sufficiently small. It follows that for some constant $\sigma' > 0$ independent of $h$ and $x^{\text{ref}}$,

$$ \sup\{ \| C^T u^h \| : u^h \in \text{SOL}(C, q^h, D^h) \} \le \sigma' (1 + \| x^{\text{ref}} \|). \quad (9.19) $$

Returning to the above string of expressions, we deduce that we may choose the constant $\sigma'$ sufficiently large so that

$$ \sup\{ v^T u^h : u^h \in \text{SOL}(C, q^h, D^h) \} \le \sigma' (1 + \| x^{\text{ref}} \|). $$

Consider the family of polyhedra with variable right-hand sides:

$$ \mathcal{P}(a, \alpha) \equiv \{ u \in \mathcal{C} : C^T u = a, v^T u = \alpha \}, \quad \text{for } (a, \alpha) \in \mathfrak{R}^n \times \mathfrak{R}_+ $$

By the renowned Hoffman bound for linear inequality systems [56], it follows that there exists a constant $\zeta > 0$, dependent on the triple $(C, C, v)$ only, such that for all pairs $(a, \alpha)$ for which the polyhedron $\mathcal{P}(a, \alpha)$ is nonempty, there exists a vector $u \in \mathcal{P}(a, \alpha)$ such that $\|u\| \le \zeta (\|a\| + \alpha)$. By [38, Corollary 2.3.13]

$$ \text{SOL}(C, q^h, D^h) = \{ u \in \mathcal{C} : C^T u = C^T u^h, v^T u = v^T u^h \}, \quad (9.20) $$

for any $u^h \in \text{SOL}(C, q^h, D^h)$, it follows that for all $h > 0$ sufficiently small and all $x^{\text{ref}} \in \mathfrak{R}^n$, $u^h \in \text{SOL}(C, q^h, D^h)$ exists such that $\|u^h\| \le 2\sigma'\zeta(1 + \|x^{\text{ref}}\|)$. From this bound, it is easy to derive a corresponding bound for $\|x^h - x^{\text{ref}}\|$. The proof that $C^T u^h$ is a single-valued Lipschitz function of $x^{\text{ref}}$ is very similar to the proof of (9.19). Indeed, for any two vectors $q + Cr^i$ for $i = 1, 2$, with $u^i \in \text{SOL}(C, q + Cr^i, D^h)$, we have $\|C^T u^1 - C^T u^2\| \le \sigma^{-1}\|r^1 - r^2\|$. The last assertion of the lemma follows. $\square$

**Remark** The CP (9.18) may have multiple solutions that are possibly unboun-
ded; yet, one exists with the much needed linear growth property.

Based on the above constant $\xi$, we can define the constants $\psi_x > 0$ and $\rho_0$ satisfying (9.10). Moreover, for each $h > 0$ sufficiently small, we can define the scalars $\rho_1, \dots, \rho_{N_h+1}$ recursively by (9.7). The rest of the proof of the following result does not need to be repeated.

**Theorem 9.5** Let $C$ be a polyhedral cone. Suppose that $q \in C\mathfrak{R}^n + C^*$. Assume further that $M+N$ is nonsingular, (9.5) holds, and $C(M+N)^{-1}(b+N\mathfrak{R}^n) \subseteq C^*$. A discrete-time BVP trajectory satisfying, for all $i = 0, 1, \dots, N_h$,

$$ \begin{align*} x^{h,i+1} - x^{h,i} &= h[p + \theta Ax^{h,i} + (1 - \theta)Ax^{h,i+1} + C^T u^{h,i+1}], \\ C \ni u^{h,i+1} &\perp q + Cx^{h,i+1} \in C^*, \\ b &= Mx^{h,0} + N x^{h,N_h+1}, \end{align*} $$

exists; moreover, a subsequence of this trajectory converges to a weak solution of
the boundary-value generalized LCS (9.17).
## 10 The linear-quadratic Nash game revisited

A major effort in the treatment of the boundary-value DVI in the previous section has been placed on the demonstration that the discrete-time subproblems (9.3) and (9.11) have solutions. Such a treatment leads to a condition such as (9.5) that restricts the terminal time $T$ based on several model constants. Once a discrete-time trajectory is known to exist, the convergence of such a trajectory to a continuous-time solution of the boundary-value DVI (9.1) turns out to be not as hard to show; it fact, it can be accomplished rather simply with the aid of the IVP analysis. The linear-quadratic differential Nash game discussed in Sect. 4.2 is an example of such a problem. In what follows, we describe a time-stepping scheme for solving this game for which the solvability of the discrete-time subproblems follows from a known result for a noncooperative Nash game in finite-dimensions (thus, no advanced fixed-point theorem is needed); more importantly, the condition (9.5) can be removed completely.

To simplify the discussion, we focus on the linear-quadratic game and assume that the data functions $d(t)$, $q(t)$, $B(t)$, $C(t)$, and $S(t)$ are all time invariant. The DVI formulation for the resulting linear-quadratic Nash game is a boundary-value DAVI:

$$ \left(\begin{array}{c} \dot{\lambda} \\ \dot{x} \end{array}\right)=\mathbf{d}+\mathbf{A}\left(\begin{array}{c} \lambda \\ \mathbf{x} \end{array}\right)+\mathbf{B} \mathbf{u}, \quad \mathbf{u}(t) \in \operatorname{SOL}(\mathbf{U}, \mathbf{q}+\mathbf{C} \mathbf{z}, \mathbf{D}) $$

with very special boundary conditions: $\mathbf{x}(0) = \mathbf{x}^0$ and $\lambda(T) = \mathbf{W}\mathbf{x}(T) + \mathbf{c}$. The time-stepping scheme computes, for $i = 0, 1, 2, \dots, N_h + 1$, the vectors $\mathbf{x}^{h,i} = (x^{v,h,i})_{v \in \mathcal{N}} \in \Re^n$, $\mathbf{u}^{h,i} = (u^{v,h,i})_{v \in \mathcal{N}} \in \Re^m$, and $\lambda^{h,i} = (\lambda^{v,h,i})_{v \in \mathcal{N}} \in \Re^n$ which are approximations of the state trajectory $\mathbf{x}(t)$, the control trajectory $\mathbf{u}(t)$, and the adjoint trajectory $\lambda(t)$, respectively, at times $t = t_{h,0}, t_{h,1}, t_{h,2}, \dots, t_{h,N_h+1}$. Respecting the initial condition $\mathbf{x}^{h,0} = \mathbf{x}^0$ and the terminal condition $\lambda(T) = \mathbf{c} + \mathbf{W}\mathbf{x}(T)$, (the latter suggests that $\lambda$ should be treated backward in time, which leads to the term $\lambda^{v,h,i}$ in (c) and the right-hand side of (a)), the discrete-time system is as follows: for every $v \in \mathcal{N}$ and for all $i = 0, 1, \dots, N_h$,

$$ (a) \quad \lambda^{v,h,i+1} - \lambda^{v,h,i} = -h \left[ \begin{aligned} & P^v + \sum_{v' \in \mathcal{N}} \left( P^{vv'} x^{v',h,i+1} + Q^{vv'} u^{v',h,i+1} \right) \\ & + P^{vv} x^{v,h,i+1} + (R^{vv})^T u^{v,h,i+1} + (G^v)^T \lambda^{v,h,i} \end{aligned} \right] $$

$$ (b) \quad x^{v,h,i+1} - x^{v,h,i} = h[r^v + G^v x^{v,h,i+1} + H^v u^{v,h,i+1}] $$

$$ (c) \quad u^{v,h,i+1} \in U^v \text{ satisfies: for all } u^v \in U^v, $$

$$ \left( u^v - u^{v,h,i+1} \right)^T \left[ \begin{aligned} & q^v + (H^v)^T \lambda^{v,h,i} + (Q^{vv})^T x^{v,h,i+1} + S^{vv} u^{v,h,i+1} \\ & \sum_{v' \in \mathcal{N}} \left( R^{vv'} x^{v',h,i+1} + S^{vv'} u^{v',h,i+1} \right) \end{aligned} \right] \geq 0, $$
$$
(d) \quad \lambda^{\nu,h,N_h+1} = c^{\nu} + W^{\nu\nu} x^{\nu,h,N_h+1} + \sum_{v' \in \mathcal{N}} W^{\nu\nu'} x^{\nu',h,N_h+1},
$$

$$
(e) \quad x^{\nu,h,0} = x^{\nu,0}.
$$

We claim that the pair $(\mathbf{x}^h, \mathbf{u}^h)$, where $\mathbf{x}^h \equiv (\mathbf{x}^{h,i})_{i=1}^{N_h+1}$ and $\mathbf{u}^h \equiv (\mathbf{u}^{h,i})_{i=1}^{N_h+1}$,
constitutes an equilibrium to a discrete-time Nash game, with $\boldsymbol{\lambda}^h \equiv (\boldsymbol{\lambda}^{h,j})_{j=0}^{N_h}$
being the vector of KKT multipliers of the players' dynamic constraints in
discrete time. The latter static game has the same number of players; player
$v$'s optimization problem is in terms of the variables $(x^{\nu,h,i}, u^{\nu,h,i})_{i=1}^{N_h+1}$, with the
rival players' variables $(x^{-\nu,h,i}, u^{-\nu,h,i})_{i=1}^{N_h+1}$ taken as exogenous:

$$
\begin{equation}
\begin{split}
\text{minimize} \quad \theta_v(\mathbf{x}^h, \mathbf{u}^h) &\equiv (x^{\nu,h,N_h+1})^T \left[ c^\nu + \sum_{v' \in \mathcal{N}} W^{\nu\nu'} x^{\nu',h,N_h+1} \right] \\
&\quad + h \sum_{i=0}^{N_h} \left[ (x^{\nu,h,i+1})^T \left\{ p^\nu + \sum_{v' \in \mathcal{N}} \left( P^{\nu\nu'} x^{\nu',h,i+1} + Q^{\nu\nu'} u^{\nu',h,i+1} \right) \right\} \right. \\
&\quad \left. + (u^{\nu,h,i+1})^T \left\{ q^\nu + \sum_{v' \in \mathcal{N}} \left( R^{\nu\nu'} x^{\nu',h,i+1} + S^{\nu\nu'} u^{\nu',h,i+1} \right) \right\} \right] \quad (10.1)
\end{split}
\end{equation}
$$

$$
\begin{array}{ll}
\text{subject to} & x^{\nu,h,0} = x^{\nu,0} \quad \text{and for } i = 0, 1, \dots, N_h, \\
& x^{\nu,h,i+1} - x^{\nu,h,i} = h \left( r^{\nu} + G^{\nu} x^{\nu,h,i+1} + H^{\nu} u^{\nu,h,i+1} \right), \\
& u^{\nu,h,i+1} \in U^{\nu}.
\end{array}
$$

When each $U^v$ is a polytope, Proposition 10.1 establishes the above claim, from
which the existence of a tuple $(\mathbf{x}^h, \mathbf{u}^h, \boldsymbol{\lambda}^h)$ satisfying the discrete-time system
(a)–(e) follows readily.

**Proposition 10.1** Let each $U^v$ be a nonempty compact polyhedron in $\mathbb{R}^{m_v}$.
Assume that for all $v \in \mathcal{N}$, $W^{vv}$ and

$$
\Xi^v \equiv \left[ \frac{2P^{vv}}{(Q^{vv})^T + R^{vv}} \quad \frac{Q^{vv} + (R^{vv})^T}{2S^{vv}} \right] \quad (10.2)
$$

are symmetric positive semidefinite. A scalar $\bar{h} > 0$ exists such for all $h \in (0, \bar{h}]$,
a tuple $(\mathbf{x}^h, \mathbf{u}^h, \boldsymbol{\lambda}^h)$ exists satisfying the discrete-time system (a)–(e).

Proof For fixed $(x^{-\nu,h,i}, u^{-\nu,h,i})_{i=1}^{N_h+1}$, (10.1) is a convex quadratic program in player $v$'s variables $(x^{\nu,h,i}, u^{\nu,h,i})_{i=1}^{N_h+1}$, which can be reformulated in terms of $(u^{\nu,h,i})_{i=1}^{N_h+1}$ only by solving for $x^{\nu,h,i+1}$ from the difference equation in the constraints of (10.1). Indeed, for all $h > 0$ sufficiently small, we have

$$
x^{\nu,h,i+1} = (I - h G^{\nu})^{-1} \left[ x^{\nu,h,i} + h (r^{\nu} + H^{\nu} u^{\nu,h,i+1}) \right], \quad i = 0, 1, \dots, N_h.
$$
Since $x^{\nu,h,0}$ is known, it follows by induction that the variables $(x^{\nu,h,i})_{i=1}^{N_h+1}$ can
be expressed as a linear function of $(u^{\nu,h,i})_{i=1}^{N_h+1}$. Hence, after substituting the
former variables into the objective function of (10.1), we obtain a quadratic
program in the variables $(u^{\nu,h,i})_{i=1}^{N_h+1}$, which remains convex and has an optimal
solution, by the compactness of each $U^{\nu}$. Therefore, by a well-known existence
result for a standard, static Nash game with compact convex strategy sets, such
as [38, Proposition 2.2.9], it follows that the discrete-time Nash problem has an
equilibrium solution $\{(x^{\nu,h,i}, u^{\nu,h,i})_{i=1}^{N_h+1}\}_{\nu \in \mathcal{N}}$. Letting $\lambda^{\nu,h,i}$ be the KKT multi-
plier of the constraint: $x^{\nu,h,i+1} - x^{\nu,h,i} = h(r^{\nu} + G^{\nu}x^{\nu,h,i+1} + H^{\nu}u^{\nu,h,i+1})$ for $i =
0, 1, \dots, N_h$, and defining $\lambda^{\nu,h,N_h+1} = c^{\nu} + W^{\nu\nu}x^{\nu,h,N_h+1} + \sum_{\nu'=\mathcal{N}} W^{\nu\nu'}x^{\nu',h,N_h+1}$,
we can readily show that the tuple $(x^h, u^h, \lambda^h)$ satisfies the desired discrete-time
system (a)-(e). $\square$

Since $U^\nu$ is bounded and $u^{\nu,h,i+1} \in U^\nu$, it follows that the assumptions of Lemma 7.2 are readily satisfied. The convergence of the above time-stepping method for solving the linear-quadratic differential Nash game therefore follows from Theorem 7.1, provided that in addition the matrix $D$ is positive semidefinite. The details are omitted.

11 Concluding remarks

In this paper, we have undertaken an extensive study on the DVI and several
of its special cases, including the generalized LCS. These problems have the
general form of a differential equation where a term in the right-hand side is a
solution of a VI/CP. Depending on the structure of the VI and the differential
equation, the regularity of the solutions and their theory can vary considerably.
In the simplest case, the VI has a (locally or globally) unique solution that
depends in a Lipschitz fashion on the data of the VI. This leads to Lipschitz
ODEs for which there is a well-known theory. If, however, there are many
solutions then we can use the theory of differential inclusions. Our theory goes
beyond these cases to situations where the VI does not always have solutions,
and where the solutions can be unbounded. Then our existence theory depends
on the joint structure of the differential equation and the VI. To prove these
results we often rely on the use of basic time-stepping schemes, whose extension
to boundary-value problems is also analyzed. There remain many issues that
require further investigation. These include the efficient numerical solution of
the discrete-time subproblems and the application of our theory to engineering
and economic problems in which dynamics is an important concern.

**Acknowledgments** The authors are very grateful to two referees and the Associate Editor for their many insightful and constructive comments. Both referees have identified numerous technical details that have helped to clarify the proofs and improve the presentation of the paper; their diligence and great care in reviewing our lengthy work is greatly appreciated. One referee has kindly reminded us of related references on the VIE and stressed that the class of DVIs treated in the paper is of index no higher than 2. As seen in Sect. 2.4, the VIE is a particular realization of the DVI. The two problems can be unified by the common formulation (2.12), which was observed
by Professor Hans Schumacher of Tilburg University, who has made other valuable comments that we have incorporated in the revision of the paper. We owe these individuals our sincere gratitude. Finally, we thank Professors Seetharama Gowda of the University of Maryland Baltimore County and Michael Ferris of the University of Wisconsin at Madison for providing us with the references [14] and [11], respectively.

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