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# Regularization of optimal control problems on stratified domains using additional controls

Anas Bouali, Alain Rapaport, Térence Bayen

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Anas Bouali, Alain Rapaport, Térence Bayen. Regularization of optimal control problems on stratified domains using additional controls. 2025. hal-04928858v2

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Regularization of optimal control problems on stratified domains
using additional controls

Anas Bouali*

Alain Rapaport†

Térence Bayen‡

April 15, 2025

Abstract

In this paper, we investigate a Mayer optimal control problem governed by a dynamics defined regionally. We consider that the state space is stratified into a family of disjoint regions with nonsmooth interfaces, and that in each region, the dynamics is given by a smooth expression. First, it is shown that this problem is equivalent to a new optimal control problem, with additional controls and a (smooth) dynamics as a convex combination of the smooth dynamics, along with a mixed control-state constraint. Next, we introduce a family of auxiliary optimal control problems. In these problems, we first regularize the nonsmooth interfaces. In addition, we consider the convex combination of smooth dynamics (only) within a boundary layer. Furthermore, we add a penalization term to the cost function to account for the mixed control-state constraint. Our main result is that solutions to these (smooth) problems converge (up to a subsequence) to a solution of the original one. It is obtained thanks to a new hypothesis related to solutions to the auxiliary problems, which is weaker than the transverse crossing condition of the literature. This technique is implemented numerically on two examples involving non-transverse crossings of interfaces showing its efficiency.

Keywords: Optimal control, Filippov solutions, discontinuous dynamics, regularization, penalty method.

2020 Mathematics Subject Classification: 34A38, 49A15, 49J53.

# 1 Introduction

**General setting.** This paper is concerned with optimal control problems governed by control systems with a discontinuous right-hand side. More precisely, we consider a stratification of the state space with open regions, and to each of them, we associate a smooth control system. The resulting dynamics can be discontinuous at each interface between two (or more) regions. The objective of the paper is to study a general Mayer optimal control problem in this setting. Problems involving discontinuous dynamics with respect to the state arise in many domains such as in mathematical biology (see, e.g., [1, 8]). The discontinuity of the dynamics makes studying these problems in general more difficult than classical<fnref n="1" /> optimal control problems. To the best of our knowledge, the approach that has been most extensively explored focuses on the necessary optimality conditions (see, e.g., [2, 3, 5, 14, 16, 17, 25], among others), while sufficient optimality conditions have also been considered in this context (see, e.g., [19]). Regarding necessary optimality conditions, the so-called *hybrid maximum principle* (HMP, in short) extends the *Pontryagin maximum principle* (PMP, in short), the main difference being the presence of jumps for the adjoint vector each time a solution crosses an interface. A general version of the HMP in this context can be found in [2]. To the best of our knowledge, a transverse hypothesis is always required when an optimal trajectory crosses an interface between two regions (see, e.g., [2, 6, 9, 17, 20, 23]). These conditions are essential to derive the HMP, whether relying on sensitivity analysis, as seen in [3, 17], or using an augmentation technique to ensure correspondence with a classical optimal control problem, as demonstrated in [4, 2].

<footnote>*MISTEA, Univ. Montpellier, INRAE, Institut Agro, 34060 Montpellier, France. anas.bouali@inrae.fr</footnote>

<footnote>†MISTEA, Univ. Montpellier, INRAE, Institut Agro, 34060 Montpellier, France. alain.rapaport@inrae.fr</footnote>

<footnote>‡Avignon Université, Laboratoire de Mathématiques d'Avignon (EA 2151) F-84018. terence.bayen@univ-avignon.fr</footnote>

<footnote n="1">¹By “classical”, we mean an optimal control problem with smooth data.</footnote>
**Regularization technique.** In this paper, we set aside the search of necessary optimality conditions in favor of a regularization technique. One motivation is to relax the transverse hypothesis made on optimal trajectories (that is unknown in advance). Of course, the price to pay is that we solve, instead of the original problem, an approximated optimal control problem, but which has the advantage to be smooth and thus eligible to classical numerical methods. In a sense, our approach is in the same spirit as the regularization technique that has been extensively studied in this context in [17]. The authors consider in [17] a general regularization of both the cost function and the discontinuous dynamics, allowing them to define smooth auxiliary optimal control problems. Their main result is that solutions to the regularized optimal control problems converge (up to a subsequence) to a solution to the original one. Interestingly, convergence of the adjoint vectors (as well as weak convergence of optimal controls in the case of control-affine systems) is also provided, in addition to convergence of trajectories. These results rely on a transverse hypothesis on an optimal solution, as mentioned above.

**Novelty of our approach and methodology.** We describe now our approach and the main differences with the one presented in [17]. Before considering a regularization, we modify the original dynamics (defined piecewise as in [2, 17]) using the Filippov's solution concept [15]. Thereby, we introduce additional controls in the dynamics (with values in $[0, 1]$), whose number is equal to the number of regions and whose role is to select the dynamics corresponding to the appropriate region. This allows us to define a new dynamics as a convex combination of all smooth dynamics, with the additional controls viewed as weights. To achieve this, we use an explicit description of each region via the intersection of sub-level sets of smooth functions. Note that the dynamics is naturally coupled with a mixed control-state constraint resulting from the introduction of the convex combination and the additional controls. In this sense, we face the additional difficulty to deal with a mixed control-state constraint, but instead we penalize this constraint with a Bolza cost. Another ingredient in our approach is to regularize the nonsmooth interfaces, which generates a boundary layer around each interface. The last step consists in the introduction of a regularized control system in such a way that it coincides with the new dynamics (arising from the convex combination) only within the boundary layer, while it coincides with the (original) dynamics outside the boundary layer. This technical modification is essential in our approach.

**Main contribution.** Our main result is to show that optimal solutions of these approximated optimal control problems converge (up to a subsequence) to an optimal solution to the original problem. Our analysis is based on a new hypothesis related to the behaviour of the optimal trajectories of the approximated problems over the boundary layer. We assume that the time spent in the boundary layer by optimal solutions to the approximated optimal control problem goes to zero as the regularization parameter goes to infinity. From a numerical viewpoint, a significant advantage is that this hypothesis can be tested numerically, whereas we cannot in general check in advance a transverse hypothesis on a solution to the original optimal control problem we are looking for. Moreover, this hypothesis does not discard solutions crossing an interface tangentially as we point out in two examples (on this question, we can also mention, e.g., [7]).

**Structure of the paper.** Section 2 introduces the optimal control problem and the corresponding state space stratification. In particular, we define a new dynamics as a convex combination of the smooth dynamics by means of the Filippov differential inclusion concept. Note that this is made possible by introducing additional controls. Furthermore, we provide an equivalence result of the original problem and an optimal control problem involving a mixed control-state constraint. In Section 3, we define the approximated optimal control problems using a penalty method. We also introduce a regularization of each interface allowing to set up a boundary layer which is a fundamental component of our approach. Finally, we prove Theorem 3.5 about convergence of solutions to the regularized problems, which is our main result. We illustrate our result in Section 4 on two examples with a discontinuous dynamics (defined on a stratification of the state space) and whose optimal solutions cross tangentially the interfaces. These examples enables the proposed method to be tested and the new hypothesis to be verified numerically.
# 2 A stratified optimal control problem and its convexification

## 2.1 State space stratification and corresponding dynamics

In this paper, we will consider a stratification<fnref n="2" /> of the state space $\mathbb{R}^n$, $n \ge 1$, into a family of disjoint open sets, referred to as regions. On this stratified state space, we will define a control system with a discontinuous right-hand side, where the dynamics is smooth within each region but has discontinuities along the boundaries. Throughout this paper, each region will be defined as an intersection of sub-level sets of smooth functions. The construction is as follows. We consider a family of functions $\phi_k: \mathbb{R}^n \to \mathbb{R}$, $1 \le k \le \ell$, of class $C^1$ and we suppose that the following hypothesis is fulfilled.

**Assumption 2.1.** For any $k \in \{1, \dots, \ell\}$, the level set $L_k := \{x \in \mathbb{R}^n; \phi_k(x) = 0\}$ is a nonempty hypersurface of $\mathbb{R}^n$, which divides $\mathbb{R}^n$ into two regions $L_k^+ := \{x \in \mathbb{R}^n; \phi_k(x) > 0\}$ and $L_k^- := \{x \in \mathbb{R}^n; \phi_k(x) < 0\}$. Furthermore, for every $k, k' \in \{1, \dots, \ell\}$ such that $k \ne k'$, every intersection $L_k^{\pm} \cap L_{k'}^{\pm}$ is empty or has a finite number of connected components.

By considering all possible intersections, we can obtain up to $N := 2^\ell$ regions (some of them possibly empty) which we can enumerate using an index $j \in \{1, \dots, N\}$. One can use for instance the *Stewart's representation*, as in [22], to characterize each region with sign matrices. Instead, to assign each specific region, we found more convenient to consider the *binary decomposition* of integers $j \in \{1, \dots, N\}$:

$$j = 1 + \sum_{k=0}^{\ell-1} a_{j,k} 2^k \quad \text{with} \quad a_{j,k} \equiv \left\lfloor \frac{j-1}{2^k} \right\rfloor \pmod{2}.$$

Note that $a_{j,k} \in \{0, 1\}$ for all $k \in \{0, \dots, \ell-1\}$. For each $j \in \{1, \dots, N\}$, we define the locally Lipschitz-continuous function $\varphi_j: \mathbb{R}^n \to \mathbb{R}$ as follows

$$\varphi_j(x) := \max \{(-1)^{a_{j,\ell-k}} \phi_k(x); 1 \le k \le \ell\} \quad (2.1)$$

together with the set

$$X_j := \{x \in \mathbb{R}^n; \varphi_j(x) < 0\}.$$

By construction, these sets provide the following stratification of $\mathbb{R}^n$:

$$\mathbb{R}^n = \bigcup_{j=1}^{N} \bar{X}_j. \quad (2.2)$$

We refer to Figure 1 for an illustration. For each $j \in \{1, \dots, N\}$, the *interface* of the set $X_j$ is given by the (nonsmooth) level set $\{x \in \mathbb{R}^n; \varphi_j(x) = 0\}$.

Note that $X_j$ is not necessarily connected (with $\ell = 2$, take for instance $L_1, L_2$ such that their intersection has three components). However, for the sake of completeness, we will briefly explain why we can assume that each region $X_j$ is connected, thanks to the Assumption 2.1. Denote by $m_j$ the number of connected components of the sets $X_j$, $j \in \{1, \dots, N\}$ (equal to 0 if $X_j$ is empty). By Assumption 2.1, the numbers $m_j$ are finite. For $j$ such that $m_j > 1$, we can write

$$X_j = \bigsqcup_{l=1}^{m_j} X_{jl},$$

where the family $(X_{jl})_{1 \le l \le m_j}$ represents the connected components of $X_j$. Next, for every $l \in \{1, \dots, m_j\}$ we can define a $C^1$-mapping $\psi_{jl}: \mathbb{R}^n \to [0, 1]$ as follows:

$$\psi_{jl}(x) := \begin{cases} 1 & \text{if } x \in X_{jl}, \\ 0 & \text{if } x \in X_{jl'} \text{ and } l' \ne l. \end{cases}$$

<footnote n="2">²A stratification of a domain $\mathcal{D}$ is a finite collection of disjoint open sets such that the union of their closures is $\mathcal{D}$.</footnote>
<figure><img src="image_2.png" /><figcaption>**Figure 1:** Examples of stratifications of $\mathbb{R}^2$ with $\ell = 2$ under Assumption 2.1. On the right, the set $X_2$ is empty because the two level sets $L_1, L_2$ do not intersect.</figcaption></figure>

Finally, we can replace each function $\varphi_j$ by the functions $\bar{\varphi}_{jl}$ defined over $\mathbb{R}^n$ as

$$ \bar{\varphi}_{jl}(x) := \psi_{jl}(x)\varphi_j(x) + \frac{1}{2}(1 - \psi_{jl}(x)), $$

for all $l \in \{1, \dots, m_j\}$, each of them being negative over the open set $X_{jl}$ only. We obtain that way a stratification of $\mathbb{R}^n$ with $N'$ connected sets where

$$ N' := 2^l + \sum_{j \in J} (m_j - 1), $$

with $J := \{j \in \{1, \dots, N\} ; m_j > 1\}$. Therefore, without any loss of generality (at the price of renumbering the function $\varphi_j$ and the sets $X_j$), we shall assume that the following hypothesis is fulfilled.

**Assumption 2.2.** For any $j \in \{1, \dots, N\}$, the function $\varphi_j$ defined in (2.1) is such that the set $X_j$ is connected.

We now turn to the definition of the controlled dynamics that will be considered throughout this paper. Hereafter, $\| \cdot \|_{\mathbb{R}^n}$ stands for the Euclidean norm in $\mathbb{R}^n$ and $\bar{B}_{\mathbb{R}^n}(x_0, r)$ denotes the closed ball in $\mathbb{R}^n$ of center $x_0$ and of radius $r > 0$. Additionally, the indicator function of a non-empty set $A \subset \mathbb{R}^n$ is defined as $\mathbf{1}_A(x) = 1$ if $x \in A$ and $\mathbf{1}_A(x) = 0$ otherwise. Let $m \in \mathbb{N}^*$, $U$ a nonempty compact subset of $\mathbb{R}^m$, and for every $j \in \{1, \dots, N\}$ consider a function $f_j : \mathbb{R}^n \times U \to \mathbb{R}^n$. Next, we suppose the following assumption to be satisfied.

**Assumption 2.3.** For every $j \in \{1, \dots, N\}$, the map $f_j$ is of class $C^1$ with linear growth with respect to $x$ uniformly in $u$, that is, there is $c_j \ge 0$ such that

$$ \forall (x, u) \in \mathbb{R}^n \times U, \quad \|f_j(x, u)\|_{\mathbb{R}^n} \le c_j (1 + \|x\|_{\mathbb{R}^n}), $$

and, for every $x \in \mathbb{R}^n$, $f_j(x, U)$ is a nonempty compact convex subset of $\mathbb{R}^n$.

We are now in a position to define the controlled dynamics $f : \mathbb{R}^n \times U \to \mathbb{R}^n$ regionally, with respect to the stratification (2.2), as follows:

$$ \forall (x, u) \in \mathbb{R}^n \times U, \quad f(x, u) := f_j(x, u), \quad \text{if } x \in X_j, \quad j \in \{1, \dots, N\}. $$

**Remark 2.1.** Some comments about the Assumptions 2.2 and 2.3 are in order.

(i) For any $j \in \{1, \dots, N\}$, the map $f_j$ is assumed to be with linear growth over $\mathbb{R}^n$ and not only over $X_j$. This property is important and will be needed in Section 3. However, if $f_j$ is with linear growth over $X_j$ only, but not over $\mathbb{R}^n$, $f_j$ can be extended outside the set $\bar{X}_j$ in such a way to fulfill Assumption 2.3.
(ii) The stratification of $\mathbb{R}^n$ may be such that $f_j = f_{j'}$ for some $j \neq j'$ such that $\partial X_j \cap \partial X_{j'} \neq \emptyset$ for some $j, j' \in \{1, \dots, N\}$ (where $\partial X_j$ denotes the boundary of $X_j$). This is precisely the case in our examples (we refer to Section 4 for further details). In such a case, one could consider another stratification with less regions. Moreover, the stratification of $\mathbb{R}^n$ as defined above may consist of less than $N$ nonempty regions. However, for sake of simplicity, we still continue with the dynamics $f$ as defined over the stratification (2.2) keeping in mind that some sets $X_j$ may be empty.

(iii) Note that the dynamics $f$ is not defined for all $x \notin \bigcup_{j=1}^N X_j$. However, we will extend the dynamics over the interfaces using Filippov's theory, which is the focus of the next section.

## 2.2 A stratified optimal control problem and its relaxed variant

Given $T > 0$, we denote by $\mathcal{U}_{[0,T]}$ the set of measurable functions $u : [0, T] \to U$ and by $\text{AC}_{[0,T]}$ the set of absolutely continuous functions from $[0, T]$ to $\mathbb{R}^n$. Given $u(\cdot) \in \mathcal{U}_{[0,T]}$, we shall consider hereafter solutions to the following control system

$$ \dot{x}(t) = f(x(t), u(t)), \quad \text{a.e. } t \in [0, T]. $$

Note that $f$ is not necessarily continuous with respect to $x$, and, in particular, the value of $f$ is not well-defined if $x(t)$ belongs to an interface between two (or more) regions. The first step is thus to specify in which sense, solutions to this system are considered. Given a control function $u(\cdot) \in \mathcal{U}_{[0,T]}$, we consider solutions to this system in the Filippov sense, that is, $x(\cdot) \in \text{AC}_{[0,T]}$ is a solution of the preceding system if and only if $x(\cdot)$ is a solution to the differential inclusion

$$ \dot{x}(t) \in F(x(t), u(t)), \quad \text{a.e. } t \in [0, T], \qquad (2.3) $$

where for each $\omega \in \mathbb{R}^m$, the multi-mapping $F(\cdot, \omega) : \mathbb{R}^n \to \mathbb{R}^n$ is defined as

$$ F(y, \omega) := \bigcap_{\delta>0} \bigcap_{\mu(E)=0} \overline{\text{co}} f(B_{\mathbb{R}^n}(y, \delta) \setminus E, \omega), $$

for all $y \in \mathbb{R}^n$. Our objective in this paper is to study the following optimal control problem of Mayer type

$$
\begin{array}{ll}
\text{minimize} & \Phi(x(T)), \\
\text{subject to} & (x(\cdot), u(\cdot)) \in \text{AC}_{[0,T]} \times \mathcal{U}_{[0,T]}, \\
& \dot{x}(t) \in F(x(t), u(t)), \quad \text{a.e. } t \in [0, T], \\
& x(0) = x_0,
\end{array}
\qquad (\text{HP}_0)
$$

where $\Phi : \mathbb{R}^n \to \mathbb{R}$ is a function of class $C^1$ (the terminal pay-off) and $x_0 \in X_{j_1}$ is a given initial condition, with $j_1 \in \{1, \dots, N\}$. The first point to have in mind is that for $x \in \mathbb{R}^n$, the set $F(x, U)$ is not necessarily convex$^3$ (although, by construction of the Filippov inclusion, the sets $F(x, u)$ are convex for $u \in U$). This can typically happen when $x$ belongs to an interface between two (or more) regions. As a consequence, this prevents the use of results about the compactness of the set of trajectories (see [13, 12]). Therefore, we introduce a “convexification” of $F$ considering the multi-mapping $\bar{F} : \mathbb{R}^n \to \mathbb{R}^n$ defined as follows:

$$ \bar{F}(x) := \overline{\text{co}} F(x, U), $$

for all $x \in \mathbb{R}^n$ where $\overline{\text{co}}$ stands for the closed convex hull operation. As a result, we can now introduce the optimal control problem

$$
\begin{array}{ll}
\text{minimize} & \Phi(x(T)), \\
\text{subject to} & x(\cdot) \in \text{AC}_{[0,T]}, \\
& \dot{x}(t) \in \bar{F}(x(t)), \quad \text{a.e. } t \in [0, T], \\
& x(0) = x_0.
\end{array}
\qquad (\text{HP})
$$

<footnote>$^3$As an example, consider the two maps $f_1, f_2 : \mathbb{R}^2 \times [0, 1] \to \mathbb{R}^2$ defined respectively as $f_1(x, u) = (u, 0)$ and $f_2(x, u) = (0, 1-u)$. It is easy to see that $(0, 1)$ and $(1, 0)$ belong to $F(x, u) = \{(uv, (1-u)(1-v)); u, v \in [0, 1]\}$, but $(\frac{1}{2}, \frac{1}{2})$ does not.</footnote>
The multi-mapping $\bar{F}$ can be represented by considering additional controls (which will be particulary useful from a numerical viewpoint). For this purpose, let us define the following control set

$$V := \{v \in [0, 1]^N : \sum_{k=1}^{N} v_k = 1\},$$

and the (smooth) dynamics $g: \mathbb{R}^n \times U^N \times V \to \mathbb{R}^n$ defined by

$$g(x, u, v) := \sum_{j=1}^{N} v_j f_j(x, u_j), \quad (x, u, v) \in \mathbb{R}^n \times U^N \times V.$$

Note that the differential inclusion in Problem (HP) will be then replaced by the ordinary differential equation

$$\dot{x}(t) = g(x(t), u(t), v(t)), \text{ a.e. } t \in [0, T],$$

with $u = (u_1, \dots, u_N)$ such that $u_j \in \mathcal{U}_{[0,T]}$ for all $j \in \{1, \dots, N\}$ and $v(t) \in V$ for almost every $t \in [0, T]$. Hence, we consider the new set of admissible controls $\mathcal{U}\mathcal{V}_{[0,T]}$ as the set of time measurable functions that take values in $U^N \times V$ almost everywhere. Finally, we consider the optimal control problem coupled with a mixed control-state constraint:

$$
\begin{array}{ll}
\text{minimize} & \Phi(x(T)), \\
\text{subject to} & (x(\cdot), u(\cdot), v(\cdot)) \in \text{AC}_{[0,T]} \times \mathcal{U}\mathcal{V}_{[0,T]}, \\
& \dot{x}(t) = g(x(t), u(t), v(t)), \quad \text{a.e. } t \in [0, T], \qquad (\text{HP}_c) \\
& (1 - 2v_j(t))\varphi_j(x(t)) = |\varphi_j(x(t))|, \quad \text{a.e. } t \in [0, T], \text{ for all } j \in \{1, \dots, N\}, \\
& x(0) = x_0.
\end{array}
$$

Due to its definition involving additional controls (or relaxed controls), we will refer to this problem as the relaxed variant of $(\text{HP}_0)$. Hereafter, we denote by $\text{int}A$ the interior of a set $A \subset \mathbb{R}^n$.

**Proposition 2.4.** Optimal control problems (HP) and $(\text{HP}_c)$ are equivalent.

*Proof.* From Assumption 2.1, one has $X_j \subset \overline{\text{int}X_j}$ so that solutions in the sense of Filippov coincide with solutions in the sense of Krasovsky, (see Lemma 2.8 in [18]). So, an admissible solution $x(\cdot)$ to $(\text{HP}_0)$ satisfies

$$\dot{x}(t) \in \tilde{F}(x(t), u(t)), \quad \text{a.e. } t \in [0, T],$$

where for $\omega \in \mathbb{R}^m$, $\tilde{F}(\cdot, \omega) : \mathbb{R}^n \to \mathbb{R}^n$ is the multi-mapping defined as

$$\tilde{F}(y, \omega) := \bigcap_{\delta>0} \overline{\text{co}} f(B_{\mathbb{R}^n}(y, \delta), \omega),$$

for all $y \in \mathbb{R}^n$ and vice-versa. From the definition of $\tilde{F}$, one also has

$$\tilde{F}(y, \omega) = \overline{\text{co}}\{f_j(y, \omega) : j \in \mathcal{J}(y)\},$$

where for $y \in \mathbb{R}^n$, $\mathcal{J}(y) := \{j : y \in \overline{X}_j\}$. We thus obtain

$$\overline{F}(y) = \overline{\text{co}} \bigcup_{j \in \mathcal{J}(y)} f_j(y, U).$$

Since each set $f_j(y, U)$ is convex, the convex hull $\overline{F}(y)$ can be expressed using the convex combination of vectors in the sets $f_j(y, U)$, that is,

$$\overline{F}(y) = \bigcup_{(\omega, v) \in U^N \times V} \left\{ \sum_{j=1}^{N} v_j f_j(y, \omega_j) ; v_j = 0, j \notin \mathcal{J}(y) \right\}. \qquad (2.4)$$
Suppose now that $x(\cdot)$ is an admissible solution to (HP). Given $t \in [0, T]$, if $1 \le j \le N$ is such that $\varphi_j(t) < 0$, then $x(t) \in X_j$ so that only the index $j$ is active in the sum defining $\bar{F}$ in (2.4). It follows that $v_j(t) = 1$ (and $v_k(t) = 0$ for all $k \ne j$) so that the mixed constraint is satisfied. If $1 \le j \le N$ is such that $\varphi_j(x(t)) = 0$, then, the mixed constraint is obviously satisfied. Finally, if $1 \le j \le N$ is such that $\varphi_j(x(t)) > 0$, then, $j \notin \mathcal{J}(x(t))$ so that $v_j(t) = 0$ by definition of $\bar{F}$ and we also conclude that the mixed constraint is verified.

Conversely, suppose that $x(\cdot)$ is an admissible solution to (HP$_c$). Given $t \in [0, T]$, suppose that $1 \le j \le N$ is such that $x(t) \notin \mathcal{J}(x(t))$. It follows that $\varphi_j(x(t)) > 0$ so that $v_j(t) = 0$ from the mixed constraint. To conclude, both problems are equivalent as their sets of admissible solutions coincide. □

**Remark 2.2.** Note that the functions $\varphi_j$ appearing in the mixed control-state constraints of Problem (HP$_c$) are nonsmooth. This is a reason why we proceed to a regularization of $\varphi_j$ in the following section.

# 3 Approximation via regularization and penalty methods

In this section, we propose a method to regularize Problem (HP$_c$) into a smooth optimal control problem, parameterized by $k \in \mathbb{N}^*$ and denoted by Problem (P$_k$). This is achieved by first applying a regularization procedure to the nonsmooth interfaces, which allows us to define, for every $k \in \mathbb{N}^*$, a boundary layer around all interfaces. Next, with the help of regularized indicator functions, we consider a smooth dynamics which coincides with maps $f_j$ away from the boundary layer and uses additional controls in the boundary layer. Finally, we introduce a penalty function to handle the mixed control-state constraints of Problem (HP$_c$). This avoids having to use the mixed control-state constraint, while making it operational through the new cost functional. We then present our main result, demonstrating that the solutions of Problem (P$_k$) converge (up to a subsequence) to an optimal solution of (HP).

## 3.1 Regularization of interfaces

We start by regularizing the nonsmooth interfaces. Since every mapping $\varphi_j : \mathbb{R}^n \to \mathbb{R}$ is locally Lipschitz, we can define by convolution a sequence of smooth functions approximating uniformly $\varphi_j$ over every compact subset of $\mathbb{R}^n$. The construction is as follows:

* First, we fix a sequence of positive real numbers $(\nu_k)$ such that $\eta_k \downarrow 0$ when $k \to \infty$.

* Second, we consider a sequence of bounded and continuous mollifiers$^4 \rho_k : \mathbb{R}^n \to \mathbb{R}_+$ satisfying $\int_{\mathbb{R}^n} \rho_k(y) dy = 1$ and $\text{supp}(\rho_k) \subset \bar{B}_{\mathbb{R}^n}(0_{\mathbb{R}^n}, \nu_k)$ for every $k \in \mathbb{N}^*$.

For every $j \in \{1, \dots, N\}$ and every $k \in \mathbb{N}^*$, the functions $\zeta_j^k : \mathbb{R}^n \to \mathbb{R}$ defined by

$$ \forall x \in \mathbb{R}^n, \quad \zeta_j^k(x) := \int_{\mathbb{R}^n} \varphi_j(y) \rho_k(x - y) dy, $$

are of class $C^\infty$ and provide a smooth approximation of $\varphi_j$. From [24, Theorem 9.67], the sequence $(\zeta_j^k)$ converges uniformly to $\varphi_j$ over every compact subset of $\mathbb{R}^n$. Now, solutions to Problem (HP$_c$) take values within a compact set $\bar{B}_{\mathbb{R}^n}(0_{\mathbb{R}^n}, R)$ for some $R > 0$. Hence, for all $j \in \{1, \dots, N\}$, the sequence $(\zeta_j^k)$ converges uniformly towards $\varphi_j$ over $\bar{B}_{\mathbb{R}^n}(0_{\mathbb{R}^n}, R)$. The next lemma follows easily by taking a subsequence and it will allow us to define a boundary layer.

**Lemma 3.1.** For every sequence of positive real numbers $(\eta_k)$ such that $\eta_k \downarrow 0$ as $k \to +\infty$, there is an extraction $\psi : \mathbb{N}^* \to \mathbb{N}^*$ such that for every $j \in \{1, \dots, N\}$ and every $k \in \mathbb{N}^*$, one has:

$$ \forall x \in \bar{B}_{\mathbb{R}^n}(0_{\mathbb{R}^n}, R), \quad |\zeta_j^{\psi(k)}(x) - \varphi_j(x)| < \eta_k. $$

<footnote>$^4$e.g., $\rho_k(y) := \frac{1}{\nu_k^\eta} \rho(\frac{y}{\nu_k})$ where $\rho(y) := C \exp(-\frac{1}{1-|y|^2})$ for $|y| < 1$ and $\rho(y) := 0$ for $|y| \ge 1$ with $C > 0$ such that $\int_{\mathbb{R}^n} \rho = 1$.</footnote>
**Definition 3.1.** Given a sequence of positive real numbers $(\eta_k)$ such that $\eta_k \downarrow 0$ as $k \to +\infty$, we define smooth regions as

$$X_j^k := \{x \in \mathbb{R}^n ; \varphi_j^k(x) < 0\},$$

where for every $j \in \{1, \dots, N\}$ and every $k \in \mathbb{N}^*$, $\varphi_j^k := \zeta_j^{\psi(k)} + 2\eta_k$. The boundary layer is given as:

$$\Lambda^k := \bigcap_{j=1}^{N} \{x \in \mathbb{R}^n ; \varphi_j^k(x) > 0\},$$

and for $j \in \{1, \dots, N\}$, the sublevel set tube (or tubular neighborhood) associated with $X_j$ is defined as

$$S_{j,k} := \{x \in \mathbb{R}^n ; |\varphi_j(x)| \le \eta_k\}.$$

**Remark 3.1.** Hereafter, we comment some aspects related to Definition 3.1.

(i) Observe that the boundary of $X_j^k$, i.e., the set $\{x \in \mathbb{R}^n ; \varphi_j^k(x) = 0\}$, is non-empty and is a smooth hypersurface of $\mathbb{R}^n$. Additionally, the sequence $(\varphi_j^k)$ satisfies:

$$\forall j \in \{1, \dots, N\}, \forall k \in \mathbb{N}^*, \forall x \in \bar{\mathbb{B}}_{\mathbb{R}^n}(0_{\mathbb{R}^n}, R), \varphi_j^k(x) \ge \varphi_j(x) + \eta_k.$$

and we thus have $X_j^k \subset X_j$ for all $j \in \{1, \dots, N\}$ and all $k \in \mathbb{N}^*$.

(ii) Following Definition 3.1, observe that $\mathbb{R}^n$ can be partitioned as follows:

$$\forall k \in \mathbb{N}^*, \quad \mathbb{R}^n = \left( \bigcup_{j=1}^{N} \overline{X_j^k} \right) \cup \Lambda^k.$$

(iii) The boundary layer $\Lambda^k$ contains all (nonsmooth) interfaces. Indeed, for every $x \in \mathbb{R}^n$ that belongs to some interface of the original partition, we have $\varphi_j(x) = 0$ for some $j \in \{1, \dots, N\}$ meaning that $\varphi_j^k(x) > 0$. Moreover, we have that $\varphi_{j'}(x) \ge 0$ whenever $j' \ne j$, this implies $\varphi_{j'}^k(x) > 0$ whenever $j' \ne j$. Hence, $x \in \Lambda^k$, and thus the boundary layer $\Lambda^k$ is a non-empty subset of $\mathbb{R}^n$ that contains all (nonsmooth) interfaces.

In the next lemma, we show that the boundary layer always contains the sublevel set tube once a sequence $(\eta_k)$ such that $\eta_k \downarrow 0$ is given.

**Lemma 3.2.** Given a sequence of positive real numbers $(\eta_k)$ such that $\eta_k \downarrow 0$ as $k \to +\infty$, one has

$$\forall j \in \{1, \dots, N\}, \forall k \in \mathbb{N}^*, \quad S_{j,k} \subset \Lambda^k.$$

*Proof.* Let us set $S_{j,k}^- := \{x \in \mathbb{R}^n ; -\eta_k \le \varphi_j(x) < 0\}$ and $S_{j,k}^+ := \{x \in \mathbb{R}^n ; 0 \le \varphi_j(x) \le \eta_k\}$ where $j \in \{1, \dots, N\}$ and $k \in \mathbb{N}^*$. Obviously, one has $S_{j,k} = S_{j,k}^- \cup S_{j,k}^+$. Now, we claim the following:

$$\forall j \in \{1, \dots, N\}, \forall k \in \mathbb{N}^*, \quad S_{j,k}^- \subset \Lambda^k \cap \overline{X}_j. \qquad (3.1)$$

Indeed, let us take $x \in S_{j,k}^-$. By definition of $S_{j,k}^-$, we have $x \in \overline{X}_j$. Moreover, since $\varphi_j(x) \in [-\eta_k, 0]$, it follows (from the definition of $\varphi_j^k$) that $0 \le \varphi_j(x) + \eta_k < \varphi_j^k(x)$. We also have $\varphi_s^k(x) > 0$ for all $s \in \{1, \dots, N\} \setminus \{j\}$, hence, $x \in \Lambda^k$ as wanted. The second step consists in showing that one has

$$\forall j \in \{1, \dots, N\}, \forall k \in \mathbb{N}^*, \quad S_{j,k}^+ \subset \Lambda^k.$$

For this purpose, it is enough to prove that

$$\forall j \in \{1, \dots, N\}, \forall k \in \mathbb{N}^*, \quad S_{j,k}^+ \subset \bigcup_{s=1}^{N} S_{s,k}^- \qquad (3.2)$$
Let $x \in S_{j,k}^+$ so that $\varphi_j(x) \in [0, \eta_k]$. To prove (3.2), it suffices to find $s \in \{1, \dots, N\}$ such that $\varphi_s(x) \in [-\eta_k, 0]$. From the definition of $\varphi_j(x)$ as given in (2.1), we have

$$ \varphi_j(x) = \max\{\delta_i\phi_i(x); 1 \le i \le \ell\}, $$

where $\delta_i \in \{-1, 1\}$ for all $i \in \{1, \dots, N\}$. The existence of an index $s$ satisfying $\varphi_s(x) \in [-\eta_k, 0)$ relies on the modification of the sign function ($\delta_i$) for all $i \in \{1, \dots, N\}$. We proceed as follows.

* Let $i^* \in \{1, \dots, \ell\}$ be such that

$$ i^* = \operatorname*{argmax}_{1 \le i \le \ell} \{\delta_i \phi_i(x)\}. $$

Since $\varphi_j(x) > 0$, we get $\delta_{i^*} \phi_{i^*}(x) \in [0, \eta_k]$. Hence, we set $\delta'_{i^*} := -\delta_{i^*}$ so that $\delta'_{i^*} \phi_{i^*}(x) \in [-\eta_k, 0]$.

* For $i \ne i^*$, if $\delta_i \phi_i(x) \le \delta'_{i^*} \phi_{i^*}(x)$, we set $\delta'_{i} := \delta_i$ so that $\delta'_{i} \phi_i(x) \le 0$.

* For $i \ne i^*$, if $\delta_i \phi_i(x) \in (-\delta_{i^*} \phi_{i^*}(x), \delta_{i^*} \phi_{i^*}(x))$:

- If $\delta_i \phi_i(x) < 0$, we set $\delta'_{i} := \delta_i$ so that $\delta'_{i} \phi_i(x) \le 0$.

- If $\delta_i \phi_i(x) \ge 0$, we set $\delta'_{i} := -\delta_i$ so that $\delta'_{i} \phi_i(x) \le 0$.

In view of (2.1), one can set $s := 1 + \sum_{i=0}^{\ell-1} a_{i,s} 2^i$ where for $i \in \{1, \dots, \ell\}$, $a_{s, \ell-i} := 0$ whenever $\delta'_{i} = 1$ and $a_{s, \ell-i} := 1$ whenever $\delta'_{i} = -1$. To conclude the proof, we claim that the function $\varphi_s$ satisfies

$$ \varphi_s(x) \in [-\eta_k, 0). $$

Indeed, since $\delta'_{i} \phi_{i}(x) \le 0$ and $\delta'_{i^*} \phi_{i^*}(x) \ge -\eta_k$ for $i \in \{1, \dots, N\}$, we have $\varphi_s(x) \in [-\eta_k, 0)$ by definition of $\varphi_s$. Hence, $x \in S_{s,k}^-$ which ends the proof. $\square$

**Remark 3.2.** Interestingly, if the level sets $L_i$ for $i \in \{1, \dots, \ell\}$ do not intersect (as illustrated in Figure 1), the boundary layer can be defined as follows:

$$ \Lambda^k := \bigcup_{j=1}^{\ell} \{x \in \mathbb{R}^n; |\phi_j(x)| \le \eta_k\}. $$

Hence, in this case, Lemma 3.2 is obviously satisfied by construction.

In the sequel, we fix a sequence of positive real numbers ($\eta_k$) such that $\eta_k \downarrow 0$ as $k \to +\infty$ which also sets the boundary layer. As highlighted before, to ensure that the dynamics $g$ coincides with the dynamics $f$ over $\mathbb{R}^n \setminus \Lambda^k$, we will proceed by regularizing the indicator functions of the sets $X_j^k$ for all $j \in \{1, \dots, N\}$ and all $k \in \mathbb{N}^*$. Using Urysohn's Lemma and a regularization process based on mollifiers, there is a sequence of smooth function $\sigma_j^k : \mathbb{R}^n \to [0, 1]$ such that

$$ \sigma_j^k(x) \in \begin{cases} \{1\}, & \text{if } x \in \overline{X_j^k}, \\ \{0\}, & \text{if } x \in \mathbb{R}^n \setminus X_j, \\ [0, 1], & \text{if } x \in \Lambda^k \cap \overline{X_j}, \end{cases} $$

for all $x \in \mathbb{R}^n$, all $j \in \{1, \dots, N\}$ and all $k \in \mathbb{N}^*$ (we refer to Figure 2 for an illustration). Note that one has $\sum_{j=1}^N \sigma_j^k(x) \le 1$ for all $x \in \mathbb{R}^n$, all $j \in \{1, \dots, N\}$ and all $k \in \mathbb{N}^*$.

We are now in a position to define a dynamics $f^k$ that will coincide with $g$ over the boundary layer $\Lambda^k$ and with $f$ over $\mathbb{R}^n \setminus \Lambda^k$. For this purpose, let $f^k : \mathbb{R}^n \times U^N \times V \to \mathbb{R}^n$ be defined as

$$ f^k(x, u, v) := \sum_{j=1}^{N} \sigma_j^k(x) f_j(x, u_j) + \left(1 - \sum_{j=1}^{N} \sigma_j^k(x)\right) g(x, u, v), $$

for all $(x, u, v) \in \mathbb{R}^n \times U^N \times V$ and all $k \in \mathbb{N}^*$. By construction, we note that $f^k$ is of class $C^1$ over $\mathbb{R}^n \times U^N \times V$. We can now introduce the regularized control system:

$$ \dot{x}(t) = f^k(x(t), u(t), v(t)), \quad \text{a.e. } t \in [0, T]. \qquad (3.3) $$
<figure><img src="image_3.png" /><figcaption>**Figure 2:** Illustration of the functions $\sigma_j^k$ (left) and $\sigma_{j'}^k$ (right) for two neighboring sets $X_j, X_{j'}$ in $\mathbb{R}^2$.</figcaption></figure>

together with the penalty function $P: \mathbb{R}^n \times \mathbb{R}^N \to \mathbb{R}$ defined by

$$P(x, v) = \sum_{j=1}^{N} ((1 - 2v_j)\varphi_j(x) - |\varphi_j(x)|)^2,$$

where $(x, v) \in \mathbb{R}^n \times \mathbb{R}^N$. It is easily seen that for each $x \in \mathbb{R}^n$, $v \mapsto P(x, v)$ is quadratic convex.

**Remark 3.3.** Some comments related to the functions $\sigma_j^k$ and to the control system (3.3) are in order.

(i) Let us emphasize that each function $\sigma_j^k$ regularizes the indicator function $1_{X_j}$ of $X_j$. Precisely, $\sigma_j^k$ coincides with $1_{X_j}$ everywhere except on $\Lambda^k \cap X_j$ (see Figure 2 for an illustration). These regularization functions satisfy two fundamental properties. First, they have a non-overlapping support: if $\sigma_j^k(x) > 0$ for some $x \in \mathbb{R}^n$, then $\sigma_{j'}^k(x) = 0$ for all indices $j' \neq j$. Second, each $\sigma_j^k$ vanishes over every interface $\partial X_{j'}$ for all $j' \in \{1, \dots, N\}$. These properties are important in our study (see [7] for a similar approach concerning the approximation of the Heaviside function for the minimal time crisis problem with semi-transverse solutions).

(ii) Based on the definition of $\sigma_j^k$, the dynamics $f^k$ is constructed in such a way to coincide with the dynamics $f$ everywhere except within the boundary layer $\Lambda^k$. Moreover, since the weight of each $\sigma_j^k$ decreases near the interfaces $\partial X_{j'}$ and vanishes entirely on $\partial X_{j'}$ for all $j' \in \{1, \dots, N\}$, the inclusion of additional controls $v_j$ and the penalty function $P$ (which will be developed in the next section) will allow us to approximate the (discontinuous) dynamics $f$ over the boundary layer $\Lambda^k$.

## 3.2 Sequence of approximated optimal control problems

For each $k \in \mathbb{N}^*$, the *approximated optimal control problem* of Bolza type we consider can be written

$$ \begin{array}{ll} \text{minimize} & \Phi(x(T)) + \frac{1}{\varepsilon_k} \int_0^T P(x(t), v(t)) \, dt, \\ \text{subject to} & (x(\cdot), u(\cdot), v(\cdot)) \in \mathrm{AC}_{[0,T]} \times \mathcal{U}\mathcal{V}_{[0,T]}, \qquad (P_k) \\ & \dot{x}(t) = f^k(x(t), u(t), v(t)), \quad \text{a.e. } t \in [0, T], \\ & x(0) = x_0, \end{array} $$

where $(\varepsilon_k)$ is such that $\varepsilon_k \downarrow 0$ as $k \to +\infty$. We refer to Remark 3.4 for a discussion on the choice of this sequence. Next, we prove the existence of an optimal solution to Problem (P_k).

**Proposition 3.3.** For all $k \in \mathbb{N}^*$, Problem (P_k) has an optimal solution $(x^k, u^k, v^k)$.
Proof. Since the proof follows directly from the application of classical existence theorems in optimal control theory, we only outline the main ingredients. First, note that $f^k$ and $\Phi$ are of class $C^1$ and that $P$ is Lipschitz continuous. Moreover, $f^k$ is with linear growth thanks to Assumption 2.3, and the admissible control set $U^N \times V$ is compact. To check that the extended velocity set

$$V_k(x) := \{(y_0, f^k(x, u, v)) ; y_0 \ge P(x, v) \text{ and } (u, v) \in U^N \times V\}$$

is convex for every $x \in \mathbb{R}^n$, observe that $f^k$ coincides with $f$ over $\mathbb{R}^n \setminus \Lambda_k$, which ensures that $V_k(x)$ is convex for $x \in \mathbb{R}^n \setminus \Lambda_k$ (remind Assumption 2.3). For $x \in \Lambda_k$, it is enough to show that $g(x, U, V)$ is convex in view of the definition of $f^k$. Using (2.4) and that $v \mapsto P(x, v)$ is convex, we obtain that $V_k(x)$ is convex for $x \in \Lambda_k$ as well. The existence of an optimal control then follows from [10, Theorem 6.3]. □

### 3.3 Convergence of solutions to (P_k) to a solution of (HP)

Before stating our main result, we make the following assumption on the behaviour of $x^k$ over the boundary layer $\Lambda^k$. Hereafter $\mu$ stands for the Lebesgue measure on the space $\mathbb{R}$.

**Assumption 3.4.** For all $k \in \mathbb{N}^*$, there is an optimal solution $(x^k, u^k, v^k)$ of (P_k) such that $(x^k)$ satisfies

$$\lim_{k \to \infty} \frac{\mu(\{t \in [0, T] ; x^k(t) \in \Lambda^k\})}{\varepsilon_k} = 0. \qquad (3.4)$$

**Remark 3.4.** Hereafter, we comment some aspects related to Assumption 3.4.

(i) Let us emphasize that we do not impose a transverse crossing assumption (such as in [2, 4, 17, 20, 23]) on the optimal solution of Problem (HP). Instead, such an assumption is replaced by (3.4) which involves the sequence of optimal trajectories $(x^k)$ of Problem (P_k). This assumption ensures that the time spent within the boundary layer converges to zero. It will allow us to handle optimal solutions that cross the interface tangentially (see Section 4 for an illustrative example).

(ii) Assumption 3.4 guarantees that the penalty function converges to zero as $k \to \infty$ (see the proof of Theorem 3.5). In the numerical examples of Section 4, we take $(\varepsilon_k)$ constant (equal to a small parameter) so that in that case (3.4) it is enough to check that $\lim_{k \to \infty} \mu(\{t \in [0, T] ; x^k(t) \in \Lambda^k\}) = 0$.

Our main result is as follows.

**Theorem 3.5.** For every $k \in \mathbb{N}^*$, let $(x^k, u^k, v^k)$ be a solution to (P_k) such that Assumption 3.4 is fulfilled. Then, there is a solution $\bar{x}$ to Problem (HP) such that, up to a subsequence, $(x^k)$ converges strongly-weakly<fnref n="5" /> to $\bar{x}$ over $[0, T]$ as $k \to +\infty$.

*Proof.* The proof is done in three steps.

**Step 1.** Since $f^k$ is with linear growth, Gronwall's Lemma implies that there is $c \ge 0$ such that

$$\forall k \in \mathbb{N}, \quad \|\dot{x}^k(t)\|_{\mathbb{R}^n} \le c \quad \text{a.e. } t \in [0, T].$$

It follows that $(x^k)$ is uniformly bounded in $L^\infty([0, T], \mathbb{R}^n)$, say by a constant $R' \ge 0$. Hence, by Arzelà-Ascoli's theorem, there is a function $\bar{x} \in AC_{[0,T]}$ such that, up to a subsequence (which we do not relabel), $(x^k)$ converges strongly-weakly to $\bar{x}$ over $[0, T]$ as $k \to +\infty$. It remains to prove that $\bar{x}$ is an admissible solution of (HP) which is the purpose of the next step.

**Step 2.** Note that for all $k \in \mathbb{N}^*$ the trajectory $x^k$ satisfy the following differential inclusion

$$\dot{x}^k(t) \in \bar{F}(x^k(t)) + G^k(x^k(t)), \quad \text{a.e. } t \in [0, T], \qquad (3.5)$$

<footnote n="5">⁵The sequence $(x_k)$ converges strongly-weakly to $\bar{x}$ over $I := [0, T]$ whenever $(x_k)$ converges to $\bar{x}$ in $L^\infty(I, \mathbb{R}^n)$ and $(\dot{x}_k)$ weakly converges in $L^2(I, \mathbb{R}^n)$ to $\dot{\bar{x}}$.</footnote>
where $\bar{F}$ is given by (2.4) and $G^k: \mathbb{R}^n \to \mathbb{R}^n$ is defined by

$$G^k(x) := \bigcup_{(u,v,\omega) \in U^N \times V \times V} \left\{ f^k(x, u, v) - \sum_{j=1}^{N} \omega_j f_j(x, u_j), \omega_j = 0, j \notin \mathcal{J}(x) \right\},$$

with $\mathcal{J}(x) := \{j; x \in \bar{X}_j\}$ for $x \in \mathbb{R}^n$. It is easy to see that $G^k(x) = \{0_{\mathbb{R}^n}\}$ for all $x \in X_j^k$ and all $j \in \{1, \dots, N\}$. Indeed, for all $x \in X_j^k$, we have $\sigma_j^k(x) = 1$, and since $j \in \mathcal{J}(x)$, we obtain $\omega_j = 1$. Moreover, since $\mathbb{R}^n = (\bigcup_{j=1}^N \overline{X_j^k}) \cup \Lambda^k$, for all $x \in \mathbb{R}^n$, every $z \in G^k(x)$ can be written

$$z = \sum_{j=1}^{N} \left[ f^k(x, u, v) - \sum_{s=1}^{N} \omega_s f_s(x, u_j) \right] \mathbf{1}_{\Lambda^k}(x),$$

for some $(u, v) \in U^N \times V$. If $M_1 := \max_{1 \le j \le N} \max_{(z',w) \in \bar{B}_{\mathbb{R}^n}(0_{\mathbb{R}^n}, R') \times U} \|f_j(z', w)\|_{\mathbb{R}^n}$, we obtain

$$\|z\|_{\mathbb{R}^n} \le 2NM_1 \mathbf{1}_{\Lambda^k}(x)$$

coming back to the expression defining $f^k$. Hence, (3.5) becomes

$$\dot{x}^k(t) \in \bar{F}(x^k(t)) + r_k(t) \bar{B}_{\mathbb{R}^n}(0_{\mathbb{R}^n}, 1),$$

where $r_k(t) := 2NM_1 \mathbf{1}_{\Lambda^k}(x^k(t))$ for all $t \in [0, T]$. Moreover, $r_k$ converges to zero in $L^2([0, T], \mathbb{R})$ as $k \to \infty$ (thanks to Assumption 3.4). We are now in a position to apply [13, Theorem 1.11], so that $\bar{x}$ satisfies $\dot{\bar{x}}(t) \in \bar{F}(\bar{x}(t))$ for almost every $t \in [0, T]$. To conclude the proof, it remains to prove that $\bar{x}$ is an optimal solution of (HP) which is addressed in the next step.

**Step 3.** Suppose by contradiction that $\bar{x}$ is not an optimal solution to Problem (HP). Then, there is an admissible solution $x^* \in AC_{[0,T]}$ of (HP) such that

$$\Phi(x^*(T)) < \Phi(\bar{x}(T)) - \eta,$$

where $\eta > 0$. Observe that $x^*$ satisfies the differential inclusion:

$$\dot{x}^*(t) \in \bar{F}(x^*(t)) \quad \text{a.e. } t \in [0, T].$$

Using a measurable selection lemma (see, e.g., [21, Lemma 3A, p. 161]), we deduce that there exist measurable control functions $u^* : [0, T] \to U^N$ and $w^* : [0, T] \to V$ such that:

$$\dot{x}^*(t) = \sum_{j=1}^{N} w_j^*(t) f_j(x^*(t), u_j^*(t)), \quad \text{a.e. } t \in [0, T]. \qquad (3.6)$$

Let us recall that for every $j \in \{1, \dots, N\}$, one has $w_j^*(t) = 1$ for almost every $t \in [0, T]$ when $x_j^*(t) \in X_j$. Our objective now is to prove that the triplet $(x^*, u^*, w^*)$ is an admissible solution of Problem (P$_k$) for every $k$ large enough. For this purpose, since $x^*(0) = x_0$, $u^*(t) \in U^N$, and $w^*(t) \in V$ for almost every $t \in [0, T]$, it is enough to prove that

$$\dot{x}^*(t) = f^k(x^*(t), u^*(t), w^*(t)), \quad \text{a.e. } t \in [0, T], \qquad (3.7)$$

for every $k \in \mathbb{N}^*$ large enough. Since $x_0 \in X_{j_1}$ where $j_1 \in \{1, \dots, N\}$ and $(\varphi_j^k)$ uniformly converges to $\varphi_j$ over compact sets, there is $k_0 \ge 1$ such that for all $k \ge k_0$, we have $x_0 \in X_{j_1}^k$. It follows that for proving (3.7), it is enough to show that

$$\sigma_j^k(x^*(t)) + \left( 1 - \sum_{s=1}^{N} \sigma_s^k(x^*(t)) \right) w_j^*(t) = w_j^*(t) \quad \text{a.e. } t \in [0, T], \qquad (3.8)$$

for all $j \in \{1, \dots, N\}$, and for all $k \ge k_0$. Observe that for all $k \ge k_0$.

$$[0, T] = A^k \cup \left( \bigcup_{j=1}^{N} B_j^k \right) \cup \left( \bigcup_{j=1}^{N} C_j^k \right),$$
where

$$
\begin{align*}
A^k &:= \{t \in [0, T] ; x^*(t) \in \bigcup_{j'=1}^N \bar{X}_{j'}^k\}, \\
B_j^k &:= \{t \in [0, T] ; x^*(t) \in \Lambda^k \cap X_j\}, \\
C_j^k &:= \{t \in [0, T] ; x^*(t) \in \partial X_j\},
\end{align*}
$$

Therefore, three cases are possible:

• Case 1. By construction, for 1 ≤ j ≤ N, w ∗ j satisfies

$$
w_j^*(t) = \begin{cases} 1, & \text{if } x^*(t) \in \bar{X}_j^k, \\ 0, & \text{if } x^*(t) \in \bar{X}_s^k, \text{ for } s \neq j, \end{cases}
$$

for almost every $t \in A^k$ and for all $k \ge k_0$. Hence, if we take $w_j^*(t) = \sigma_j^k(x^*(t))$ for almost every $t \in A^k$,
(3.8) is automatically satisfied since in that case $\sum_{s=1}^N \sigma_s^k(x^*(t)) = 1$.

• Case 2. Let $j \in \{1, \dots, N\}$. Note that for all $k \ge k_0$ and for almost every $t \in B_j^k$, one has $w_j^*(t) = 1$. On the other hand, one can write

$$
\sigma_j^k(x^*(t)) + \left(1 - \sum_{s=1}^N \sigma_s^k(x^*(t))\right) w_j^*(t) = \sigma_j^k(x^*(t)) + (1 - \sigma_j^k(x^*(t))) w_j^*(t) = w_j^*(t),
$$

for almost every $t \in B_j^k$ and all $k \ge k_0$ (since $\sigma_s^k(x^*(t)) = 0$ for every $s \ne j$). Hence (3.8) is satisfied.

• Case 3. Let $j \in \{1, \dots, N\}$. For $k \ge k_0$ and for almost every $t \in C_j^k$, one has $w_j^* \in [0, 1]$. Moreover, we have $\sigma_s^k(x^*(t)) = 0$ for all $t \in C_j^k$ and all $s \in \{1, \dots, N\}$. Hence (3.8) is automatically satisfied.

A this step, we have thus shown that the triplet $(x^*, u^*, w^*)$ is admissible for Problem (P$_k$) for $k \ge k_0$. To conclude the proof, we claim that

$$
\lim_{k \to \infty} \Phi(x^k(T)) + \frac{1}{\varepsilon_k} \int_0^T P(x^k(t), v^k(t)) \, dt = \Phi(\bar{x}(T)). \quad (3.9)
$$

Indeed, when $x \notin \Lambda^k$, $f^k(x, u, v)$ does not depend on the control function $v$ and $\sigma_j^k(x) = 1$ if $x \in X_j^k$ and thus $\sigma_{j'}^k(x) = 0$ for $j' \ne j$. So, for almost every $t \in [0, T]$ such that $x^k(t) \notin \Lambda_k$, one has $P(x^k(t), v^k(t)) = 0$ with $v^k(t) := (\sigma_1^k(x^k(t)), \dots, \sigma_N^k(x^k(t)))$. We deduce that

$$
\left| \frac{1}{\varepsilon_k} \int_0^T P(x^k(t), v^k(t)) \, dt \right| = \left| \frac{1}{\varepsilon_k} \int_{\{t \in [0, T] ; x^k(t) \in \Lambda^k\}} P(x^k(t), v^k(t)) \, dt \right| \le M \frac{\mu(\{t \in [0, T] ; x^k(t) \in \Lambda^k\})}{\varepsilon_k},
$$

where $M > 0$ is an upper bound of the map $(x, v) \mapsto |P(x, v)|$ over the compact set $\bar{B}_{\mathbb{R}^n}(0_{\mathbb{R}^n}, R') \times V$. Hence, using Assumption 3.4 and the continuity of $\Phi$, we obtain (3.9) letting $k \to +\infty$. Finally, since for $k \ge k_0$, $(x^k, u^k, v^k)$ is an optimal solution of (P$_k$) and $(x^*, u^*, w^*)$ is admissible for (P$_k$), we have

$$
\Phi(x^k(T)) \leq \Phi(x^*(T)) < \Phi(\bar{x}(T)) - \eta.
$$

But, from (3.9), $\Phi(x^k(T)) \to \Phi(\bar{x}(T))$ as $k \to +\infty$ which gives a contradiction, whence the result.

<figure><img src="image_4.png" /></figure>

Let us now define the *value function* $\mathcal{V}: (-\infty, T] \times \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ (associated with problem (HP)) as:

$$
\mathcal{V}(t_0, x_0) := \inf_{u \in \mathcal{U}[0, T]} \Phi(x(T)),
$$

where $(t_0, x_0) \in (-\infty, T] \times \mathbb{R}^n$ and $x$ is a solution of (2.3) such that $x(0) = x_0$. Moreover, for $k \in \mathbb{N}^*$, let us define the value function $\mathcal{V}_k : (-\infty, T] \times \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ (associated with problem (P$_k$)) as follows:

$$
\mathcal{V}_k(t_0, x_0) := \inf_{(u,v) \in \mathcal{U}\mathcal{V}_{[0,T]}} \Phi(x(T)) + \frac{1}{\varepsilon_k} \int_0^T P(x(t), v(t)) \, dt,
$$

where $(t_0, x_0) \in (-\infty, T] \times \mathbb{R}^n$ and $x$ is a solution of (3.3) such that $x(t_0) = x_0$. The next result is a simple consequence of Theorem 3.5.
**Corollary 3.6.** If Assumption 3.4 is fulfilled, one has (up to a subsequence) for every $x_0 \in \mathbb{R}^n$:

$$ \lim_{k \to +\infty} \mathcal{V}_k(0, x_0) = \mathcal{V}(0, x_0). $$

## 3.4 Absence of sliding mode

The objective of this section is to prove that $\bar{x}$ has no sliding mode over $[0, T]$ (where $\bar{x}$ is given by Theorem 3.5).

**Proposition 3.7.** If Assumption 3.4 is fulfilled, then, the solution $\bar{x}$ to (HP) satisfies $\mu(E) = 0$ where

$$ E := \bigcup_{j=1}^{N} \{t \in [0, T] ; \varphi_j(\bar{x}(t)) = 0\}. $$

*Proof.* Suppose that $\mu(E) > 0$. Then, using the subadditivity of the Lebesgue measure, there is $j \in \{1, \dots, N\}$ such that the set $E_j := \{t \in [0, T] : \varphi_j(\bar{x}(t)) = 0\}$ is with positive measure. Since $\varphi_j$ is uniformly continuous over $\bar{B}_{\mathbb{R}^n}(0_{\mathbb{R}^n}, R)$, for all $k \in \mathbb{N}^*$, there is $\alpha_k > 0$ such that

$$ \|x - x'\|_{\mathbb{R}^n} \le \alpha_k \implies |\varphi_j(x) - \varphi_j(x')| \le \eta_k. $$

Now, $(x^k)$ converges uniformly to $\bar{x}$ over $[0, T]$, hence, one can show, by extracting a subsequence, that there is an extraction $s : \mathbb{N}^* \to \mathbb{N}^*$ such that for every $k \in \mathbb{N}^*$, one has:

$$ \forall t \in [0, T], \quad \|x^{s(k)}(t) - \bar{x}(t)\|_{\mathbb{R}^n} \le \alpha_k. $$

For all $t \in E_j$, we have $\varphi_j(\bar{x}(t)) = 0$. Hence, from the uniform continuity of $\varphi_j$ over $\bar{B}_{\mathbb{R}^n}(0_{\mathbb{R}^n}, R)$, one has

$$ \forall k \in \mathbb{N}^*, \forall t \in E_j, \quad |\varphi_j(x^{s(k)}(t))| = |\varphi_j(x^{s(k)}(t)) - \varphi_j(\bar{x}(t))| \le \eta_k. $$

We deduce that for all $k \in \mathbb{N}^*$,

$$ E_j \subset \{t \in [0, T] ; |\varphi_j(x^{s(k)}(t))| \le \eta_k\} \subset \{t \in [0, T] ; x^{s(k)}(t) \in S_{j,k}\}. $$

From Lemma 3.2, we have $S_{j,k} \subset \Lambda_k$, hence $E_j \subset \{t \in [0, T] ; x^{s(k)}(t) \in \Lambda^k\}$. Thanks to Assumption 3.4, we deduce (letting $k \to +\infty$) that $\mu(E_j) = 0$, which leads to a contradiction. Hence $\mu(E) = 0$ as wanted. $\square$

Even though $\bar{x}$ has no sliding mode, $\bar{x}$ can have several crossing points (or crossings to shorten) that can be transverse or not. We refer to [2, 5] for a precise definition of a crossing point and a presentation of the various transverse crossing hypotheses.

# 4 Illustration of the penalization approach on two examples involving non-transverse crossings

In this section, we provide two examples<fnref n="6" /> illustrating numerically the convergence of approximated optimal solutions (to the penalized problem) to an optimal solution of the original problem. We solve the penalized optimal control problem associated with Problem (Pex) using the Julia package OptimalControl.jl (we refer to [11] for more details). The solution process involves reformulating the optimal control problem as a finite-dimensional optimization problem, which is then solved using the IPOPT solver with a precision tolerance set to $10^{-8}$. A batch resolution of the penalized problem is performed for various values of $k \in \{30, 40, 50, 150\}$ where $\varepsilon_k = \varepsilon = 10^{-2}$ for example 1 and $\varepsilon_k = \varepsilon = 10^{-3}$ for example 2 (as illustrated in Figures 3 and 5).

<footnote n="6">⁶The scripts for reproducing the numerical experiments of this paper are published in the repository: https://github.com/AnasXbouali/Academic-example</footnote>
## 4.1 Example 1: non-transverse crossing at a smooth point of the interface

By smooth point, we mean a point that belongs to an interface which is locally of class $C^1$ (and thus it belongs to a single interface in our context). Here, we consider the following controlled dynamics $g: \mathbb{R}^2 \times [-1, 1] \to \mathbb{R}^2$ that is defined regionally as follows

$$g(x, u) := \begin{cases} g_1(x, u), & \text{if } x_2 > 0, \\ g_2(x, u), & \text{if } x_1 > 0 \text{ and } x_2 < 0, \\ g_3(x, u), & \text{if } x_1 < 0 \text{ and } x_2 < 0, \end{cases}$$

where

$$g_1(x, u) := \begin{pmatrix} 0 \\ -2 - u \end{pmatrix}, \quad g_2(x, u) := \begin{pmatrix} -2 + u \\ x_1 - 1 \end{pmatrix}, \quad g_3(x, u) := \begin{pmatrix} -\left(x_2 + \frac{1}{2}\right)^2 \\ -2 + u \end{pmatrix},$$

for all $(x, u) \in \mathbb{R}^2 \times [-1, 1]$. The stratification is defined as follows. Let $\phi_1, \phi_2: \mathbb{R}^2 \to \mathbb{R}$ be defined as $\phi_1(x) := x_1$ and $\phi_2(x) := x_2$ for $x \in \mathbb{R}^2$. Following Section 2.1, four regions can be defined thanks to $\phi_1, \phi_2$:

$$X_1 := \{x \in \mathbb{R}^2; x_1 < 0, x_2 < 0\}, X_2 := \{x \in \mathbb{R}^2; x_1 < 0, x_2 > 0\},$$

$$X_3 := \{x \in \mathbb{R}^2; x_1 > 0, x_2 < 0\}, X_4 := \{x \in \mathbb{R}^2; x_1 > 0, x_2 > 0\}.$$

This leads to the stratification of the plane with $N = 2^2 = 4$ regions: $\mathbb{R}^2 = \bar{X}_1 \cup \bar{X}_2 \cup \bar{X}_3 \cup \bar{X}_4$, and we can write the corresponding controlled dynamics as

$$f(x, u) := \begin{cases} f_1(x, u) := g_3(x, u), & \text{if } x \in X_1, \\ f_2(x, u) := g_1(x, u), & \text{if } x \in X_2, \\ f_3(x, u) := g_2(x, u), & \text{if } x \in X_3, \\ f_4(x, u) := g_1(x, u), & \text{if } x \in X_4, \end{cases}$$

for all $(x, u) \in \mathbb{R}^2 \times [-1, 1]$. Note that in this example, the dynamics $f_3$ and $f_4$ are identical (which amounts to consider only the three regions $\bar{X}_1, \bar{X}_2$, and $\bar{X}_3 \cup \bar{X}_4$). This can happen, as we mentioned in Remark 2.1. We are now in a position to state the problem of interest:

$$
\begin{array}{ll}
\text{minimize} & \displaystyle\int_0^{\frac{\tau}{2}} x_2^2(t) \, dt, \\
\text{subject to} & (x(\cdot), u(\cdot)) \in \mathrm{AC}_{[0, \frac{\tau}{2}]} \times \mathcal{U}_{[0, \frac{\tau}{2}]}, \\
& \dot{x}(t) = f(x(t), u(t)), \quad \text{a.e. } t \in [0, \frac{7}{2}], \\
& x(0) = (1, 1).
\end{array}
\qquad (\mathrm{P}_{\mathrm{ex}})
$$

Using the penalization approach presented in Section 3.2, we now solve the approximated optimal control problems associated with Problem $(\mathrm{P}_{\mathrm{ex}})$. We observe that the optimal pair $(x^k, u^k)$ has two crossing points at times $\tau_1^k \approx 1$ and $\tau_2^k \approx 2$. Moreover, each trajectory $x^k$ (for $k \in \{50, 150\}$) has a crossing point at a smooth point of the interface which is non-transverse (see Figure 3). Moreover, it is worth noting that the optimal values of the additional controls $v_1, v_2$, and $v_3$ are restricted to the set $\{0, 1\}$, following the penalty function's effect (see Figure 4). Finally, we compute the time spent by $x^k$ in the boundary layer $\Lambda^k$ for every $k \in \{10, 20, 50, 150\}$, showing that Assumptions 3.4 (for a constant $\varepsilon = 10^{-2}$, see Remark 3.4 Item (ii)) is satisfied numerically (see Table 1).

## 4.2 Example 2: non-transverse crossing at a corner point of the interface

In this example, we consider the following controlled dynamics $g: \mathbb{R}^2 \times [-1, 1] \to \mathbb{R}^2$ that is defined regionally as follows

$$g(x, u) := \begin{cases} g_1(x, u) & x_1 > 0, \text{ and } x_2 + \frac{1}{2} > 0, \\ g_2(x, u) & x_1 > 0, \text{ and } x_2 + \frac{1}{2} < 0, \\ g_3(x, u) & x_1 < 0, \end{cases}$$
<figure><img src="image_5.png" /><figcaption>Figure 3: Optimal pairs $(x^k, u^k)$ of the approximated problem associated to Problem (P$_{ex}$).</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 4: Optimal controls $v^k$ of the approximated problem associated to Problem (P$_{ex}$).</figcaption></figure>

where

$$g_1(x, u) := \begin{pmatrix} -2+u \\ x_1-1 \end{pmatrix}, \quad g_2(x, u) := \begin{pmatrix} u(t) \\ -10 \end{pmatrix}, \quad g_3(x, u) := \begin{pmatrix} -\left(x_2+\frac{1}{2}\right)^2 \\ -2+u \end{pmatrix},$$
<table><caption>**Table 1:** Summary of results for different values of k, with ε = 10⁻².</caption><thead><tr><th>k</th><th>μ({t ∈ [0, T]; x<sup>k</sup>(t) ∈ Λ<sup>k</sup>}</th><th>Objective</th></tr></thead><tbody><tr><td>10</td><td>0.3124</td><td>29.7478</td></tr><tr><td>20</td><td>0.1323</td><td>47.9891</td></tr><tr><td>50</td><td>0.2496</td><td>3.0546</td></tr><tr><td>150</td><td>0.1454</td><td>3.0502</td></tr></tbody></table>

for $(x, u) \in \mathbb{R}^2 \times [-1, 1]$. The stratification is defined as follows. Let $\phi_1, \phi_2: \mathbb{R}^2 \to \mathbb{R}$ be defined as $\phi_1(x) := x_1$ and $\phi_2(x) := x_2 + \frac{1}{2}$ for $x \in \mathbb{R}^2$. Following Section 2.1, four regions can be defined thanks to $\phi_1, \phi_2$:

$$X_1 := \{x \in \mathbb{R}^2; x_1 < 0, x_2 + \frac{1}{2} < 0\}, \quad X_2 := \{x \in \mathbb{R}^2; x_1 < 0, x_2 + \frac{1}{2} > 0\}, \\ X_3 := \{x \in \mathbb{R}^2; x_1 > 0, x_2 + \frac{1}{2} < 0\}, \quad X_4 := \{x \in \mathbb{R}^2; x_1 > 0, x_2 + \frac{1}{2} > 0\}.$$

This leads to the stratification of the plane with $N = 2^2 = 4$ regions: $\mathbb{R}^2 = \bar{X}_1 \cup \bar{X}_2 \cup \bar{X}_3 \cup \bar{X}_4$, and we can write the corresponding controlled dynamics as

$$f(x, u) := \begin{cases} f_1(x, u) := g_3(x, u), & \text{if } x \in X_1, \\ f_2(x, u) := g_3(x, u), & \text{if } x \in X_2, \\ f_3(x, u) := g_2(x, u), & \text{if } x \in X_3, \\ f_4(x, u) := g_1(x, u), & \text{if } x \in X_4, \end{cases}$$

for all $(x, u) \in \mathbb{R}^2 \times [-1, 1]$. We are now in a position to state the problem of interest:

$$\begin{array}{ll} \text{minimize} & \displaystyle\int_0^2 x_2^2(t) \, dt, \\ \text{subject to} & (x(\cdot), u(\cdot)) \in \mathrm{AC}_{[0,2]} \times \mathcal{U}_{[0,2]}, \\ & \dot{x}(t) = f(x(t), u(t)), \quad \text{a.e. } t \in [0, 2], \\ & x(0) = (1, 0). \end{array} \qquad (P'_{\mathrm{ex}})$$

Using the penalization approach presented in Section 3.2, we solve the approximated optimal control problems associated with Problem $(P'_{\mathrm{ex}})$. We observe that the optimal pair $(x^k, u^k)$ has a single crossing point at a time $\tau^k \approx 1$. Moreover, each trajectory $x^k$ (for $k \in \{10, 20, 50, 150\}$) has a crossing point (which is non-transverse) at a corner point of the interface (see Figure 5). So, the crossing point of $x^k$ occurs at some point of the interface where it is nonsmooth. Moreover, it is worth noting that the optimal values of the additional controls $v_1, v_2$, and $v_3$ are restricted to the set $\{0, 1\}$, following the penalty function's effect (see Figure 6). Finally, we compute the time spent in the boundary layer $\Lambda^k$ for every $k \in \{10, 20, 50, 150\}$, showing that Assumptions 3.4 (for a constant $\varepsilon = 10^{-3}$, see Remark 3.4 Item (ii)) is satisfied (see Table 2).

<table><caption>**Table 2:** Summary of results for different values of k, with ε = 10⁻³.</caption><thead><tr><th>k</th><th>μ({t ∈ [0, T]; x<sup>k</sup>(t) ∈ Λ<sup>k</sup>}</th><th>Objective</th></tr></thead><tbody><tr><td>10</td><td>0.292</td><td>1.38</td></tr><tr><td>20</td><td>0.1801</td><td>1.1689</td></tr><tr><td>50</td><td>0.1098</td><td>1.1362</td></tr><tr><td>150</td><td>0.0068</td><td>1.1335</td></tr></tbody></table>

**Remark 4.1.** Hereafter, we provide comments about the numerical solutions of the penalized problems associated to both Problems $(P_{\mathrm{ex}})$ and $(P'_{\mathrm{ex}})$.
<figure><img src="image_7.png" /><figcaption>Figure 5: Optimal pairs $(x^k, u^k)$ of the approximated problem associated to Problem (P'$_{ex}$).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 6: Optimal controls $v^k$ of the approximated problem associated to Problem (P'$_{ex}$).</figcaption></figure>

(i) Note that optimal solutions to Problems (P$_{ex}$) and (P'$_{ex}$) satisfy Assumption 3.4 (as detailed in Tables 1 and 2). However, a (standard) transverse crossing condition is not satisfied. Instead, only a semi-transverse condition is satisfied numerically (see Figures 3 and 5).

(ii) Let us highlight that the method presented in this work using a penalization approach performs (slightly) better than a (standard) regularization of the discontinuous dynamics (we refer to Figure 7). Here, one
can show that the optimal solution of Problem (P<sub>ex</sub>) is given by

$$
u^*(t) = \begin{cases} -1 & \text{for a.e. } t \in [0, \tau_1^*], \\ 1 & \text{for a.e. } t \in [\tau_1^*, \frac{7}{2}], \end{cases}
$$

where $\tau_1^* \in (0, T)$ corresponds to the time when the first crossing occurs at $x_2^*(\tau_1^*) = 0$. Hence, we get that

$$
x_1^*(t) = \begin{cases} 1 & \text{for all } t \in [0, \tau_1^*], \\ 2-t & \text{for all } t \in [\tau_1^*, \tau_2^*], \\ -\frac{t^3}{3} + 2t^2 - 4t + \frac{8}{3} & \text{for all } t \in [\tau_2^*, \frac{7}{2}], \end{cases} \quad x_2^*(t) = \begin{cases} 1-t & \text{for all } t \in [0, \tau_1^*], \\ -\frac{t^2}{2} + t - \frac{1}{2} & \text{for all } t \in [\tau_1^*, \tau_2^*], \\ -t + \frac{3}{2} & \text{for all } t \in [\tau_2^*, \frac{7}{2}], \end{cases}
$$

with $\tau_1^* = 1$ and $\tau_2^* = 2$ (the time corresponding to the second crossing point of $x_2^*$).

<figure><img src="image_9.png" /><figcaption>**Figure 7:** Comparison with a (standard) regularization approach in solving Problem (P<sub>ex</sub>).</figcaption></figure>

(iii) Note that in the case where an optimal trajectory crosses a corner point, as illustrated in Problem (P'_{ex}), the (standard) regularization technique struggles in capturing the dynamics near the corner point. On the other hand, the penalization method provides more flexibility due to the additional control v^k (see Figure 8). Here, the optimal solution of Problem (P'_{ex}) is given by

$$
u^*(t) = \begin{cases} 1 & \text{for a.e. } t \in [0, \tau^*], \\ -1 & \text{for a.e. } t \in [\tau^*, 2], \end{cases}
$$

where $\tau^* \in (0, T)$ corresponds to the time when the first crossing occurs at the corner point such that $(x_1^*(\tau^*), x_2^*(\tau^*)) = (0, -\frac{1}{2})$. Hence, we get that

$$
x_1^*(t) = \begin{cases} -t+1 & \text{for all } t \in [0, \tau^*], \\ -\frac{(t-1)^3}{3} & \text{for all } t \in [\tau^*, 2], \end{cases} \quad x_2^*(t) = \begin{cases} -\frac{t^2}{2} & \text{for all } t \in [0, \tau^*], \\ -(t-1) - \frac{1}{2} & \text{for all } t \in [\tau^*, 2], \end{cases}
$$

with $\tau^* = 1$.

# 5 Conclusion and perspectives

In this work, we have proposed an approximation scheme for optimal control problems on stratified domains. An originality of our approach is the embedding of the original discontinuous dynamics into a standard one via the reformulation of the Filippov mapping associated with the original dynamics. This approach required introducing a boundary layer to bypass the nonsmooth interfaces and to introduce additional control in the dynamics allowing to regularize the dynamics near the interface. These additional control are chosen in order to minimize a penalty function associated to a mixed control-state constraint. Our method offers more
<figure><img src="image_10.png" /><figcaption>**Figure 8:** Comparison with a (standard) regularization approach in solving Problem (P'_{ex}).</figcaption></figure>

flexibility near the interface compared to a standard regularization technique. Indeed, when considering a regularization technique, the problem is that near the interface, the choice of dynamics depends on the type of regularization chosen and the position of the state ([7, 17]). On the other hand, regularization using an additional control is not constrained by the position of the state, which better approximates the dynamics near the interface. The approach we have proposed allows optimal solutions to cross tangentially interfaces between regions. This feature is not covered by the existing literature on necessary optimality conditions for such problems, which require non tangent crossing on at least one side of the interface (see, e.g., the particular case of the minimal time of crisis with semi-transverse crossing [7]). Moreover, we do not require this property to be a priori satisfied on an optimal solution (which is usually not known in advance), but we simply require that the time spent on the boundary layer by the sequence of approximated solutions goes to zero as the “thickness” of the boundary layer goes to zero. The advantage is that this condition can be tested numerically. This is a significant improvement with regards to the more usual transverse condition to be encountered in the literature. Finally, we have seen that this implies that optimal solutions obtained at the limit do not have sliding modes. Our approach allows to apply the necessary optimality conditions on the regularized sequence of optimal control problems, using the Pontryagin Maximin Principle (in the smooth case) and to get informations on the optimal solution to the original problem when passing to the limit. Due to space limitations, this analysis is not included in the present work but will considered in a forthcoming work.

## Acknowledgment

The postdoctoral position of the first author was funded by the French National Research Agency (ANR) under the NOCIME project (ANR-23-CE48-0004-03). The authors are grateful to Florent Nacry for helpful discussions on the subject.

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