_i+1}^{p_i} \left(\left(M^{1/2}(Z^n) - M^{1/2}(Z^{n+1})\right)V^n, w_\alpha(\bar{u})\right) \\
&\quad - \sum_{n=n_i+1}^{p_i} h_{\varphi(i)} \left(M^{1/2}(Z^n)F^n, w_\alpha(Z^n)\right) \\
&\quad + \sum_{n=n_i+1}^{p_i} \left(M^{1/2}(Z^n)(V^{n-1} - V^n), w_\alpha(\bar{u}) - w_\alpha(Z^n)\right).
\end{aligned}
$$

Then, using (20), we obtain

$$ 
\begin{aligned}
|\bar{u} - Z^n| &\le |u(\bar{t}) - u_{h_{\varphi(i)}}(\bar{t})| + |u_{h_{\varphi(i)}}(\bar{t}) - u_{h_{\varphi(i)}}(nh_{\varphi(i)})| + |U^n - Z^n| \\
&\le \|u - u_{h_{\varphi(i)}}\|_{C^0([0, \tau]; \mathbb{R}^d)} + C_0|\bar{t} - nh_{\varphi(i)}| + \frac{h_{\varphi(i)}}{1+e}|V^n - eV^{n-1}| \\
&\le \|u - u_{h_{\varphi(i)}}\|_{C^0([0, \tau]; \mathbb{R}^d)} + C_0(\varepsilon + h_{\varphi(i)}),
\end{aligned}
$$

for all $n \in \{n_i + 1, \dots, p_i\}$ and we estimate $|w_\alpha(\bar{u}) - w_\alpha(Z^n)|$ as

$$ L_{\bar{u}} \left( \|u - u_{h_{\varphi(i)}}\|_{C^0([0, \tau]; \mathbb{R}^d)} + C_0(\varepsilon + h_{\varphi(i)}) \right). $$

Hence, with the estimate of the discrete accelerations obtained at Proposition 4 and (21), we get

$$ 
\begin{aligned}
& \left| \left(M^{1/2}(Z^{n_i+1})V^{n_i} - M^{1/2}(Z^{p_i+1})V^{p_i}, w_\alpha(\bar{u})\right) \right| \\
&\le \sum_{n=n_i+1}^{p_i} \nu C_2' C_{\bar{u}} C_0 \lambda_{\max, B_1} L_{M^{-1}} h_{\varphi(i)} \\
&\quad + \sum_{n=n_i+1}^{p_i} C_0 C_{\bar{u}} L_{M^{1/2}} |Z^n - Z^{n+1}| + \sum_{n=n_i+1}^{p_i} h_{\varphi(i)} C_{\bar{u}} C_F \sqrt{\lambda_{\max, B_1}} \\
&\quad + \sqrt{\lambda_{\max, B_1}} L_{\bar{u}} \left( \|u - u_{h_{\varphi(i)}}\|_{C^0([0, \tau]; \mathbb{R}^d)} + C_0(\varepsilon + h_{\varphi(i)}) \right) \sum_{n=n_i+1}^{p_i} |V^n - V^{n-1}|
\end{aligned}
$$
$$
\le (2\epsilon + h_{\varphi(i)}) C_{\bar{u}} \left( \nu C_2' C_0 \lambda_{\max, B_1} L_{M^{-1}} + C_0^2 L_{M^{1/2}} + C_F \sqrt{\lambda_{\max, B_1}} \right) \\
+ C_0' \sqrt{\lambda_{\max, B_1}} L_{\bar{u}} \left( \|u - u_{h_{\varphi(i)}}\|_{C^0([0, \tau]; \mathbb{R}^d)} + C_0 (\epsilon + h_{\varphi(i)}) \right). \quad (47)
$$

But $V^{n_i} = v_{h_{\varphi(i)}}(t_{\varepsilon}^{-})$ and $V^{p_i} = v_{h_{\varphi(i)}}(t_{\varepsilon}^{+})$ and

$$
\begin{align*}
& \left| \begin{aligned}[t]
    & (M^{1/2} (u_{h_{\varphi(i)}}(t_{\varepsilon}^{-})) v_{h_{\varphi(i)}}(t_{\varepsilon}^{-}) - M^{1/2} (u_{h_{\varphi(i)}}(t_{\varepsilon}^{+})) v_{h_{\varphi(i)}}(t_{\varepsilon}^{+}), w_{\alpha}(\bar{u}) \\
    & \quad - (M^{1/2}(Z^{n_i+1})V^{n_i} - M^{1/2}(Z^{p_i+1})V^{p_i}, w_{\alpha}(\bar{u}))
    \end{aligned} \right| \\
\le & \left( \begin{aligned}[t]
    & |M^{1/2} (u_{h_{\varphi(i)}}(t_{\varepsilon}^{-})) - M^{1/2}(Z^{n_i+1})| \\
    & + |M^{1/2} (u_{h_{\varphi(i)}}(t_{\varepsilon}^{+})) - M^{1/2}(Z^{p_i+1})| \end{aligned} \right) C_0 |w_{\alpha}(\bar{u})| \\
\le & C_0 C_{\bar{u}} L_{M^{1/2}} \left( \begin{aligned}[t]
    & |u_{h_{\varphi(i)}}(t_{\varepsilon}^{-}) - u_{h_{\varphi(i)}}((n_i+1)h_{\varphi(i)})| + |U^{n_i+1} - Z^{n_i+1}| \\
    & + |u_{h_{\varphi(i)}}(t_{\varepsilon}^{+}) - u_{h_{\varphi(i)}}((p_i+1)h_{\varphi(i)})| + |U^{p_i+1} - Z^{p_i+1}| \end{aligned} \right) \\
\le & 4C_0^2 C_{\bar{u}} L_{M^{1/2}} h_{\varphi(i)}. \tag{48}
\end{align*}
$$

Finally, from (47)–(48) we obtain

$$
\left| \left( M^{1/2} (u_{h_i}(t_{\epsilon}^{-})) v_{h_{\phi(i)}}(t_{\epsilon}^{-}) - M^{1/2} (u_{h_i}(t_{\epsilon}^{+})) v_{h_{\phi(i)}}(t_{\epsilon}^{+}), w_{\alpha}(\bar{u}) \right) \right| \\
\le \mathcal{O} \left( h_{\phi(i)} + \epsilon + \|u - u_{h_{\phi(i)}}\|_{C^0([0, \tau]; \mathbb{R}^d)} \right)
$$

which enables us to conclude. □

Let us prove now that

**Lemma 8.** For all $\alpha \in J(\bar{u})$ such that $\mu_{\alpha} \neq 0$, we have

$$
(M^{1/2}(\bar{u})\dot{u}^{+}, v_{\alpha}(\bar{u})) \le -e(M^{1/2}(\bar{u})\dot{u}^{-}, v_{\alpha}(\bar{u})).
$$

**Proof.** Let $\varepsilon \in (0, \bar{\varepsilon}]$, thus $\bar{t} \pm \varepsilon \in (0, \tau)$. We apply the previous Lemma, with
$\varepsilon_1 = \varepsilon/2$: there exists $h_{\varepsilon/2} \in (0, \min(h_2^*, \varepsilon/6)]$ such that, for all $h_i \in (0, h_{\varepsilon/2}]$,
there exists $n h_i \in [\bar{t} - \varepsilon/2, \bar{t} + \varepsilon/2]$ such that $f_{\alpha}(Z^n) \le 0$, that is, $\alpha \in J(Z^n)$.
Let $h_i \in (0, h_{\varepsilon/2}]$. We define $P_i$, $p_i$ and $n_i$ by

$$
P_i = \left\lfloor \frac{\bar{t} + \varepsilon/2}{h_i} \right\rfloor, \quad p_i = \left\lfloor \frac{\bar{t} + \varepsilon}{h_i} \right\rfloor, \quad n_i = \left\lfloor \frac{\bar{t} - \varepsilon}{h_i} \right\rfloor.
$$

Let us recall that $\bar{\varepsilon} \in (0, \min(\bar{t}, \tau - \bar{t})/2)$ so $\bar{t} - \varepsilon \ge \bar{\varepsilon}$ and $\tau - \bar{t} - \varepsilon \ge \bar{\varepsilon}$. Thus
$h_i \in (0, \varepsilon/6] \subset (0, \bar{\varepsilon}/6] \subset (0, \min(\bar{t} - \varepsilon, \tau - \bar{t} - \varepsilon)/2]$. It follows that

$$
\begin{align*}
& 0 < n_i h_i \le \bar{t} - \varepsilon < (n_i + 1) h_i < \bar{t} < P_i h_i \le \bar{t} + \frac{\varepsilon}{2} < (P_i + 1) h_i < p_i h_i \\
& \le \bar{t} + \varepsilon < (p_i + 1) h_i < \tau
\end{align*}
$$

and $v_{h_i}(\bar{t} - \varepsilon) = V^{n_i}$, $v_{h_i}(\bar{t} + \frac{\varepsilon}{2}) = V^{p_i}$.
Let $N_i$ be the last time step in $[\bar{t} - \frac{\epsilon}{2}, \bar{t} + \frac{\epsilon}{2}]$ such that $f_\alpha(Z^n) \le 0$, that is,

$$N_i = \max \left\{ n \in \mathbb{N} ; nh_i \in \left[ \bar{t} - \frac{\epsilon}{2}, \bar{t} + \frac{\epsilon}{2} \right] \text{ and } f_\alpha(Z^n) \le 0 \right\}.$$

Then, recalling that $h_i \in (0, \epsilon/6]$ we get $N_i \ge n_i + \frac{\epsilon}{2h_i} \ge n_i + 3$ and with the same computations as in Proposition 1, we obtain that

$$(\nabla f_{\alpha}(Z^{N_i}), V^{N_i} + eV^{N_i-2}) \le \frac{L_f h_i}{2(1+e)} |V^{N_i} + eV^{N_i-2}|^2 \le L_f h_i C_0^2$$

and with Lemma 15

$$(v_{\alpha}(Z^{N_i}), M^{1/2}(Z^{N_i})(V^{N_i} + eV^{N_i-2})) \le \frac{L_f C_0^2}{m_{B_1}} h_i.$$

It follows that

$$(v_{\alpha}(Z^{N_i}), M^{1/2}(Z^{N_i})V^{N_i}) \le -e (v_{\alpha}(Z^{N_i-2}), M^{1/2}(Z^{N_i-2})V^{N_i-2}) \\ + \frac{L_f C_0^2}{m_{B_1}} h_i + e |M^{1/2}(Z^{N_i-2})v_{\alpha}(Z^{N_i-2}) \\ - M^{1/2}(Z^{N_i})v_{\alpha}(Z^{N_i})| C_0.$$

Thus, using the estimate (21) twice and the Lipschitz property of $M^{1/2}$ on $B_1$ and $v_\alpha$ on $\bar{B}(\bar{u}, r_{\bar{u}})$, we obtain

$$(v_{\alpha}(Z^{N_i}), M^{1/2}(Z^{N_i})V^{N_i}) \le -e (v_{\alpha}(Z^{N_i-2}), M^{1/2}(Z^{N_i-2})V^{N_i-2}) + C_3' h_i$$

with

$$C_3' = \frac{L_f C_0^2}{m_{B_1}} + 2C_0^2(L_{M^{1/2}} + L_{\bar{u}}\sqrt{\lambda_{\max,B_1}}).$$

By definition of $N_i$, we have $f_\alpha(Z^n) > 0$ for all $n \in \{N_i + 1, \dots, P_i\}$, thus

$$\begin{aligned} & \left| (M^{1/2}(Z^{P_i})V^{P_i}, v_\alpha(Z^{P_i})) - (M^{1/2}(Z^{N_i})V^{N_i}, v_\alpha(Z^{N_i})) \right| \\ & \le \left| (M^{1/2}(Z^{N_i})V^{N_i}, v_\alpha(Z^{P_i}) - v_\alpha(Z^{N_i})) \right| \\ & \quad + \sum_{n=N_i+1}^{P_i} \left| (M^{1/2}(Z^n)V^n - M^{1/2}(Z^{n-1})V^{n-1}, v_\alpha(Z^{P_i})) \right| \\ & \le \left| (M^{1/2}(Z^{N_i})V^{N_i}, v_\alpha(Z^{P_i}) - v_\alpha(Z^{N_i})) \right| \\ & \quad + \sum_{n=N_i+1}^{P_i} \left| ((M^{1/2}(Z^n) - M^{1/2}(Z^{n-1}))V^{n-1}, v_\alpha(Z^{P_i})) \right| \\ & \quad + \sum_{n=N_i+1}^{P_i} h_i \left| (M^{1/2}(Z^n)F^n, v_\alpha(Z^{P_i})) \right| \\ & \quad + \sum_{n=N_i+1}^{P_i} \left| (M^{1/2}(Z^n)(V^n - V^{n-1} - h_i F^n), v_\alpha(Z^{P_i})) \right|. \end{aligned}$$
With the definition of $P_i$ and $N_i$, we have immediately $P_i h_i \le \tilde{t} + \varepsilon/2$ and $N_i h_i \ge \tilde{t} - \varepsilon/2$, so $(P_i - N_i)h_i \le \varepsilon$ and with (21), we get $|Z^{P_i} - Z^{N_i}| \le C_0 \varepsilon$. Furthermore, using the Lipschitz property of $M^{1/2}$ on $B_1$ and recalling that the mappings $v_\alpha$ ($\alpha \in \{1, \dots, \nu\}$) are $L_{\tilde{u}}$-Lipschitz continuous on $\bar{B}(\tilde{u}, r_{\tilde{u}})$, we get

$$ \begin{aligned} & \left| (M^{1/2}(Z^{P_i})V^{P_i}, v_\alpha(Z^{P_i})) - (M^{1/2}(Z^{N_i})V^{N_i}, v_\alpha(Z^{N_i})) \right| \\ & \le C_0^2 \sqrt{\lambda_{\max,B_1}} L_{\tilde{u}} \varepsilon + \varepsilon (L_{M^{1/2}} C_0^2 + \sqrt{\lambda_{\max,B_1}} C_F) \\ & + \sum_{n=N_i+1}^{P_i} \left| (M^{1/2}(Z^n)(V^n - V^{n-1} - h_i F^n), v_\alpha(Z^{P_i})) \right|. \end{aligned} \quad (49) $$

Moreover, for all $n \in \{n_i + 1, \dots, p_i\}$ we have

$$ M(U^n)(V^{n-1} - V^n + h_i F^n) = \sum_{\beta \in J(Z^n)} \mu_{\beta}^n M^{1/2}(Z^n) v_{\beta}(Z^n) \quad (50) $$

with (see (44))

$$ 0 \le -\mu_{\beta}^{n} \le \frac{\lambda_{\max, B_1}}{\sqrt{\lambda_{\min, B_1}}} (|V^n - V^{n-1}| + h_i C_F) \le C_2' = 3C_0 \frac{\lambda_{\max, B_1}}{\sqrt{\lambda_{\min, B_1}}} \quad (51) $$

and the last term of (49) can be estimated as

$$ \begin{aligned} & \sum_{n=N_i+1}^{P_i} \left| (M^{1/2}(Z^n)(V^n - V^{n-1} - h_i F^n), v_\alpha(Z^{P_i})) \right| \\ & \le \sum_{n=N_i+1}^{P_i} \sum_{\beta \in J(Z^n)} \left| -\mu_\beta^n \right| \left| (M^{1/2}(Z^n)M^{-1}(U^n)M^{1/2}(Z^n)v_\beta(Z^n), v_\alpha(Z^{P_i})) \right| \\ & \le \sum_{n=N_i+1}^{P_i} \sum_{\beta \in J(Z^n)} \left| -\mu_\beta^n \right| \left| (v_\beta(Z^n), v_\alpha(Z^{P_i})) \right| \\ & \qquad + \sum_{n=N_i+1}^{P_i} \sum_{\beta \in J(Z^n)} \left| -\mu_\beta^n \right| \left\| M^{1/2}(Z^n) \right\|^2 \left\| M^{-1}(U^n) - M^{-1}(Z^n) \right\| \\ & \le \sum_{n=N_i+1}^{P_i} \sum_{\beta \in J(Z^n)} \frac{\lambda_{\max, B_1}}{\sqrt{\lambda_{\min, B_1}}} (|V^n - V^{n-1}| + h_i C_F) \left| (v_\beta(Z^n), v_\alpha(Z^{P_i})) \right| \\ & \qquad + \sum_{n=N_i+1}^{P_i} \sum_{\beta \in J(Z^n)} 3C_0 \frac{\lambda_{\max, B_1}^2}{\sqrt{\lambda_{\min, B_1}}} L_{M^{-1}} |U^n - Z^n|. \end{aligned} $$

But, with (20) we have

$$ |U^n - Z^n| \le C_0 h_i \quad (52) $$
and, recalling that $\beta \in J(Z^n)$ and $\alpha \in J(Z^{N_i})$, we infer from (8) and Lemma 15 that

$$ \begin{aligned} \left| (v_{\beta}(Z^n), v_{\alpha}(Z^{P_i})) \right| &\le \left| (v_{\beta}(Z^n), v_{\alpha}(Z^{N_i})) \right| + \left| (v_{\beta}(Z^n), v_{\alpha}(Z^{P_i}) - v_{\alpha}(Z^{N_i})) \right| \\ &\le C_{B_1} |Z^n - Z^{N_i}| + C'_{B_1} |Z^{P_i} - Z^{N_i}| \le 2C_{e, B_1} C_0 \varepsilon \end{aligned} $$

with $C_{e, B_1} = \max(C_{B_1}, C'_{B_1})$. Using the estimate of the discrete accelerations obtained at Proposition 4, we obtain

$$ \begin{aligned} & \sum_{n=N_i+1}^{P_i} \left| (M^{1/2}(Z^n)(V^n - V^{n-1} - h_i F^n), v_\alpha(Z^{P_i})) \right| \\ &\le 2\nu C_{e,B_1} C_0 \varepsilon \frac{\lambda_{\max,B_1}}{\sqrt{\lambda_{\min,B_1}}} (C'_0 + \varepsilon C_F) + 3\nu \frac{\lambda_{\max,B_1}^2}{\sqrt{\lambda_{\min,B_1}}} C_0^2 L_{M^{-1}} \varepsilon. \end{aligned} $$

Hence

$$ (M^{1/2}(Z^{P_i})V^{P_i}, v_{\alpha}(Z^{P_i})) \le -e (M^{1/2}(Z^{N_i-2})V^{N_i-2}, v_{\alpha}(Z^{N_i-2})) + \mathcal{O}(\varepsilon + h_i) \quad (53) $$

But $N_i - 2 \ge n_i + 1$ and

$$ \begin{aligned} & (M^{1/2}(Z^{N_i-2})V^{N_i-2}, v_{\alpha}(Z^{N_i-2})) - (M^{1/2}(Z^{n_i})V^{n_i}, v_{\alpha}(Z^{n_i})) \\ &= \sum_{n=n_i+1}^{N_i-2} \sum_{\beta \in J(Z^n)} -\mu_{\beta}^n (M^{1/2}(Z^n)M^{-1}(U^n)M^{1/2}(Z^n)v_{\beta}(Z^n), v_{\alpha}(Z^n)) \\ &\quad + \sum_{n=n_i+1}^{N_i-2} \left( (M^{1/2}(Z^n) - M^{1/2}(Z^{n-1}))V^{n-1}, v_{\alpha}(Z^n) \right) \\ &\quad + \sum_{n=n_i+1}^{N_i-2} (M^{1/2}(Z^{n-1})V^{n-1}, v_{\alpha}(Z^n) - v_{\alpha}(Z^{n-1})) \\ &\quad + \sum_{n=n_i+1}^{N_i-2} h_i (M^{1/2}(Z^n)F^n, v_{\alpha}(Z^n)) \\ &\ge \sum_{n=n_i+1}^{N_i-2} \sum_{\beta \in J(Z^n)} -\mu_{\beta}^n (M^{1/2}(Z^n)M^{-1}(U^n)M^{1/2}(Z^n)v_{\beta}(Z^n), v_{\alpha}(Z^n)) \\ &\quad - (L_{M^{1/2}} C_0^2 + \sqrt{\lambda_{\max,B_1}} L_{\tilde{U}} C_0^2 + \sqrt{\lambda_{\max,B_1}} C_F) \frac{3}{2} \varepsilon. \end{aligned} \quad (54) $$
We rewrite the first term of the right-hand side as follows

$$
\begin{align*}
& \sum_{n=n_i+1}^{N_i-2} \sum_{\beta \in J(Z^n)} -\mu_{\beta}^n (M^{1/2}(Z^n) M^{-1}(U^n) M^{1/2}(Z^n) v_{\beta}(Z^n), v_{\alpha}(Z^n)) \\
&= \sum_{n=n_i+1}^{N_i-2} \sum_{\beta \in J(Z^n)} -\mu_{\beta}^n (v_{\beta}(Z^n), v_{\alpha}(Z^n)) \\
&\quad + \sum_{n=n_i+1}^{N_i-2} \sum_{\beta \in J(Z^n)} -\mu_{\beta}^n (M^{1/2}(Z^n) (M^{-1}(U^n) \\
&\qquad - M^{-1}(Z^n)) M^{1/2}(Z^n) v_{\beta}(Z^n), v_{\alpha}(Z^n)).
\end{align*}
$$

Then assumption (H6) [see (8)] implies that, for all $n \in \{n_i + 1, \dots, N_i - 2\}$

$$
(v_{\beta}(Z^n), v_{\alpha}(Z^n)) = \begin{cases} 1 & \text{if } \alpha = \beta, \\ 0 & \text{otherwise} \end{cases}
$$

if $\alpha \in J(Z^n)$, and with (8) and Lemma 15

$$
\begin{align*}
|(v_{\beta}(Z^n), v_{\alpha}(Z^n))| &\le |(v_{\beta}(Z^n), v_{\alpha}(Z^{N_i}))| + |(v_{\beta}(Z^n), v_{\alpha}(Z^n) - v_{\alpha}(Z^{N_i}))| \\
&\le C_{B_1}|Z^n - Z^{N_i}| + C'_{B_1}|Z^n - Z^{N_i}| \le 2C_{e, B_1}|Z^n - Z^{N_i}| \\
&\le 3C_{e, B_1}C_{0\epsilon}
\end{align*}
$$

if $\alpha \notin J(Z^n)$. From (52) it follows that

$$
\begin{align*}
& \sum_{n=n_i+1}^{N_i-2} \sum_{\beta \in J(Z^n)} -\mu_{\beta}^n (M^{1/2}(Z^n) M^{-1}(U^n) M^{1/2}(Z^n) v_{\beta}(Z^n), v_{\alpha}(Z^n)) \\
\ge {} & -3v \frac{\lambda_{\max, B_1}}{\sqrt{\lambda_{\min, B_1}}} \left(C_0' + \frac{3}{2}\epsilon C_F\right) C_{e, B_1} C_{0\epsilon} - v C_2' \lambda_{\max, B_1} L M^{-1} C_0 \frac{3}{2}\epsilon. \quad (55)
\end{align*}
$$

Thus, combining (53), (54) and (55)

$$
(M^{1/2}(Z^{P_i})V^{P_i}, v_{\alpha}(Z^{P_i})) \le -e(M^{1/2}(Z^{n_i})V^{n_i}, v_{\alpha}(Z^{n_i})) + \mathcal{O}(\epsilon + h_i).
$$

Recalling that

$$
V^{P_i} = v_{h_i} \left( \bar{t} + \frac{\epsilon}{2} \right), \quad V^{n_i} = v_{h_i} \left( \bar{t} - \epsilon \right)
$$

and

$$
|Z^{P_i} - u_{h_i} \left( \bar{t} + \frac{\epsilon}{2} \right)| \le 2C_0 h_i, \quad |Z^{n_i} - u_{h_i} \left( \bar{t} - \epsilon \right)| \le 2C_0 h_i,
$$

we can pass to the limit as $h_i$ tends to zero, then as $\epsilon$ tends to zero, we get

$$
(M^{1/2}(\bar{u})\dot{u}^{+}, v_{\alpha}(\bar{u})) \le -e(M^{1/2}(\bar{u})\dot{u}^{-}, v_{\alpha}(\bar{u})).
$$

□
It remains for us to prove that, for all $\alpha \in J(\bar{u})$ such that $\mu_{\alpha} \neq 0$, we have

$$ (M^{1/2}(\bar{u})\dot{u}^{+}, v_{\alpha}(\bar{u})) \geq -e (M^{1/2}(\bar{u})\dot{u}^{-}, v_{\alpha}(\bar{u})). $$

Let us observe that the result is immediate if $(M^{1/2}(\bar{u})\dot{u}^{-}, v_{\alpha}(\bar{u})) = 0$ since we have, already, $(M^{1/2}(\bar{u})\dot{u}^{+}, v_{\alpha}(\bar{u})) \geq 0$.

So we assume now that $\alpha \in J(\bar{u})$ such that $\mu_{\alpha} \neq 0$, $(M^{1/2}(\bar{u})\dot{u}^{-}, v_{\alpha}(\bar{u})) < 0$ and we adopt the same notations as in the previous lemma. Since $f_{\alpha}(\bar{u}) = 0$, $(\nabla f_{\alpha}(\bar{u}), \dot{u}^{-}) < 0$ and $\lim_{t \to \bar{t}^{-}} \dot{u}(t) = \dot{u}^{-}$, we infer that, possibly decreasing $\bar{\epsilon}$, we have

$$ f_{\alpha}(u(t)) \geq \frac{|(\nabla f_{\alpha}(\bar{u}), \dot{u}^{-})|}{2} (\bar{t} - t) \quad \forall t \in [\bar{t} - \bar{\epsilon}, \bar{t}). $$

Then, the Lipschitz property of $f_{\alpha}$ on $B_1$ implies that

$$ f_{\alpha}(u_{h_i}(t)) \geq \frac{|(\nabla f_{\alpha}(\bar{u}), \dot{u}^{-})|}{2} (\bar{t} - t) - L_f \|u - u_{h_i}\|_{C^0([0, \tau]; \mathbb{R}^d)} $$

for all $t \in [\bar{t} - \bar{\epsilon}, \bar{t})$. As a consequence, for all $\epsilon \in (0, \bar{\epsilon}]$, for all $t \in [\bar{t} - \epsilon, \bar{t} - \epsilon/2]$ we have

$$ f_{\alpha}(u_{h_i}(t)) \geq |(\nabla f_{\alpha}(\bar{u}), \dot{u}^{-})| \frac{\epsilon}{4} - L_f \|u - u_{h_i}\|_{C^0([0, \tau]; \mathbb{R}^d)} $$

and, for all $nh_i \in [\bar{t} - \epsilon, \bar{t} - \epsilon/2]$

$$ f_{\alpha}(Z^n) \geq |(\nabla f_{\alpha}(\bar{u}), \dot{u}^{-})| \frac{\epsilon}{4} - L_f \|u - u_{h_i}\|_{C^0([0, \tau]; \mathbb{R}^d)} - L_f C_0 h_i. $$

It follows that, for all $\epsilon \in (0, \bar{\epsilon}]$ there exists $\tilde{h}_{\epsilon} \in (0, h_2^*)$ such that

$$ f_{\alpha}(Z^n) > 0 \quad \forall nh_i \in [\bar{t} - \epsilon, \bar{t} - \epsilon/2], \quad \forall h_i \in (0, \tilde{h}_{\epsilon}). $$

Now, we consider $\epsilon \in (0, \bar{\epsilon}]$. Let $h'_{\epsilon} = \min(\tilde{h}_{\epsilon}, h_{\epsilon}/2)$ and let $h_i \in (0, h'_{\epsilon}]$. We define $p_i$, $n_i$ and $m_i$ by

$$ p_i = \left\lfloor \frac{\bar{t} + \epsilon}{h_i} \right\rfloor, \quad n_i = \left\lfloor \frac{\bar{t} - \epsilon}{h_i} \right\rfloor, \quad m_i = \left\lfloor \frac{\bar{t} - \epsilon/2}{h_i} \right\rfloor. $$

Thus

$$ \begin{aligned} 0 < n_i h_i \leq \bar{t} - \epsilon < (n_i + 1)h_i < m_i h_i \leq \bar{t} - \frac{\epsilon}{2} < (m_i + 1)h_i < p_i h_i \\ \leq \bar{t} + \epsilon < (p_i + 1)h_i < \tau. \end{aligned} $$

Now let $N'_i$ be the first time step in $[\bar{t} - \frac{\epsilon}{2}, \bar{t} + \frac{\epsilon}{2}]$ such that $f_{\alpha}(Z^n) \leq 0$, that is,

$$ N'_i = \min \left\{ n \in \mathbb{N} ; nh_i \in \left[ \bar{t} - \frac{\epsilon}{2}, \bar{t} + \frac{\epsilon}{2} \right] \text{ and } f_{\alpha}(Z^n) \leq 0 \right\}. $$

Then we have $f_{\alpha}(Z^n) > 0$ for all $n \in \{n_i + 1, \dots, N'_i - 1\}$. Moreover,
**Lemma 9.** There exists a constant $C'_4 > 0$ such that, for all $\varepsilon \in (0, \bar{\varepsilon}]$ and for all $h_i \in (0, h'_\varepsilon]$, we have:

$$ \left| (M^{1/2}(Z^{N'_i})(V^{n_i} - V^{N'_i-1}), v_\alpha(Z^{N'_i})) \right| \le C'_4 \varepsilon. $$

**Proof.** With (43) we have

$$ \begin{aligned} & (M^{1/2}(Z^{N'_i})(V^{n_i} - V^{N'_i-1}), v_\alpha(Z^{N'_i})) \\ &= \sum_{n=n_i+1}^{N'_i-1} (M^{1/2}(Z^{N'_i})(V^{n-1} - V^n), v_\alpha(Z^{N'_i})) \\ &= -h_i \sum_{n=n_i+1}^{N'_i-1} (M^{1/2}(Z^{N'_i})F^n, v_\alpha(Z^{N'_i})) \\ &\quad + \sum_{n=n_i+1}^{N'_i-1} \sum_{\beta \in J(Z^n)} \mu_\beta^n (M^{1/2}(Z^{N'_i})M^{-1}(U^n)M^{1/2}(Z^n)v_\beta(Z^n), v_\alpha(Z^{N'_i})). \end{aligned} $$

The first term of the right-hand side can be easily estimated:

$$ \begin{aligned} \left| h_i \sum_{n=n_i+1}^{N'_i-1} (M^{1/2}(Z^{N'_i})F^n, v_\alpha(Z^{N'_i})) \right| &\le h_i (N'_i - n_i - 1) C_F \sqrt{\lambda_{\max, B_1}} \\ &\le \frac{3}{2} C_F \sqrt{\lambda_{\max, B_1}} \varepsilon. \end{aligned} \quad (56) $$

We rewrite the second term of the right-hand side as follows:

$$ \begin{aligned} & \left| \sum_{n=n_i+1}^{N'_i-1} \sum_{\beta \in J(Z^n)} \mu_\beta^n (M^{1/2}(Z^{N'_i})M^{-1}(U^n)M^{1/2}(Z^n)v_\beta(Z^n), v_\alpha(Z^{N'_i})) \right| \\ &\le \sum_{n=n_i+1}^{N'_i-1} \sum_{\beta \in J(Z^n)} \left| \mu_\beta^n \right| \left\| M^{1/2}(Z^{N'_i})M^{-1}(U^n)M^{1/2}(Z^n) - \mathrm{Id} \right\| \\ &\quad + \sum_{n=n_i+1}^{N'_i-1} \sum_{\beta \in J(Z^n)} \left| \mu_\beta^n \right| \left| (v_\beta(Z^n), v_\alpha(Z^{N'_i})) \right|. \end{aligned} \quad (57) $$

Since $\alpha \notin J(Z^n)$ for all $n \in \{n_i+1, \dots, N'_i-1\}$, assumption (H6) and (21) imply that, for all $\beta \in J(Z^n)$

$$ \begin{aligned} \left| (v_\beta(Z^n), v_\alpha(Z^{N'_i})) \right| &\le C_{e,B_1} |Z^n - Z^{N'_i}| \le C_{e,B_1} \sum_{k=n+1}^{N'_i} \frac{h}{1+e} |V^k + eV^{k-2}| \\ &\le \frac{3}{2} C_{e,B_1} C_0 \varepsilon. \end{aligned} $$
Using estimate (51) for $|\mu_{\beta}^n|$ and the estimate of the discrete accelerations obtained in Proposition 4, we get

$$
\begin{align}
& \sum_{n=n_i+1}^{N_i'-1} \sum_{\beta \in J(Z^n)} |\mu_{\beta}^n| |(v_{\beta}(Z^n), v_{\alpha}(Z^{N_i'}))| \nonumber \\
& \le \sum_{n=n_i+1}^{N_i'-1} \frac{3}{2} \nu C_{e,B_1} C_0 \varepsilon \frac{\lambda_{\max,B_1}}{\sqrt{\lambda_{\min,B_1}}} (|V^n - V^{n-1}| + h_i C_F) \nonumber \\
& \le \frac{3}{2} \nu C_{e,B_1} C_0 \varepsilon \frac{\lambda_{\max,B_1}}{\sqrt{\lambda_{\min,B_1}}} (C_0' + \frac{3\varepsilon}{2} C_F). \tag{58}
\end{align}
$$

It remains to estimate the first term of the right-hand side of (57). Using the Lipschitz
property of $M^{1/2}$ and $M^{-1}$, we get

$$
\begin{align*}
& \sum_{n=n_i+1}^{N_i'-1} \sum_{\beta \in J(Z^n)} |\mu_{\beta}^n| \left\| M^{1/2}(Z^{N_i'}) M^{-1}(U^n) M^{1/2}(Z^n) - \text{Id} \right\| \\
& \le \sum_{n=n_i+1}^{N_i'-1} \sum_{\beta \in J(Z^n)} |\mu_{\beta}^n| \left( \begin{aligned}[t]
& \left\| M^{1/2}(Z^{N_i'}) - M^{1/2}(Z^n) \right\| \left\| M^{-1}(U^n) M^{1/2}(Z^n) \right\| \\
& + \left\| M^{-1}(U^n) - M^{-1}(Z^n) \right\| \left\| M^{1/2}(Z^n) \right\|^2 \end{aligned} \right) \\
& \le \sum_{n=n_i+1}^{N_i'-1} \sum_{\beta \in J(Z^n)} |\mu_{\beta}^n| \left( L M^{1/2} C_0 h_i (N_i' - n) \frac{\sqrt{\lambda_{\max,B_1}}}{\lambda_{\min,B_1}} + \lambda_{\max,B_1} L M^{-1} C_0 h_i \right) \\
& \le \sum_{n=n_i+1}^{N_i'-1} \sum_{\beta \in J(Z^n)} \frac{3}{2} \varepsilon L M^{1/2} C_0 \left( \frac{\lambda_{\max,B_1}}{\lambda_{\min,B_1}} \right)^{3/2} (|V^n - V^{n-1}| + h_i C_F) \\
& \phantom{\le} + \frac{3}{2} \varepsilon \nu C_2' \lambda_{\max,B_1} L M^{-1} C_0 \\
& \le \frac{3}{2} \varepsilon L M^{1/2} C_0 \nu \left( \frac{\lambda_{\max,B_1}}{\lambda_{\min,B_1}} \right)^{3/2} \left( C_0' + \frac{3\varepsilon}{2} C_F \right) + \frac{3}{2} \varepsilon \nu C_2' \lambda_{\max,B_1} L M^{-1} C_0.
\end{align*}
$$

Combining this last estimate with (56), (57) and (58), we may conclude. □

Now let us prove that

**Lemma 10.** There exists a constant $C'_5 > 0$ such that, for all $\varepsilon \in (0, \bar{\varepsilon}]$ and for all $h_i \in (0, h'_\varepsilon]$, we have:

$$
f_{\alpha}(Z^{N_i'+1}) \leq C_5 h_i^2.
$$
**Proof.** If $\alpha \in J(Z^{N'_i+1})$ the result is obvious. Otherwise,

$$
\begin{aligned}
& \left| (V^{N'_i} - V^{N'_i+1} + h_i F^{N'_i+1}, M^{1/2}(Z^{N'_i+1}) v_\alpha(Z^{N'_i+1})) \right| \\
& \le \sum_{\beta \in J(Z^{N'_i+1})} |\mu_\beta^{N'_i+1}| \left| (M^{-1}(U^{N'_i+1}) M^{1/2}(Z^{N'_i+1}) v_\beta(Z^{N'_i+1}), \right. \\
& \qquad \left. M^{1/2}(Z^{N'_i+1}) v_\alpha(Z^{N'_i+1})) \right| \\
& \le \sum_{\beta \in J(Z^{N'_i+1})} |\mu_\beta^{N'_i+1}| \left\| M^{-1}(U^{N'_i+1}) - M^{-1}(Z^{N'_i+1}) \right\| \left\| M^{1/2}(Z^{N'_i+1}) \right\|^2 \\
& \qquad + \sum_{\beta \in J(Z^{N'_i+1})} |\mu_\beta^{N'_i+1}| \left| (v_\beta(Z^{N'_i+1}), v_\alpha(Z^{N'_i+1})) \right|.
\end{aligned}
$$

Since $\alpha \in J(Z^{N'_i})$ and $\alpha \notin J(Z^{N'_i+1})$, assumption (H6) and Lemma 15 imply that

$$
\begin{aligned}
\left| (v_{\beta}(Z^{N'_{i}+1}), v_{\alpha}(Z^{N'_{i}+1})) \right| &\le \left| (v_{\beta}(Z^{N'_{i}+1}), v_{\alpha}(Z^{N'_{i}))) \right| \\
&\quad + \left| v_{\alpha}(Z^{N'_{i}}) - v_{\alpha}(Z^{N'_{i}+1}) \right| \\
&\le 2C_{e,B_1}C_0h_i.
\end{aligned}
$$

Using the Lipschitz property of $M^{-1}$, we get finally

$$
\begin{aligned}
& \left| (V^{N'_i} - V^{N'_i+1}, M^{1/2}(Z^{N'_i+1}) v_\alpha(Z^{N'_i+1})) \right| \\
& \leq h_i \sqrt{\lambda_{\max,B_1} C_F + h_i \nu C'_2 C_0 (\lambda_{\max,B_1} L_{M^{-1}} + 2C_{e,B_1})}.
\end{aligned}
\qquad (59)
$$

With exactly the same computations, we also obtain

$$
\begin{aligned}
& \left| (V^{N'_i-2} - V^{N'_i-1}, M^{1/2}(Z^{N'_i-1}) v_\alpha(Z^{N'_i-1})) \right| \\
& \le h_i \sqrt{\lambda_{\max,B_1} C_F + h_i \nu C'_2 C_0 (\lambda_{\max,B_1} L_{M^{-1}} + 2C_{e,B_1})} \quad (60)
\end{aligned}
$$

since (recall that $\alpha \in J(Z^{N'_i})$ and $\alpha \notin J(Z^{N'_i-1})$)

$$
\begin{aligned}
\left| (v_{\beta}(Z^{N'_{i}-1}), v_{\alpha}(Z^{N'_{i}-1})) \right| &\le \left| (v_{\beta}(Z^{N'_{i}-1}), v_{\alpha}(Z^{N'_{i}})) \right| \\
&\quad + \left| (v_{\alpha}(Z^{N'_{i}}) - v_{\alpha}(Z^{N'_{i}-1})) \right| \\
&\le 2C_{e,B_1}C_0h_i.
\end{aligned}
$$

We define

$$
C'_6 = \sqrt{\lambda_{\max,B_1} C_F + \nu C'_2 C_0 (\lambda_{\max,B_1} L_{M^{-1}} + 2C_{e,B_1})}.
$$
Then, using a Taylor's formula, we get

$$ 
\begin{aligned}
f_{\alpha}(Z^{N'_{i}+1}) = f_{\alpha}(Z^{N'_{i}}) &+ (\nabla f_{\alpha}(Z^{N'_{i}}), Z^{N'_{i}+1} - Z^{N'_{i}}) \\ 
&+ \int_{0}^{1} (\nabla f_{\alpha}(Z^{N'_{i}} + t(Z^{N'_{i}+1} - Z^{N'_{i}})) \\ 
&- \nabla f_{\alpha}(Z^{N'_{i}}), Z^{N'_{i}+1} - Z^{N'_{i}}) \, dt.
\end{aligned}
$$

But $f_{\alpha}(Z^{N'_{i}}) = 0$ since $\alpha \in J(Z^{N'_{i}})$ and

$$ Z^{N'_{i}+1} - Z^{N'_{i}} = \frac{h_i}{1+e} (V^{N'_{i}+1} + eV^{N'_{i}-1}). $$

Moreover,

$$ (\nabla f_{\alpha}(Z^{N'_{i}}), V^{N'_{i}+1} + eV^{N'_{i}-1}) = (\nabla f_{\alpha}(Z^{N'_{i}}), V^{N'_{i}+1} - V^{N'_{i}}) \\ + (\nabla f_{\alpha}(Z^{N'_{i}}), V^{N'_{i}} + eV^{N'_{i}-2}) \\ + e (\nabla f_{\alpha}(Z^{N'_{i}}), V^{N'_{i}-1} - V^{N'_{i}-2}). \quad (61) $$

The second term of the right-hand side is estimated with Proposition 1:

$$ (\nabla f_{\alpha}(Z^{N'_{i}}), V^{N'_{i}} + eV^{N'_{i}-2}) \leq \frac{L_f h_i}{2(1+e)} |V^{N'_{i}} + eV^{N'_{i}-2}|^2 \leq L_f h_i C_0^2. $$

The first and third terms of the right-hand side of (61) can be estimated by using the Lipschitz property of $\nabla f_{\alpha}$ on $B_1$ and (59)–(60). Indeed,

$$ 
\begin{aligned}
(\nabla f_{\alpha}(Z^{N'_{i}}), V^{N'_{i}+1} - V^{N'_{i}}) &= (\nabla f_{\alpha}(Z^{N'_{i}}) - \nabla f_{\alpha}(Z^{N'_{i}+1}), V^{N'_{i}+1} - V^{N'_{i}}) \\ 
&\quad + |M^{-1/2}(Z^{N'_{i}+1})\nabla f_{\alpha}(Z^{N'_{i}+1})| \\ 
&\quad \times (M^{1/2}(Z^{N'_{i}+1})v_{\alpha}(Z^{N'_{i}+1}), V^{N'_{i}+1} - V^{N'_{i}}) \\ 
&\leq 2L_f C_0^2 h_i + \frac{C_{\bar{u}}}{\sqrt{\lambda_{\min,B_1}}} C_6' h_i
\end{aligned}
$$

and similarly

$$ 
\begin{aligned}
(\nabla f_{\alpha}(Z^{N'_{i}}), V^{N'_{i}-1} - V^{N'_{i}-2}) &= (\nabla f_{\alpha}(Z^{N'_{i}}) - \nabla f_{\alpha}(Z^{N'_{i}-1}), V^{N'_{i}-1} - V^{N'_{i}-2}) \\ 
&\quad + |M^{-1/2}(Z^{N'_{i}-1})\nabla f_{\alpha}(Z^{N'_{i}-1})| \\ 
&\quad \times (M^{1/2}(Z^{N'_{i}-1})v_{\alpha}(Z^{N'_{i}-1}), V^{N'_{i}-1} - V^{N'_{i}-2}) \\ 
&\leq 2L_f C_0^2 h_i + \frac{C_{\bar{u}}}{\sqrt{\lambda_{\min,B_1}}} C_6' h_i.
\end{aligned}
$$
Moreover,

$$ \left| \int_0^1 \left( \nabla f_\alpha \left( Z^{N'_i} + t(Z^{N'_i+1} - Z^{N'_i}) \right) - \nabla f_\alpha(Z^{N'_i}), Z^{N'_i+1} - Z^{N'_i} \right) dt \right| \\ \le \int_0^1 L_f |Z^{N'_i+1} - Z^{N'_i}|^2 t dt \le \frac{C_0^2 L_f}{2} h_i^2. $$

Thus

$$ f_{\alpha}(Z^{N'_{i}+1}) \le C'_{5}h_{i}^{2} \quad \forall h_{i} \in (0, h_{2}^{*}] $$

with

$$ C'_{5} = \frac{11}{2} C_{0}^{2} L_{f} + 2 \frac{C_{\bar{u}}}{\sqrt{\lambda_{\min, B_{1}}}} C'_{6}. $$

It follows that, for all $\varepsilon \in (0, \bar{\varepsilon}]$ and for all $h_i \in (0, h'_e]$, we have

$$ 0 \le f_{\alpha}(Z^{N'_{i}+1}) - f_{\alpha}(Z^{N'_{i}}) = f_{\alpha}(Z^{N'_{i}+1}) \le C'_{5}h_{i}^{2} $$

since $Z^{N'_i+1} \in K$ by construction. But, with the same computations as in Proposition 1, we also have

$$ f_{\alpha}(Z^{N'_{i}+1}) - f_{\alpha}(Z^{N'_{i}}) = \frac{h_i}{1+e} (\nabla f_{\alpha}(Z^{N'_{i}}), V^{N'_{i}+1} + eV^{N'_{i}-1}) \\ + \frac{h_i}{1+e} \int_0^1 (\nabla f_{\alpha}(Z^{N'_{i}} + t(Z^{N'_{i}+1} - Z^{N'_{i}})) \\ - \nabla f_{\alpha}(Z^{N'_{i}}), V^{N'_{i}+1} + eV^{N'_{i}-1}) dt. $$

We can estimate the last term of the right-hand side by

$$ \left| \int_0^1 \left( \nabla f_\alpha (Z^{N'_i} + t(Z^{N'_i+1} - Z^{N'_i})) - \nabla f_\alpha(Z^{N'_i}), V^{N'_i+1} + eV^{N'_i-1} \right) dt \right| \\ \le L_f h_i \frac{|V^{N'_i+1} + eV^{N'_i-1}|^2}{2(1+e)} \le L_f C_0^2 h_i $$

and thus

$$ -L_f C_0^2 h_i \le (\nabla f_\alpha(Z^{N'_i}), V^{N'_i+1} + eV^{N'_i-1}) \le C'_5(1+e)h_i + L_f C_0^2 h_i. $$

Using Lemma 15, we finally get

$$ \begin{aligned} \left| (M^{1/2}(Z^{N'_i+1})v_\alpha(Z^{N'_i+1}), V^{N'_i+1} + eV^{N'_i-1}) \right| & \le \frac{L_f C_0^2 + C'_5(1+e)}{m_{B_1}} h_i \\ &= C'_7 h_i. \end{aligned} \quad (62) $$

Now we will estimate $(M^{1/2}(Z^{p_i+1})v_\alpha(Z^{p_i+1}), V^{N'_i+1} - V^{p_i})$.
**Lemma 11.** There exists a constant $C'_8 > 0$ such that, for all $\varepsilon \in (0, \bar{\varepsilon}]$ and for all $h_i \in (0, h'_\varepsilon]$, we have:

$$ (M^{1/2}(Z^{p_i+1})v_\alpha(Z^{p_i+1}), V^{N'_i+1} - V^{p_i}) \le C'_8(\varepsilon + h_i). $$

**Proof.** Let $h_i \in (0, h'_\varepsilon]$. First we observe that $N'_i+1 < p_i$ since $h_i \le h'_\varepsilon \le h_{\varepsilon/2} \le \frac{\varepsilon}{6}$. With (50) we have

$$ V^{N'_i+1} - V^{p_i} = -h_i \sum_{l=N'_i+1}^{p_i-1} F^{l+1} + \sum_{l=N'_i+1}^{p_i-1} \sum_{\beta \in J(Z^{l+1})} \mu_\beta^{l+1} M^{-1}(U^{l+1}) M^{1/2}(Z^{l+1}) v_\beta(Z^{l+1}) $$

and

$$ \begin{aligned} & (V^{N'_i+1} - V^{p_i}, M^{1/2}(Z^{p_i+1})v_\alpha(Z^{p_i+1})) \\ &= -h_i \sum_{l=N'_i+1}^{p_i-1} (F^l, M^{1/2}(Z^{p_i+1})v_\alpha(Z^{p_i+1})) + \sum_{l=N'_i+1}^{p_i-1} \sum_{\beta \in J(Z^{l+1})} \mu_\beta^{l+1} \\ & \quad \times (M^{-1}(U^{l+1})M^{1/2}(Z^{l+1})v_\beta(Z^{l+1}), M^{1/2}(Z^{p_i+1})v_\alpha(Z^{p_i+1})) \\ & \le \sum_{l=N'_i+1}^{p_i-1} \sum_{\beta \in J(Z^{l+1})} |\mu_\beta^{l+1}| \|M^{1/2}(Z^{p_i+1}) - M^{1/2}(Z^{l+1})\| \\ & \quad \times \|M^{-1}(U^{l+1})\| \|M^{1/2}(Z^{l+1})\| \\ & \quad + \sum_{l=N'_i+1}^{p_i-1} \sum_{\beta \in J(Z^{l+1})} |\mu_\beta^{l+1}| \|M^{1/2}(Z^{l+1})\|^2 \\ & \quad \times \|M^{-1}(U^{l+1}) - M^{-1}(Z^{l+1})\| \\ & \quad + \sum_{l=N'_i+1}^{p_i-1} \sum_{\beta \in J(Z^{l+1})} \mu_\beta^{l+1} (v_\beta(Z^{l+1}), v_\alpha(Z^{p_i+1})) \\ & \quad + h_i(p_i - N'_i - 1)\sqrt{\lambda_{\max, B_1}} C_F \\ & \le \frac{3}{2}\varepsilon\sqrt{\lambda_{\max, B_1}} C_F + \sum_{l=N'_i+1}^{p_i-1} \sum_{\beta \in J(Z^{l+1})} |\mu_\beta^{l+1}| \frac{\sqrt{\lambda_{\max, B_1}}}{\lambda_{\min, B_1}} L M^{1/2} C_0 \frac{3}{2}\varepsilon \end{aligned} $$
$$
\begin{equation}
\begin{aligned}
&+ \sum_{l=N_i'+1}^{p_i-1} \nu C_2^{l'} \lambda_{\max, B_1} L_{M^{-1}} C_0 h_i \\
&+ \sum_{l=N_i'+1}^{p_i-1} \sum_{\beta \in J(Z^{l+1})} \mu_{\beta}^{l+1} (v_{\beta}(Z^{l+1}), v_{\alpha}(Z^{p_i+1})).
\end{aligned}
\tag{63}
\end{equation}
$$

We estimate the second term of the right-hand side by using (51) and Proposition 4,
that is,

$$
\begin{align*}
& \sum_{l=N_i'+1}^{p_i-1} \sum_{\beta \in J(Z^{l+1})} \left| \mu_{\beta}^{l+1} \right| \frac{\sqrt{\lambda_{\max, B_1}}}{\lambda_{\min, B_1}} L_{M^{1/2}} C_0 \frac{3}{2} \epsilon \\
& \le \sum_{l=N_i'+1}^{p_i-1} C_0 \nu \left( \frac{\lambda_{\max, B_1}}{\lambda_{\min, B_1}} \right)^{3/2} L_{M^{1/2}} \frac{3}{2} \epsilon (|V^{l+1} - V^l| + h_i C_F) \\
& \le \frac{3}{2} \epsilon C_0 \nu \left( \frac{\lambda_{\max, B_1}}{\lambda_{\min, B_1}} \right)^{3/2} L_{M^{1/2}} \left( C_0' + \frac{3}{2} \epsilon C_F \right). \tag{64}
\end{align*}
$$

For the last term, we observe that, if $\alpha \in J(Z^{l+1})$, then $\mu_{\alpha}^{l+1} \le 0$ and

$$
\begin{align*}
\mu_{\alpha}^{l+1} (v_{\alpha}(Z^{l+1}), v_{\alpha}(Z^{p_i+1})) &\le \mu_{\alpha}^{l+1} + |\mu_{\alpha}^{l+1}| |v_{\alpha}(Z^{p_i+1}) - v_{\alpha}(Z^{l+1})| \\
&\le |\mu_{\alpha}^{l+1}| |C_{e,B_1}| |Z^{p_i+1} - Z^{l+1}| \\
&\le \frac{\lambda_{\max,B_1}}{\sqrt{\lambda_{\min,B_1}}} \times (|V^{l+1} - V^l| + h_i C_F) C_{e,B_1} C_0 (p_i - l) h_i \\
&\le \frac{\lambda_{\max,B_1}}{\sqrt{\lambda_{\min,B_1}}} (|V^{l+1} - V^l| + h_i C_F) C_{e,B_1} C_0 \frac{3}{2} \epsilon.
\end{align*}
$$

On the other hand, if $\beta \in J(Z^{l+1}) \setminus \{\alpha\}$, recalling that $\alpha \in J(Z^{N'_i})$, assumption (H6) implies that

$$
\begin{align*}
|\mu_{\beta}^{l+1}(v_{\beta}(Z^{l+1}), v_{\alpha}(Z^{p_i+1}))| &\le |\mu_{\beta}^{l+1}| (|v_{\alpha}(Z^{p_i+1}) - v_{\alpha}(Z^{N'_i})| \\
&\quad + |(v_{\alpha}(Z^{N'_i}), v_{\beta}(Z^{l+1}))|) \\
&\le |\mu_{\beta}^{l+1}| C_{e,B_1} (|Z^{p_i+1} - Z^{N'_i}| + |Z^{N'_i} - Z^{l+1}|) \\
&\le \frac{\lambda_{\max,B_1}}{\sqrt{\lambda_{\min,B_1}}} 2C_{e,B_1} C_0 \left(\frac{3}{2}\epsilon + h_i\right) \\
&\quad \times (|V^{l+1} - V^l| + h_i C_F).
\end{align*}
$$
Finally,

$$
\begin{align}
& \sum_{l=N_i^\prime+1}^{p_i-1} \sum_{\beta \in J(Z^{l+1})} \mu_\beta^{l+1} (v_\beta(Z^{l+1}), v_\alpha(Z^{p_i+1})) \nonumber \\
& \le \frac{\lambda_{\max, B_1}}{\sqrt{\lambda_{\min, B_1}}} 2C_{e,B_1} C_0 \nu \left(\frac{3}{2}\varepsilon + h_i\right) \left(C_0' + \frac{3}{2}\varepsilon C_F\right) \tag{65}
\end{align}
$$

and, combining (63), (64) and (65) we may conclude with

$$
C_8' = \frac{3\sqrt{\lambda_{\max, B_1}}}{2} (C_F + \nu C_2' \sqrt{\lambda_{\max, B_1}} L_{M^{-1}} C_0) \\
\qquad + \frac{3}{2} \nu C_0 \frac{\lambda_{\max, B_1}}{\sqrt{\lambda_{\min, B_1}}} \left( \frac{\sqrt{\lambda_{\max, B_1}}}{\lambda_{\min, B_1}} L_{M^{1/2}} + 2C_{e, B_1} \right) \left( C_0' + \frac{3}{2} \bar{\epsilon} C_F \right).
\quad \Box
$$

Now, for all $\varepsilon \in (0, \bar{\varepsilon}]$ and for all $h_i \in (0, h'_\varepsilon]$, we get, using estimate (62),

$$
\begin{align*}
(V^{p_i}, M^{1/2}(Z^{p_i+1})v_\alpha(Z^{p_i+1})) &\ge (V^{N_i'+1}, M^{1/2}(Z^{p_i+1})v_\alpha(Z^{p_i+1})) - C_8'(\varepsilon + h_i) \\
&\ge (V^{N_i'+1}, M^{1/2}(Z^{N_i'+1})v_\alpha(Z^{N_i'+1})) \\
&\quad + (V^{N_i'+1}, (M^{1/2}(Z^{p_i+1}) - M^{1/2}(Z^{N_i'+1}))v_\alpha(Z^{N_i'+1})) \\
&\quad + (V^{N_i'+1}, M^{1/2}(Z^{p_i+1})(v_\alpha(Z^{p_i+1}) - v_\alpha(Z^{N_i'+1}))) - C_8'(\varepsilon + h_i) \\
&\ge -e(V^{N_i'-1}, M^{1/2}(Z^{N_i'+1})v_\alpha(Z^{N_i'+1})) \\
&\quad - (L_{M^{1/2}} + \sqrt{\lambda_{\max B_1}} C_{e,B_1}) C_0 |Z^{p_i+1} - Z^{N_i'+1}| \\
&\quad - C_8'(\varepsilon + h_i) - C_7' h_i
\end{align*}
$$

and

$$
\begin{align*}
& (V^{p_i}, M^{1/2}(Z^{p_i+1})v_\alpha(Z^{p_i+1})) \\
& \ge -e(V^{N_i'-1}, M^{1/2}(Z^{N_i'})v_\alpha(Z^{N_i'})) - C_8'(\varepsilon + h_i) - C_7'h_i \\
& \quad - (L_{M^{1/2}} + \sqrt{\lambda_{\max,B_1}} C_{e,B_1}) C_0 (e|Z^{N_i'+1} - Z^{N_i'}| + |Z^{p_i+1} - Z^{N_i'+1}|) \\
& \ge -e(V^{N_i'-1}, M^{1/2}(Z^{N_i'})v_\alpha(Z^{N_i'})) - C_8'(\varepsilon + h_i) - C_7'h_i \\
& \quad - (L_{M^{1/2}} + \sqrt{\lambda_{\max,B_1}} C_{e,B_1}) C_0^2 (p_i - N_i' + 1)h_i.
\end{align*}
$$

So, with Lemma 9

$$
\begin{align}
& (V^{p_i}, M^{1/2}(Z^{p_i+1})v_\alpha(Z^{p_i+1})) \nonumber \\
& \ge -e(V^{n_i}, M^{1/2}(Z^{n_i'})v_\alpha(Z^{n_i'})) - C_8'(\varepsilon + h_i) - C_7'h_i \nonumber \\
& \quad -eC_4'\varepsilon - (L_{M^{1/2}} + \sqrt{\lambda_{\max,B_1}} C_{e,B_1}) C_0^2 \left(\frac{3}{2}\varepsilon + h_i\right), \tag{66}
\end{align}
$$
for all $\varepsilon \in (0, \bar{\varepsilon}]$ and for all $h_i \in (0, h'_\varepsilon]$. Observing that $V^{p_i} = v_{h_i}(\bar{t} + \varepsilon)$, $V^{n_i} = v_{h_i}(\bar{t} - \varepsilon)$ and

$$ \begin{aligned} |Z^{p_i+1} - \bar{u}| &\le |Z^{p_i+1} - U^{p_i+1}| + |u_{h_i}((p_i+1)h_i) - u_{h_i}(\bar{t})| + |u_{h_i}(\bar{t}) - u(\bar{t})| \\ &\le C_0h_i + C_0|(p_i+1)h_i - \bar{t}| + \|u_{h_i} - u\|_{C^0([0,\tau];\mathbb{R}^d)} \\ &\le \|u_{h_i} - u\|_{C^0([0,\tau];\mathbb{R}^d)} + C_0(\varepsilon + 2h_i) \end{aligned} $$

and similarly

$$ |Z^{N'_i} - \bar{u}| \le \|u_{h_i} - u\|_{C^0([0, \tau]; \mathbb{R}^d)} + C_0 \left( \frac{1}{2}\varepsilon + h_i \right) $$

we can pass to the limit in (66) as $h_i$ tends to zero. Then as $\varepsilon$ tends to zero, we get

$$ (M^{1/2}(\bar{u})\dot{u}^+, v_\alpha(\bar{u})) \ge -e (M^{1/2}(\bar{u})\dot{u}^-, v_\alpha(\bar{u})). $$

Combining this inequality with the result of Lemma 8 we obtain

$$ (M^{1/2}(\bar{u})\dot{u}^+, v_\alpha(\bar{u})) = -e (M^{1/2}(\bar{u})\dot{u}^-, v_\alpha(\bar{u})) $$

for all $\alpha \in J(\bar{u})$ such that $\mu_\alpha \ne 0$, which concludes the proof of the impact law.

### 3.3. Study of the initial conditions

Let us prove now that property (P4) holds. We begin with the following lemma:

**Lemma 12.** For all $k \in \{0, \dots, v\}$, we have

$$ \lim_{h_i \to 0} V^k = v_0. $$

**Proof.** Observing that the definitions of $U^0$ and $U^1$ are the same as in the inelastic case, we obtain immediately that (see Lemma 5 in [11])

$$ \lim_{h_i \to 0} V^0 = \lim_{h_i \to 0} v_{h_i}(0) = v(0) = v_0. $$

As in the previous subsection, we define $r_{u_0} > 0$ such that $J(q) \subset J(u_0)$ for all $q \in \bar{B}(u_0, r_{u_0})$. Possibly decreasing $r_{u_0}$, we may assume without loss of generality that $r_{u_0} \le \min(r_{B_1}, r'_{B_1})/2$. From the continuity of $u$ and the uniform convergence of $(u_{h_i})_{h_i \ge h_i > 0}$ to $u$ on $[0, \tau]$, we infer that there exist $\bar{\varepsilon} \in (0, \tau/2)$ and $\tilde{h}_1^* \in (0, \min(h_1^*, \frac{\bar{\varepsilon}}{\nu+1}, \frac{r_{u_0}}{6C_0}))$ such that

$$ u(t) \in \bar{B} \left( \bar{u}_0, \frac{r_{u_0}}{3} \right) \quad \forall t \in [0, \bar{\varepsilon}], $$

$$ \|u - u_{h_i}\|_{C^0([0, \tau]; \mathbb{R}^d)} \le \frac{r_{u_0}}{3} \quad \forall h_i \in (0, \tilde{h}_1^*]. $$

It follows that

$$ U^{n+1}, U^n, Z^n, Z^{n+1} \in \bar{B}(u_0, r_{u_0}) \quad \forall nh_i \in [0, \bar{\varepsilon}], \quad \forall h_i \in (0, \tilde{h}_1^*]. $$
Let us now distinguish between the two cases $u_0 \in \text{Int}(K)$ and $u_0 \in \partial K$.

Case 1: $u_0 \in \text{Int}(K)$.

Then, $\bar{B}(u_0, r_{u_0}) \subset \text{Int}(K)$. With Lemma 1 we infer that, for all $nh_i \in (0, \bar{\varepsilon}]$, for all $h_i \in (0, \tilde{h}_1^*)$

$$M(U^n)(V^{n-1} - V^n + h_i F^n) \in N_K(Z^n) = \{0\}$$

and thus

$$V^n = V^{n-1} + h_i F^n.$$

It follows that

$$|V^k - V^0| \le \sum_{l=1}^{k} h_i |F^l| \le k h_i C_F$$

for all $k \in \{1, \dots, \nu\}$ and for all $h_i \in (0, \tilde{h}_1^*)$ (let us recall that $\tilde{h}_1^* \le \bar{\varepsilon}/(\nu + 1)$), and the conclusion follows.

Case 2: $u_0 \in \partial K$.

Let us prove by induction on $k$ that

$$\lim_{h_i \to 0} V^k = v_0$$

for all $k \in \{0, \dots, \nu\}$. More precisely, let $k \in \{1, \dots, \nu\}$ and assume that

$$\lim_{h_i \to 0} V^l = v_0 \quad \forall l \in \{0, \dots, k-1\}. \qquad (67)$$

Since $v_0 \in T_K(u_0)$ and $\tilde{T}_K(u_0)$ is dense in $T_K(u_0)$, we may consider a sequence $(v_p)_{p \in \mathbb{N}^*}$ converging to $v_0$ and such that

$$v_p \in \tilde{T}_K(u_0) = \{w \in \mathbb{R}^d; (\nabla f_\alpha(u_0), w) > 0 \quad \forall \alpha \in J(u_0)\} \quad \forall p \in \mathbb{N}^*.$$

As in Lemma 5 in [11], we infer that there exist $t_p > 0$ such that

$$u_0 + t v_p \in K \cap B_1 \quad \forall t \in (0, t_p].$$

By definition of the scheme

$$\|W^k - Z^k\|_{M(U^k)} \le \|W^k - Z\|_{M(U^k)} \quad \forall Z \in K.$$

Let $h_p = \min(h^*, \frac{t_p(1+e)}{2+(1+e)(k-1)})$. By choosing $Z = u_0 + (\frac{2}{1+e} + (k-1))h_i v_p$, we get for all $h_i \in (0, h_p]$

$$\begin{aligned} W^k - Z &= \frac{2U^k - (1-e)U^{k-1} + h_i^2 F^k}{1+e} - \left(u_0 + \left(\frac{2}{1+e} + (k-1)\right)h_i v_p\right) \\ &= \frac{h_i}{1+e} \left(2V^{k-1} + (1+e) \sum_{l=0}^{k-2} V^l + h_i F^k - (2+(1+e)(k-1))v_p\right) \end{aligned}$$
and with (10)

$$
\begin{aligned}
& \|V^{k-1} - V^k + hF^k\|_{M(U^k)} \\
& \le \left\| 2V^{k-1} + (1+e) \sum_{l=0}^{k-2} V^l + h_i F^k - (2+(1+e)(k-1)) v_p \right\|_{M(U^k)}
\end{aligned}
$$

with the convention that $\sum_{l=0}^{k-2} V^l = 0$ if $k=1$, and thus

$$
\begin{aligned}
\|v_0 - V^k\|_{M(U^k)} \le {}& 2h_i \|F^k\|_{M(U^k)} + \|V^{k-1} - v_0\|_{M(U^k)} \\
& + (2+(1+e)(k-1)) (\|w_k - v_0\|_{M(U^k)} + \|v_0 - v_p\|_{M(U^k)})
\end{aligned}
$$

with

$$
w_k = \frac{1}{2 + (1 + e)(k - 1)} \left( 2V^{k-1} + (1 + e) \sum_{l=0}^{k-2} V^l \right).
$$

It follows that

$$
|v_0 - V^k| \le \sqrt{\frac{\lambda_{\max, B_1}}{\lambda_{\min, B_1}}} \left( 2h_i |F^k| + |V^{k-1} - v_0| \right. \\
\left. \qquad + (2+(1+e)(k-1)) (|w_k - v_0| + |v_0 - v_p|) \right) \quad \forall h_i \in (0, h_p].
$$

But the assumption (67) implies that $\lim_{h_i \to 0} w_k = v_0$ and we get

$$
\limsup_{h_i \to 0} |V^k - v_0| \le \sqrt{\frac{\lambda_{\max, B_1}}{\lambda_{\min, B_1}}} (2 + (1+e)(k-1)) |v_0 - v_p|.
$$

This last inequality holds for all $p \in \mathbb{N}^*$, and passing to the limit as $p$ tends to $+\infty$, we obtain

$$
\limsup_{h_i \to 0} |V^k - v_0| \le 0
$$

which implies $\lim_{h_i \to 0} V^k = v_0$. $\square$

Now we can prove that

**Lemma 13.** The initial conditions $(u_0, v_0)$ are satisfied in the following sense

$$
u(0) = u_0, \quad \dot{u}(0+0) = v_0.
$$

**Proof.** The first equality is trivial, since

$$
u(0) = \lim_{h_i \to 0} u_{h_i}(0) \quad \text{and} \quad u_{h_i}(0) = U^0 = u_0 \quad \forall i \in \mathbb{N}.
$$

In order to establish the second equality, we first observe that we already know with
Proposition 6 that

$$
M(u_0)(\dot{u}(0+0) - \dot{u}(0-0)) \in -N_K(u_0).
$$
Moreover, with convention (37) and the previous Lemma, we have

$$ \dot{u}(0-0) = \dot{u}(0) = v(0) = v_0 $$

and thus

$$ M(u_0)(\dot{u}(0+0) - v_0) \in -N_K(u_0). $$

If $u_0 \in \text{Int}(K)$, then $N_K(u_0) = \{0\}$ and we may conclude.

Let us assume now that $u_0 \in \partial K$. Since $v_0$ and $\dot{u}(0+0)$ belong to $T_K(u_0)$ we obtain

$$ \begin{aligned} (\dot{u}(0+0) - v_0, \dot{u}(0+0))_{M(u_0)} &\ge 0, \\ (\dot{u}(0+0) - v_0, v_0)_{M(u_0)} &\ge 0 \end{aligned} $$

and thus

$$ \|v_0\|_{M(u_0)}^2 \le (\dot{u}(0+0), v_0)_{M(u_0)} \le \|\dot{u}(0+0)\|_{M(u_0)}^2. \quad (68) $$

But with Proposition 3 we obtain

$$ \|V^n\|_{M(Z^n)}^2 \le \max \left( \|V^0\|_{M(Z^0)}^2, \dots, \|V^v\|_{M(Z^v)}^2 \right) + C_{B,C_0} nh $$

for all $nh \in [0, \tilde{\tau}]$, for all $h \in (0, \tilde{h}_1^*]$, where $\tilde{\tau} = \min(\delta/(2C_0), T)$ is defined at Proposition 2.

Let $t \in (0, \tilde{\tau})$ and $h_i \in (0, \tilde{h}_1^*)$. We define $n = \lfloor t/h_i \rfloor$ and with the Lipschitz property of $M$ on $B_1$ we get

$$ \begin{aligned} \|v_{h_i}(t)\|_{M(u_0)}^2 &= \|V^n\|_{M(u_0)}^2 \le \|V^n\|_{M(Z^n)}^2 + L_M |Z^n - u_0| C_0^2 \\ &\le \max \left( \|V^0\|_{M(u_0)}^2, \dots, \|V^v\|_{M(u_0)}^2 \right) + C_{B,C_0} nh \\ &\quad + \max_{1 \le l \le v} \left( L_M |Z^l - u_0| C_0^2 \right) + L_M |Z^n - u_0| C_0^2. \end{aligned} $$

But, for all $k \in \{1, \dots, n\}$

$$ \begin{aligned} |Z^k - u_0| &= \left| \frac{U^{k+1} + eU^{k-1}}{1+e} - U^0 \right| \le \frac{1}{1+e} \left( |U^{k+1} - U^0| + e|U^{k-1} - U^0| \right) \\ &\le (k+1)C_0h_i \end{aligned} \quad (69) $$

and $Z^0 = U^0$ by convention. So

$$ \begin{aligned} \|v_{h_i}(t)\|_{M(u_0)}^2 &\le \max \left( \|V^0\|_{M(u_0)}^2, \dots, \|V^v\|_{M(u_0)}^2 \right) + (v+1)C_0^3 L_M h_i \\ &\quad + C_{B,C_0} t + L_M C_0^3 (t+h_i). \end{aligned} $$

By passing to the limit as $h_i$ tends to zero, we get

$$ \|v(t)\|_{M(u_0)}^2 \le \|v_0\|_{M(u_0)}^2 + C_{B,C_0} t + L_M C_0^3 t \quad \forall t \in (0, \tilde{\tau}) $$
and, by passing to the limit as $t$ tends to zero

$$ \|\dot{u}(0+0)\|_{M(u_0)}^2 = \|v(0+0)\|_{M(u_0)}^2 \le \|v_0\|_{M(u_0)}^2. $$

With (68) it follows that

$$ \|v_0\|_{M(u_0)}^2 = (\dot{u}(0+0), v_0)_{M(u_0)} = \|\dot{u}(0+0)\|_{M(u_0)}^2 $$

and thus $\dot{u}(0+0) = v_0$. $\square$

Finally, with these results, we can state the following theorem.

**Theorem 2.** Let us assume that there exist $C_0 > 0$, $\tau_0 > 0$, $h_0^* \in (0, h^*]$ and a subsequence of the approximate positions defined by (4)–(7) such that

$$ |V^n| = \left| \frac{U^{n+1} - U^n}{h_i} \right| \le C_0 \quad \forall nh_i \in [0, \min(\tau_0, T)], \quad \forall h_i \in (0, h_0^*], $$

with $(h_i)_{i \in \mathbb{N}}$ decreasing to zero. Let $u_h$ and $v_h$ be defined by (32) and (33). Then, there exist a subsequence, still denoted $(h_i)_{i \in \mathbb{N}}$, and $(u, v) \in C^0([0, \min(\tau_0, T)]; \mathbb{R}^d) \times BV(0, \min(\tau_0, T); \mathbb{R}^d)$ such that

$$ u_{h_i} \to u \quad \text{strongly in } C^0([0, \min(\tau_0, T)]; \mathbb{R}^d), $$

$$ v_{h_i} \to v \quad \text{pointwise in } [0, \min(\tau_0, T)], $$

with

$$ u(t) = u_0 + \int_0^t v(s) \, ds \quad \text{for all } t \in [0, \min(\tau_0, T)], $$

and $u$ is a solution of problem (P) on $[0, \min(\tau_0, T)]$.

Combining Theorem 2 with the a priori estimate of the discrete velocities obtained in Theorem 1, we immediately obtain a local convergence result for the numerical scheme, and thus a local existence result for problem (P).

## 4. Energy estimates and global results

As in the inelastic case, we observe now that we have some energy estimates for the solutions of problem (P).

**Proposition 7.** Let $C > \|v_0\|_{M(u_0)}$. Then there exists $\tau(C) > 0$ such that, for any solution $u$ of problem (P) defined on $[0, \tau]$ (with $\tau \in (0, T]$), we have

$$ \begin{gathered} |u(t) - u_0| \le C \quad \forall t \in [0, \min(\tau(C), \tau)], \\ \|\dot{u}(t)\|_{M(u(t))} \le C \quad \text{dt almost everywhere on } [0, \min(\tau(C), \tau)]. \end{gathered} $$

**Proof.** Since property (P3) implies that

$$ \|\dot{u}(t+0)\|_{M(u(t))} \le \|\dot{u}(t-0)\|_{M(u(t))} \quad \forall t \in (0, \tau) $$

the proof remains the same as in [11] Proposition 6. $\square$
As a consequence, we can prove a “global” convergence result. More precisely, for any given energy level $C > ||v_0||_{M(u_0)}$, we will prove the convergence of the time-stepping scheme on a time interval $[0, \tau(C)]$ depending only on C and the data.

**Theorem 3.** Let $C > ||v_0||_{M(u_0)}$ and $\tau(C) > 0$ such that, for any solution $u$ of problem (P) defined on $[0, \tau]$ (with $\tau \in (0, T]$), we have

$$|u(t) - u_0| \le C \quad \forall t \in [0, \min(\tau(C), \tau)], \\ ||\dot{u}(t)||_{M(u(t))} \le C \quad \text{dt almost everywhere on } [0, \min(\tau(C), \tau)].$$

Let $u_h$ and $v_h$ be the approximate positions and velocities defined by (32) and (33). Then, there exists a subsequence $(u_{h_i}, v_{h_i})_{i \in \mathbb{N}}$, $\tau \in [\min(\tau(C), T), T]$ and $(u, v) \in C^0([0, \tau]; \mathbb{R}^d) \times BV(0, \tau; \mathbb{R}^d)$ such that

$$u_{h_i} \to u \quad \text{strongly in } C^0([0, \tau]; \mathbb{R}^d), \\ v_{h_i} \to v \quad \text{pointwise in } [0, \tau],$$

with

$$u(t) = u_0 + \int_0^t v(s) \, ds \quad \forall t \in [0, \tau]$$

and $u$ is a solution of problem (P) on $[0, \tau]$.

**Proof.** The proof is analogous to the Proof of Theorem 3 in [11]. More precisely, let $C > ||v_0||_{M(u_0)}$. We define $\mathcal{B} = \bar{B}(u_0, C+1)$ and $B_0, \lambda_{\min}, \lambda_{\max}$ by (11) and (12), respectively. Let $C_F$ be defined by (13), $C_0^*$ and $C'$ be given by

$$C_0^* = \max \left( 2 \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} (|v_0| + 1), C' \right) \quad (70)$$

$$C' = (C + 1) \sup \left\{ \|M^{-1/2}(q)\| ; q \in \bar{B}(u_0, C+1) \right\} \quad (71)$$

and $C_0$ be chosen as in Theorem 1, that is,

$$C_0 > \left( 4 \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} \right)^{v+1} C_0^* \quad (72)$$

Then, with Theorem 1, we know that there exist $h_0^* \in (0, h^*]$ and $\tau_0 > 0$, depending only on $\mathcal{B}, C_0, C_0^*$ and the data, such that

$$|V^n| = \left| \frac{U^{n+1} - U^n}{h} \right| \le C_0 \quad \forall nh \in [0, \min(\tau_0, T)], \quad \forall h \in (0, h_0^*]. \quad (73)$$

Moreover, with Proposition 2, we also know that for all $t_{0h} \in [0, T)$ and for all $(\hat{U}^0, \hat{U}^1) \in (\mathcal{B} \cap K) \times K$ such that

$$|\hat{U}^1 - \hat{U}^0| \le hC_0^* \quad \forall h \in (0, h_0^*],$$
the approximate positions defined by

$$ \hat{U}^{n+1} = -e \hat{U}^{n-1} + (1+e) \hat{Z}^n $$

with

$$ \hat{Z}^n \in \operatorname{Argmin}_{Z \in K} \|\hat{W}^n - Z\|_{M(\hat{U}^n)} $$

and

$$ \hat{W}^n = \frac{2\hat{U}^n - (1-e)\hat{U}^{n-1} + h^2 \hat{F}^n}{1+e}, \quad \hat{F}^n = F\left(t_{0h} + nh, \hat{U}^n, \frac{\hat{U}^n - \hat{U}^{n-1}}{h}, h\right) $$

for all $n \in \{1, \dots, \lfloor \frac{T-t_{0h}}{h} \rfloor\}$ and for all $h \in (0, h_0^*]$, satisfy

$$ \left| \frac{\hat{U}^{n+1} - \hat{U}^n}{h} \right| \le C_0 \quad \forall nh \in [0, \min(\tau_0, T - t_{0h})), \forall h \in (0, h_0^*]. $$

Let $\tau(h) = m(h)h$ be the maximal discrete time step such that estimate (73) holds, that is, for all $h \in (0, h_0^*]$

$$ m(h) = \max \left\{ n \in \{0, \dots, \lfloor T/h \rfloor\}; |V^k| \le C_0 \quad \forall k \in \{0, \dots, n\} \right\}. $$

We define $\tau_1 = \liminf_{h \to 0} \tau(h) = \liminf_{h \to 0} m(h)h$. Theorem 1 implies that $\tau_1 \ge \tau' = \min(\tau_0, T)$.

Let $\varepsilon \in (0, \tau'/8)$. Then, there exists a subsequence $(h_i)_{i \in \mathbb{N}}$, decreasing to zero, such that $(\tau(h_i))_{i \in \mathbb{N}}$ converges to $\tau_1$ and there exists $h_0^* \in (0, \min(h_0^*, \tau'/8)]$ such that $m(h_i)h_i \ge \tau_1 - \varepsilon$ for all $h_i \in (0, h_0^*]$. We may apply Theorem 2 with $h_0^* := h_0^*$ and $\tau_0 := \tau_1 - \varepsilon$: we infer that there exists a subsequence, still denoted $(h_i)_{i \in \mathbb{N}}$, such that $(u_{h_i}, v_{h_i})_{i \in \mathbb{N}}$ converges to a solution of problem (P) on $[0, \tau_1 - \varepsilon]$. Thus, with Proposition 7 we get

$$ \begin{gathered} |u(t) - u_0| \le C \quad \forall t \in [0, \min(\tau(C), \tau_1 - \varepsilon)], \\ \|\dot{u}(t)\|_{M(u(t))} \le C \quad \text{dt almost everywhere on } [0, \min(\tau(C), \tau_1 - \varepsilon)]. \end{gathered} $$

Now we prove that:

**Lemma 14.** We have

$$ \begin{align*} & \limsup_{h_i \to 0^+} \sup_{t \to 0^+} \left\{ \|V^n\|_{M(Z^n)}^2, \quad 0 \le nh_i \le \tau_1 - \varepsilon \right\} \\ & \le \ess \sup_{t \to 0^+} \left\{ \|\dot{u}(t)\|_{M(u(t))}^2, \quad 0 \le t \le \tau_1 - \varepsilon \right\}. \end{align*} $$
**Proof.** The proof is quite similar to the proof of Lemma 6 in [11]. More precisely, we prove this result by contradiction. Assume that

$$ \limsup_{h_i \to 0^+} \sup \left\{ \|V^n\|_{M(Z^n)}^2, 0 \le nh_i \le \tau_1 - \varepsilon \right\} > S $$

with $S = \ess \sup \left\{ \|\dot{u}(t)\|_{M(u(t))}^2, 0 \le t \le \tau_1 - \varepsilon \right\}$. Then, there exist $\gamma > 0$, $\tilde{h}_\varepsilon^* \in (0, \tilde{h}_\varepsilon^*]$ and a subsequence $(h_{\varphi(i)})_{i \in \mathbb{N}}$ decreasing to zero such that

$$ \sup \left\{ \|V^n\|_{M(Z^n)}^2, 0 \le nh_{\varphi(i)} \le \tau_1 - \varepsilon \right\} \ge S + \gamma \quad \forall h_{\varphi(i)} \in (0, \tilde{h}_\varepsilon^*]. $$

It follows that there exists $n_{\varphi(i)} \in \{0, \dots, \lfloor(\tau_1 - \varepsilon)/h_{\varphi(i)}\rfloor\}$ such that

$$ \|V^{n_{\varphi(i)}}\|_{M(Z^{n_{\varphi(i)}})}^2 \ge S + \gamma \quad \forall h_{\varphi(i)} \in (0, \tilde{h}_\varepsilon^*]. $$

Possibly extracting another subsequence, still denoted $(h_{\varphi(i)})_{i \in \mathbb{N}}$, we may assume without loss of generality that the sequence $(n_{\varphi(i)}h_{\varphi(i)})_{i \in \mathbb{N}}$ converges to a limit $\tilde{\tau} \in [0, \tau_1 - \varepsilon]$.

First, we observe that $\tilde{\tau} > 0$. Indeed, with the same computations as in Proposition 3, we can prove that there exists a constant $C_{B_1, C_0} > 0$, depending only on $B_1$ and $C_0$, such that, for all $h_i \in (0, \tilde{h}_\varepsilon^*]$

$$ \|V^n\|_{M(Z^n)}^2 \le \max \left( \|V^{n-1}\|_{M(Z^{n-1})}^2, \dots, \|V^{n-\nu-1}\|_{M(Z^{n-\nu})}^2 \right) + C_{B_1, C_0} h_i \quad (74) $$

for all $n \in \{\nu + 1, \dots, \lfloor(\tau_1 - \varepsilon)/h_i\rfloor\}$, thus

$$ \|V^n\|_{M(Z^n)}^2 \le \max \left( \|V^0\|_{M(Z^0)}^2, \dots, \|V^\nu\|_{M(Z^\nu)}^2 \right) + C_{B_1, C_0} nh_i $$

for all $nh_i \in [0, \tau_1 - \varepsilon]$.

With the Lipschitz property of $M$ on $B_1$ and (69), we infer that, for all $h_{\varphi(i)} \in (0, \tilde{h}_\varepsilon^*]$

$$ \begin{aligned} S + \gamma &\le \|V^{n_{\varphi(i)}}\|_{M(Z^{n_{\varphi(i)}})}^2 \\ &\le \max \left( \|V^0\|_{M(u_0)}^2, \dots, \|V^\nu\|_{M(u_0)}^2 \right) \\ &\quad + C_{B_1, C_0} n_{\varphi(i)} h_{\varphi(i)} + (\nu + 1) C_0^3 L_M h_{\varphi(i)}. \end{aligned} $$

At the limit when $i$ tends to $+\infty$, we get with Lemma 12

$$ S + \gamma \le \|v_0\|_{M(u_0)}^2 + C_{B_1, C_0} \tilde{\tau}. $$

But the right continuity of $\|\dot{u}(\cdot + 0)\|_{M(u)}$ implies that $\|v_0\|_{M(u_0)}^2 = \|\dot{u}(0 + 0)\|_{M(u_0)}^2 \le S$ (see Lemma 13), and finally $\tilde{\tau} > 0$.

We infer that, possibly reducing $\tilde{h}_\varepsilon^*$, $n_{\varphi(i)}h_{\varphi(i)} \ge \tilde{\tau}/2 \ge 2\nu h_{\varphi(i)}$ for all $h_{\varphi(i)} \in (0, \tilde{h}_\varepsilon^*]$. Then, once again using estimate (74), we obtain

$$ S + \gamma \le \max \left( \|V^{n_{\varphi(i)}-p}\|_{M(Z^{n_{\varphi(i)}-p})}^2, \dots, \|V^{n_{\varphi(i)}-p-\nu}\|_{M(Z^{n_{\varphi(i)}-p-\nu})}^2 \right) \\ + C_{B_1, C_0} p h_{\varphi(i)} $$
for all $p \in \{0, \dots, n_{\varphi(i)} - \nu\}$, for all $h_{\varphi(i)} \in (0, \tilde{h}_{\varepsilon}^*]$. It follows that

$$S + \frac{\gamma}{2} \le \max \left( \|V^k\|_{M(Z^k)}^2, \dots, \|V^{k-\nu}\|_{M(Z^{k-\nu})}^2 \right)$$

for all $kh_{\varphi(i)} \in [\max(n_{\varphi(i)}h_{\varphi(i)} - \frac{\gamma}{2C_{B_1}C_0}, \nu h_{\varphi(i)}), n_{\varphi(i)}h_{\varphi(i)}]$. Moreover, using (20), we infer that for all $t \in [kh_{\varphi(i)}, (k+1)h_{\varphi(i)})$, for all $l \in \{0, \dots, \nu\}$

$$\begin{aligned} \left| \|v_{h_{\varphi(i)}}(t - lh_{\varphi(i)})\|_{M(Z^{k-l})}^2 - \|v_{h_{\varphi(i)}}(t -lh_{\varphi(i)})\|_{M(u_{h_{\varphi(i)}}(t))}^2 \right| & \le \|M(Z^{k-l}) - M(u_{h_{\varphi(i)}}(t))\|_{V^{k-l}}^2 \le L_M |Z^{k-l} - u_{h_{\varphi(i)}}(t)| C_0^2 \\ & \le L_M C_0^2 \left( |Z^{k-l} - U^{k-l}| + |u_{h_{\varphi(i)}}((k-l)h_{\varphi(i)}) - u_{h_{\varphi(i)}}(t)| \right) \\ & \le L_M C_0^3 (\nu + 2)h_{\varphi(i)}. \end{aligned}$$

Since $(n_{\varphi(i)}h_{\varphi(i)})_{i \in \mathbb{N}}$ converges to $\tilde{\tau} > 0$, there exists an interval $I \subset [0, \tau_1 - \varepsilon]$, with a non-empty interior, such that

$$I \subset \left[ \max \left( (n_{\varphi(i)} + 1)h_{\varphi(i)} - \frac{\gamma}{2C_{B_1}C_0}, \nu h_{\varphi(i)} \right), n_{\varphi(i)}h_{\varphi(i)} \right]$$

and

$$S + \frac{\gamma}{4} \le \max \left( \|v_{h_{\varphi(i)}}(t)\|_{M(u_{h_{\varphi(i)}}(t))}^2, \dots, \|v_{h_{\varphi(i)}}(t - \nu h_{\varphi(i)})\|_{M(u_{h_{\varphi(i)}}(t))}^2 \right) \quad (75)$$

for all $t \in I$. But the sequence $(v_{h_i})_{i \in \mathbb{N}}$ converges pointwise to $v$ on $[0, \tau_1 - \varepsilon]$ and $v = \dot{u}$ almost everywhere on $[0, \tau_1 - \varepsilon]$. Thus, Lebesgue's theorem implies that $(v_{h_i})_{i \in \mathbb{N}}$ converges strongly to $\dot{u}$ in $L^1(0, \tau_1 - \varepsilon; \mathbb{R}^d)$. We may extend $v_{h_i}$ ($i \in \mathbb{N}$) and $\dot{u}$ by zero outside $[0, \tau_1 - \varepsilon]$ and we denote $\tilde{v}_{h_i}$ and $\tilde{\dot{u}}$ the extended functions. Of course $\|\tilde{v}_{h_i} - \tilde{\dot{u}}\|_{L^1(\mathbb{R}; \mathbb{R}^d)} = \|v_{h_i} - \dot{u}\|_{L^1(0, \tau_1 - \varepsilon; \mathbb{R}^d)}$ and $(\tilde{v}_{h_i})_{i \in \mathbb{N}}$ converges strongly to $\tilde{\dot{u}}$ in $L^1(\mathbb{R}; \mathbb{R}^d)$. The compactness criteria in $L^1(\mathbb{R}; \mathbb{R}^d)$ (see [3]) yields that

$$\limsup_{\theta \to 0} \int_{\mathbb{R}} |\tilde{v}_{h_i}(t - \theta) - \tilde{v}_{h_i}(t)| dt = 0.$$

Thus, $(\tilde{v}_{h_i}(\cdot - lh_i))_{i \in \mathbb{N}}$ converges to $\tilde{\dot{u}}$ strongly in $L^1(\mathbb{R}; \mathbb{R}^d)$ for all $l \in \{0, \dots, \nu\}$. It follows that $(\tilde{v}_{h_{\varphi(i)}}(\cdot - lh_{\varphi(i)}))_{i \in \mathbb{N}}$ also converges to $\tilde{\dot{u}}$ strongly in $L^1(\mathbb{R}; \mathbb{R}^d)$ for all $l \in \{0, \dots, \nu\}$ and, possibly extracting another subsequence, still denoted $(h_{\varphi(i)})_{i \in \mathbb{N}}$, we have for all $l \in \{0, \dots, \nu\}$

$$\tilde{v}_{h_{\varphi(i)}}(t - lh_{\varphi(i)}) \to \tilde{\dot{u}}(t) \quad \text{almost everywhere in } \mathbb{R}.$$

By passing to the limit in (75), we get

$$S + \frac{\gamma}{4} \le \|\dot{u}(t)\|_{M(u(t))}^2 \quad \text{almost everywhere in } I.$$
Recalling that $I$ is an interval included in $[0, \tau_1 - \varepsilon]$ with a non-empty interior, we get

$$S \geq \ess \sup \left\{ \|\dot{u}(t)\|_{M(u(t))}^2 ; t \in I \right\}$$

which yields a contradiction. □

With the previous lemma, possibly decreasing $h_{\varepsilon}^*$ we get

$$U^n \in \bar{B}(u_0, C+1), \quad \|V^n\|_{M(Z^n)} \leq C+1 \quad \forall nh_i \in [0, \min(\tau_1 - \varepsilon, \tau(C))], \\ \forall h_i \in (0, h_{\varepsilon}^*].$$

Let us now distinguish two subcases.

Case 1: $\tau(C) < T$.

Let us prove that $\tau_1 > \tau(C) = \min(\tau(C), T)$ by contradiction. Hence, assume that $\tau_1 \leq \tau(C)$. It follows that $\tau' = \min(\tau_0, T) \leq \tau_1 \leq \tau(C) < T$ and $\tau' = \tau_0$. Furthermore, recalling that $Z^n = \frac{U^{n+1}+eU^{n-1}}{1+e}$ if $n \geq 1$ and $Z^0 = U^0$, we infer that $Z^n \in \bar{B}(u_0, C+1)$ for all $nh_i \in [0, \tau_1 - \varepsilon - h_1]$ and

$$|V^n| \leq (C+1) \sup \left\{ \|M^{-1/2}(q)\| ; q \in \bar{B}(u_0, C+1) \right\} = C' \leq C_0^* < C_0$$

for all $nh_i \in [0, \tau_1 - \varepsilon - h_i]$, for all $h_i \in (0, h_{\varepsilon}^*]$, where $C', C_0^*$ and $C_0$ are given by (70), (71) and (72). We now choose $l(h_i) \in \{0, \dots, \lfloor T/h_i \rfloor\}$ such that

$$l(h_i)h_i \in \left[ \tau_1 - \frac{\tau'}{2}, \tau_1 - \frac{\tau'}{4} \right] \quad \forall h_i \in (0, h_{\varepsilon}^*]$$

and let

$$\hat{U}^0 = U^{l(h_i)}, \quad \hat{U}^1 = U^{l(h_i)+1}, \quad t_{0h_i} = l(h_i)h_i.$$

We have

$$|\hat{U}^1 - \hat{U}^0| = h_i |V^{l(h_i)}| \leq C_0^* h_i$$

and $\hat{U}^0, \hat{U}^1$ belong to $B \cap K$, for all $h_i \in (0, h_{\varepsilon}^*]$.

Then, for all $n \in \{l(h_i), \dots, \lfloor \frac{T}{h_i} \rfloor\}$, $U^n = \hat{U}^{n-l(h_i)}$ and, with Proposition 2, we obtain

$$|V^n| = \left| \frac{\hat{U}^{n-l(h_i)+1} - \hat{U}^{n-l(h_i)}}{h_i} \right| \leq C_0$$

for all $(n-l(h_i)) h_i \in [0, \min(\tau_0, T-l(h_i)h_i)]$, for all $h_i \in (0, h_{\varepsilon}^*]$. Thus

$$m(h_i)h_i > l(h_i)h_i + \min(\tau_0, T-l(h_i)h_i) - h_i \geq \min\left(\tau_1 + \frac{\tau'}{2}, T\right) - h_i.$$

It follows that $\lim_{h_i \to 0} m(h_i)h_i = \tau_1 \geq \min\left(\tau_1 + \frac{\tau'}{2}, T\right)$ which gives a contradiction.
Thus $\tau_1 > \tau(C)$, and there exists $\tilde{h}_0^* \in (0, h_0^*]$ such that

$$m(h_i)h_i = \tau(h_i) \ge \tau_1 - \frac{\tau_1 - \tau(C)}{2} = \frac{\tau_1 + \tau(C)}{2} > \tau(C) \quad \forall h_i \in (0, \tilde{h}_0^*].$$

Then, we apply Theorem 2 with $\tau_0$ replaced by $\tau = \frac{\tau_1+\tau(C)}{2}$ and $h_0^*$ by $\tilde{h}_0^*$, which yields the announced result.

Case 2: $\tau(C) \ge T$.

Since $m(h)h = \tau(h) \le T$, we have $\tau_1 \le T$. We choose as previously $l(h_i) \in \{0, \dots, \lfloor T/h_i \rfloor\}$ such that

$$l(h_i)h_i \in \left[ \tau_1 - \frac{\tau'}{2}, \tau_1 - \frac{\tau'}{4} \right] \quad \forall h_i \in (0, h_e^*].$$

Then, we again have

$$m(h_i)h_i > l(h_i)h_i + \min(\tau_0, T - l(h_i)h_i) - h_i \ge \min\left(\tau_1 + \frac{\tau'}{2}, T\right) - h_i \quad (76)$$

for all $h_i \in (0, h_e^*]$. Thus, if $\tau_1 < T$, $\min(\tau_1 + \frac{\tau'}{2}, T) \in (\tau_1, T]$ and (76) yields a contradiction with the definition of $\tau_1 = \lim_{h_i \to 0} m(h_i)h_i$. We infer that $\tau_1 = T$ and (76) implies that $m(h_i)h_i > T - h_i$, that is, $m(h_i) \ge \lfloor T/h_i \rfloor$ for all $h_i \in (0, h_e^*]$. Hence we may apply Theorem 2 to obtain the convergence of a subsequence of $(u_{h_i}, v_{h_i})_{i \in \mathbb{N}}$, still denoted $(u_{h_i}, v_{h_i})_{i \in \mathbb{N}}$, to a solution of problem (P) on $[0, T]$. $\square$

## Appendix

**Lemma 15.** For all compact subsets $\mathcal{B}$ of $\mathbb{R}^d$, there exist $m_B > 0$, $C_B > 0$ and $r_B' > 0$ such that for all $q \in K \cap \mathcal{B}$ and for all $\alpha \in J(q)$ we have

$$|M^{-1/2}(q')\nabla f_{\alpha}(q')| \ge m_B \quad \forall q' \in \bar{B}(q, r_B')$$

and the mappings

$$q' \mapsto \frac{M^{-1/2}(q')\nabla f_{\alpha}(q')}{|M^{-1/2}(q')\nabla f_{\alpha}(q')|}$$

are defined and $C_B$-Lipschitz continuous on $\bar{B}(q, r_B')$.

**Proof.** Let $\mathcal{B}$ be a given compact subset of $\mathbb{R}^d$. With the same arguments as in [11] Lemma 7, we first obtain that there exist $m_B > 0$ and $r_B' > 0$ such that for all $(q, q') \in (K \cap \mathcal{B}) \times \mathbb{R}^d$ such that $|q - q'| \le r_B'$, and for all $\alpha \in J(q)$ we have

$$|M^{-1/2}(q')\nabla f_{\alpha}(q')| \ge m_B.$$

Hence, for all $q \in K \cap \mathcal{B}$, the mappings

$$q' \mapsto \frac{M^{-1/2}(q')\nabla f_{\alpha}(q')}{|M^{-1/2}(q')\nabla f_{\alpha}(q')|}, \quad \alpha \in J(q)$$

are defined on $\bar{B}(q, r_B')$.
Now let us prove that there exists $C'_B > 0$ such that, for all $(q', q'') \in \bar{B}(q, r'_B)^2$, and for all $\alpha \in J(q)$, we have

$$ \left| \frac{M^{-1/2}(q') \nabla f_{\alpha}(q')}{|M^{-1/2}(q') \nabla f_{\alpha}(q')|} - \frac{M^{-1/2}(q'') \nabla f_{\alpha}(q'')}{|M^{-1/2}(q'') \nabla f_{\alpha}(q'')|} \right| \le C'_B |q' - q''|. $$

We define $\tilde{B} = \{q \in \mathbb{R}^d; \text{dist}(q, B) \le r'_B\}$. The set $\tilde{B}$ is also compact. Moreover, from assumptions (H2) and (H4) we know that the mappings $\nabla f_\alpha$ ($1 \le \alpha \le \nu$) and $M^{-1/2}$ are locally Lipschitz continuous on $\mathbb{R}^d$. Thus, for all $\alpha \in \{1, \dots, \nu\}$ there exists $C_\alpha > 0$ such that

$$ |M^{-1/2}(q')\nabla f_{\alpha}(q') - M^{-1/2}(q'')\nabla f_{\alpha}(q'')| \le C_{\alpha}|q' - q''| \quad \forall (q', q'') \in \tilde{B}^2. $$

Let $q \in B \cap K$ and $(q', q'') \in \bar{B}(q, r'_B)^2$. Then

$$ \begin{aligned} & \left| \frac{M^{-1/2}(q') \nabla f_{\alpha}(q')}{|M^{-1/2}(q') \nabla f_{\alpha}(q')|} - \frac{M^{-1/2}(q'') \nabla f_{\alpha}(q'')}{|M^{-1/2}(q'') \nabla f_{\alpha}(q'')|} \right| \\ & \le \left| \frac{M^{-1/2}(q') \nabla f_{\alpha}(q') - M^{-1/2}(q'') \nabla f_{\alpha}(q'')}{|M^{-1/2}(q') \nabla f_{\alpha}(q')|} \right. \\ & \qquad \left. + \left( \frac{M^{-1/2}(q'') \nabla f_{\alpha}(q'')}{|M^{-1/2}(q') \nabla f_{\alpha}(q')|} - \frac{M^{-1/2}(q'') \nabla f_{\alpha}(q'')}{|M^{-1/2}(q'') \nabla f_{\alpha}(q'')|} \right) \right| \\ & \le \frac{|M^{-1/2}(q') \nabla f_{\alpha}(q') - M^{-1/2}(q'') \nabla f_{\alpha}(q'')|}{|M^{-1/2}(q') \nabla f_{\alpha}(q')|} \\ & \qquad + \left| M^{-1/2}(q'') \nabla f_{\alpha}(q'') \right| \frac{|M^{-1/2}(q'') \nabla f_{\alpha}(q'')| - |M^{-1/2}(q') \nabla f_{\alpha}(q')|}{|M^{-1/2}(q') \nabla f_{\alpha}(q')| |M^{-1/2}(q'') \nabla f_{\alpha}(q'')|} \\ & \le \frac{C_{\alpha}}{m_B} |q' - q''| + \frac{C_{\alpha}}{m_B} |q' - q''|. \end{aligned} $$

By defining

$$ C'_{B} = \max_{1 \le \alpha \le \nu} \frac{2C_{\alpha}}{m_{B}} $$

we may conclude. $\square$

**Lemma 16.** Let us recall the definition of $T_K(q)$:

$$ T_K(q) = \{v \in \mathbb{R}^d; (\nabla f_\alpha(q), v) \ge 0 \quad \forall \alpha \in J(q)\} \quad \forall q \in \mathbb{R}^d $$

with

$$ J(q) = \{\alpha \in \{1, \dots, \nu\}; f_{\alpha}(q) \le 0\}. $$

Then, for all $q_0 \in K$, there exist $\delta > 0$, $r > 0$ and $a \in \mathbb{R}^d$ such that, for all $q \in \bar{B}(q_0, 2\delta)$:

$$ \bar{B}(a, r) \subset T_K(q). $$

**Proof.** See Lemma 8 in [11]. $\square$
*Acknowledgments*. The author would like to thank the referee for the numerous relevant remarks which have contributed to improving the final form of the manuscript.

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