Optimal Positional Strategies in Differential Games for Neutral-Type Systems

Abstract

The paper deals with a zero-sum differential game for a dynamical system described by neutral-type functional-differential equations in Hale’s form with initial conditions determined by piecewise continuous functions. It is proved that the differential game has a value and optimal positional (feedback) players’ strategies. If the value functional satisfies certain smoothness conditions, the optimal strategies are constructed based on its gradient. In the general case, such strategies are described using quasi-gradient constructions. The fact that the quasi-gradients under consideration require looking for extremum points only on a finite-dimensional set is the crucial contribution of this paper.

1 Introduction

In differential games for dynamical systems described by ordinary differential equations satisfying the Isaacs condition [11] (or the saddle point condition in a small game in other terminology [14]), it is known (see, e.g., [14, 25]) that the value of the game exists and can be achieved by positional (feedback) players’ strategies. If the value function is continuously differentiable, then the optimal positional strategies can be obtained utilizing its gradient. If the differentiability of this function is not assumed, they can be constructed by various regularizing tools [2, 6, 14, 22, 25, 27]. In particular, under fairly general conditions, the optimal positional strategies can utilize the quasi-gradients of the value function [25]. This paper aims to describe optimal positional strategies for more general differential games in which dynamical systems are described by a functional-differential equation of neutral-type in Hale’s from [10].

Such equations represent a fairly general class of functional-differential systems which contain not only a delay in the state vector but also a delay in its derivative. They arise in studying, for example, transmission line nonlinear oscillators [4, 5], torsional motions of driven drill strings [1] and other applications (see [13]). As a quality index for the differential games under consideration, we choose a Boltz cost functional, which is quite a typical choice for the differential games [14, 25] and natural for applications. In a particular and often used case, it estimates the distance from the target point at the terminal time and the integral of the players’ control cost.

Note that the usage of the previously developed optimal positional strategies in differential games for both time-delay [15,16,17, 23] and neutral-type [9, 18] functional-differential systems is problematic since their application requires looking for extremum points on infinite-dimensional sets of continuous functions (possible histories of system motions). However, recent papers [20, 21] devoted to optimal control problems and differential games for time-delay systems established that it is possible to look for extremum points on finite-dimensional sets if motion histories are piecewise continuous functions. Therefore, starving to obtain a similar result in differential games for neutral-type systems, we also consider a motion history space with jumps. Namely, following [24], we choose the space of piecewise Lipschitz continuous functions. In particular, paper [24] established the uniqueness of a generalized (minimax and viscosity) solution of the Hamilton-Jacobi equations arising from control problems for neural-type systems in this space. Thus, proving that the upper and lower values of the game in the class of non-anticipatory strategies are the viscosity solution of the corresponding Hamilton-Jacobi equation, we obtain the value of the game exists (Theorem 1).

Next, we describe the smoothness conditions under which optimal positional strategies can be constructed based on the gradient of the value functional (Theorem 2). Note that, in contrast to time-delay systems (see [16, 17]), we cannot use the ci-smoothness condition since, even in the simplest case of neutral-type systems, the gradient is not continuous (see [8]). Instead, we use conditions considering possible discontinuities of the gradient following [8]. However, the choice of the motion history space with jumps allows us to obtain more general smoothness conditions than in [8].

Finally, thanks to such a choice of the motion history space, for the general case when the value functional is not smooth, we prove the optimality of positional strategies based, in fact, on the classical quasi-gradient definition [25] (Theorem 3). Let us emphasize again that, unlike previous papers [9, 18] devoted to optimal positional strategies for neutral-type systems, such quasi-gradient constructions requires looking for extremum points only on a finite-dimensional set, which is the crucial contribution of this paper.

Note also that the usage of the motion history space with jumps creates various additional difficulties in proofs. Namely, motions of neutral-type systems on such space have a certain periodicity of jumps during the control interval (see Remark 1), which is not typical for time-delay systems, for example. In addition, the value functional has rather specific continuity properties (see condition $(\rho _1)$ and $(\rho _2)$) that are different from [9, 18], where Lipschitz motion histories were considered. Nonetheless, the accounting of this specificity in the proofs allows us to obtain the above results.

2 Results

2.1 Functional Spaces

Let $\mathbb R^n$ be the n-dimensional Euclidean space with the inner product $\langle \cdot , \cdot \rangle$ and the norm $\Vert \cdot \Vert$. A function $x(\cdot ) :[a,b) \mapsto \mathbb R^n$ (or $x(\cdot ) :[a,b] \mapsto \mathbb R^n$) is called piecewise Lipschitz continuous if there exist points $a = \xi _0< \xi _1< \ldots < \xi _{I+1} = b$ such that the function $x(\cdot )$ is Lipschitz continuous on the interval $[\xi _i,\xi _{i+1})$ for each $i \in \overline{0,I}$. Note that such a function $x(\cdot )$ is right continuous on [a, b) and has a finite left limit $x(\xi -0)$ for any $\xi \in (a,b]$. Denote by $\textrm{PLip}([a,b),\mathbb R^n)$ and $\textrm{Lip}([a,b),\mathbb R^n)$ the linear spaces of piecewise Lipschitz and Lipschitz continuous functions $x(\cdot ) :[a,b) \mapsto \mathbb R^n$, respectively.

Let $\vartheta , h > 0$. Without loss of generality of the results presented below, we can suppose the existence of $J \in \mathbb N$ such that $\vartheta = J h$. For the sake of brevity, we set $\textrm{PLip} = \textrm{PLip}([-h,0),\mathbb R^n)$ and, for any $w(\cdot ) \in \textrm{PLip}$, we denote

$$\begin{aligned} \Vert w(\cdot )\Vert _1 = \int _{-h}^0 \Vert w(\xi )\Vert \textrm{d} \xi ,\quad \Vert w(\cdot )\Vert _\infty = \sup \limits _{\xi \in [-h,0)} \Vert w(\xi )\Vert ,\quad w(-0) = w(0 - 0). \end{aligned}$$

Following [24], define the space $\textrm{PLip}_*$ of functions $w(\cdot ) \in \textrm{PLip}$ continuously differentiable on $[-h,-h+\delta _w]$ for some $\delta _w > 0$ and the spaces

$$\begin{aligned} \mathbb G = [0,\vartheta ] \times \mathbb R^n \times \textrm{PLip},\quad \mathbb G_* = \cup _{j=0}^{J-1} (j h, (j+1) h) \times \mathbb R^n \times \textrm{PLip}_*. \end{aligned}$$

(1)

2.2 Differential Game

For each $(\tau ,z,w(\cdot )) \in \mathbb G$, consider a zero-sum differential game for a dynamical system described by the neutral-type differential equation in Hale’s form [10]

$$\begin{aligned} \begin{array}{c} \displaystyle \frac{d}{d t}\Big (x(t) - g(t,x(t-h))\Big ) = f(t,x(t),x(t-h),u(t),v(t)),\quad t \in [\tau ,\vartheta ],\\ x(t) \in \mathbb R^n,\quad u(t) \in \mathbb U \subset \mathbb R^l,\quad v(t) \in \mathbb V \subset \mathbb R^m, \end{array} \end{aligned}$$

(2)

with the initial condition

$$\begin{aligned} x(\tau ) = z,\quad x(t) = w(t - \tau ),\quad t \in [\tau -h,\tau ), \end{aligned}$$

(3)

and the quality index

$$\begin{aligned} \gamma = \sigma (x(\vartheta ),x_\vartheta (\cdot )) + \int \limits _\tau ^\vartheta f^0(t,x(t),x(t-h),u(t),v(t)) d t, \end{aligned}$$

(4)

Here t is the time variable; x(t) is the state vector at the time t; u(t) and v(t) are control actions of the first and second players, respectively; $\mathbb U$ and $\mathbb V$ are compact sets. Hereinafter, the symbol $x_t(\cdot )$ denotes the function on the interval $[-h,0)$ defined by $x_t(\xi ) = x(t + \xi )$, $\xi \in [-h,0)$.

In this differential game, the first player aims to minimize $\gamma$, while the second player aims to maximize it.

We assume that the following conditions hold:

$(g_1)$:

The function g is continuously differentiable.

$(g_2)$:

There exists a constant $c_g > 0$ such that

$$\begin{aligned} \Vert g(t,x)\Vert \le c_g \big (1 + \Vert x\Vert \big ),\quad (t,x) \in [0,\vartheta ] \times \mathbb R^n. \end{aligned}$$

$(f_1)$:

The functions f and $f^0$ are continuous.

$(f_2)$:

There exists a constant $c_f > 0$ such that

$$\begin{aligned} \big \Vert f(t,x,y,u,v)\big \Vert + \big \vert f^0(t,x,y,u,v) \big \vert \le c_f \big (1 + \Vert x\Vert + \Vert y\Vert \big ) \end{aligned}$$

for any $t \in [0,\vartheta ]$, $x,y \in \mathbb R^n$, $u \in \mathbb U$, and $v \in \mathbb V$.

$(f_3)$:

For every $\alpha > 0$, there exists a number $\lambda _f = \lambda _f(\alpha ) > 0$ such that

$$\begin{aligned} \begin{array}{l} \big \Vert f(t,x,y,u,v) - f(t,x',y',u,v)\big \Vert + \big \vert f^0(t,x,y,u,v) - f^0(t,x',y',u,v) \big \vert \\ \hspace{3cm} \le \lambda _f \big (\Vert x - x'\Vert + \Vert y - y'\Vert \big ) \end{array} \end{aligned}$$

for any $t \in [0,\vartheta ]$, $x,y,x',y' \in B(\alpha ) = \{x \in \mathbb R^n :\Vert x\Vert \le \alpha \}$, $u \in \mathbb U$, and $v \in \mathbb V$.

$(f_4)$:

The equality

$$\begin{aligned} \min \limits _{u \in \mathbb U} \max \limits _{v \in \mathbb V} \chi (t,x,y,u,v,s) = \max \limits _{v \in \mathbb V} \min \limits _{u \in \mathbb U} \chi (t,x,y,u,v,s) \end{aligned}$$

holds for any $t \in [0,\vartheta ]$ and $x,y,s \in \mathbb R^n$, where

$$\begin{aligned} \chi (t,x,y,u,v,s) = \langle f(t,x,y,u,v),s \rangle + f^0(t,x,y,u,v). \end{aligned}$$

(5)

$(\sigma _1)$:

For every $\alpha > 0$, there exists $\lambda _\sigma = \lambda _\sigma (\alpha ) > 0$ such that

$$\begin{aligned} \big \vert \sigma (x,r(\cdot )) - \sigma (x',r'(\cdot )) \big \vert \le \lambda _\sigma \big (\Vert x - x'\Vert + \Vert r(\cdot ) - r'(\cdot )\Vert _1\big ) \end{aligned}$$

for any $(x,r(\cdot )),(x',r'(\cdot )) \in P(\alpha )$, where

$$\begin{aligned} P(\alpha ) = \big \{(x,r(\cdot )) \in \mathbb R^n \times \textrm{PLip} :\Vert x\Vert \le \alpha ,\, \Vert r(\cdot )\Vert _\infty \le \alpha \big \}. \end{aligned}$$

(6)

$(\sigma _2)$:

There exists $c_\sigma > 0$ such that

$$\begin{aligned} \big \vert \sigma (x,r(\cdot )) \big \vert \le c_\sigma \big (1 + \Vert x\Vert + \Vert r(\cdot )\Vert _\infty \big ),\quad (x,r(\cdot )) \in \mathbb R^n \times \textrm{PLip}. \end{aligned}$$

Note that these conditions are quite typical for differential games theory [11, 14, 25]. In particular, condition $(f_4)$, called the Isaacs’s condition [11] or the saddle point condition in a small game in other terminology [14, 25], is crucial for proving the existence of a value (see Theorem 1 below).

Define the set of piecewise Lipschitz continuous right extensions from the point $(\tau ,z,w(\cdot ))$ as follows:

$$\begin{aligned} \Lambda (\tau ,z,w(\cdot )) = \big \{x(\cdot ) \in \textrm{PLip}([\tau -h,\vartheta ],\mathbb R^n) :x(\tau ) = z,\, x_\tau (\cdot ) = w(\cdot )\big \}. \end{aligned}$$

By admissible control realizations of the first and second players, we mean Lebesgue measurable functions $u(\cdot ) :[\tau ,\vartheta ] \mapsto \mathbb U$ and $v(\cdot ) :[\tau ,\vartheta ] \mapsto \mathbb V$, respectively. Denote by $\mathcal {U}_\tau$ and $\mathcal {V}_\tau$ the sets of admissible control realizations of the first and second players. Under conditions $(g_1)$ and $(f_1)-(f_3)$, following, for example, the scheme from [7, Section 7] (see also [12, Section 4.2]), one can show that each pair of realizations $u(\cdot ) \in \mathcal {U}_\tau$ and $v(\cdot ) \in \mathcal {V}_\tau$ uniquely generates the motion $x(\cdot ) = x(\cdot \,\vert \,\tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$ of system (2), (3) that is the function from $\Lambda (\tau ,z,w(\cdot ))$ such that the function $x(t) - g(t,x(t-h))$, $t \in [\tau ,\vartheta ]$ is Lipschitz continuous and $x(\cdot )$ satisfies Eq. (2) almost everywhere.

Remark 1

Note that the motions of system (2), (3) have a certain structure of Lipschitz continuous pieces (and discontinuity points). Namely, if $-h = \xi _0< \xi _1< \ldots < \xi _{I+1} = 0$ such that the function $w(\cdot )$ is Lipschitz continuous on $[\xi _i,\xi _{i+1})$, $i \in \overline{0,I}$ then the motion $x(\cdot ) = x(\cdot \,\vert \,\tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$ is Lipschitz continuous on the intervals $[\tau + \xi _i + j h, \tau + \xi _{i+1} + j h) \cap [\tau ,\vartheta ]$, $i \in \overline{0,I}$, $j \in \overline{0,J}$. This fact can be proved similar to Proposition 8.

We first consider differential game (2)–(4) in classes of non-anticipative strategies of players (see, e.g. [2, Chapter VIII, Section 1]) or quasi-strategies in another terminology (see, e.g. [25, Chapter III, Section 14.2]).

By a non-anticipative strategy of the first player, we mean a mapping $Q^u_\tau :\mathcal {V}_\tau \mapsto \mathcal {U}_\tau$ such that, for each $v(\cdot ),v'(\cdot ) \in \mathcal {V}_\tau$ and $t \in [\tau ,\vartheta ]$, if the equality $v(\xi ) = v'(\xi )$ is valid for a.e. $\xi \in [\tau ,t]$ then the equality $Q^u_\tau [v(\cdot )](\xi ) = Q^u_\tau [v'(\cdot )](\xi )$ holds for a.e. $\xi \in [\tau ,t]$.

A non-anticipative strategy of the first player $Q^u_\tau$ and a control realization of the second player $v(\cdot ) \in \mathcal {V}_\tau$ define the control realization of the first player $u(\cdot ) = Q^u_\tau [v(\cdot )](\cdot )$, the motion $x(\cdot ) = x(\cdot \,\vert \,\tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$ and the value $\gamma = \gamma (\tau ,z,w(\cdot ),Q^u_\tau ,v(\cdot ))$ of quality index (4). The lower value of differential game (2)–(4) is defined by

$$\begin{aligned} \rho ^u(\tau ,z,w(\cdot )) = \inf \limits _{Q^u_\tau } \sup \limits _{v(\cdot ) \in \mathcal {V}_\tau } \gamma (\tau ,z,w(\cdot ),Q^u_\tau ,v(\cdot )). \end{aligned}$$

(7)

The functional $\mathbb G \ni (\tau ,z,w(\cdot )) \mapsto \rho ^u = \rho ^u(\tau ,z,w(\cdot )) \in \mathbb R$ is the lower value functional of differential game (2)–(4).

Similarly, a non-anticipative strategy of the second player is a mapping $Q^v_\tau :\mathcal {U}_\tau \mapsto \mathcal {V}_\tau$ such that, for each $u(\cdot ),u'(\cdot ) \in \mathcal {U}_\tau$ and $t \in [\tau ,\vartheta ]$, if the equality $u(\xi ) = u'(\xi )$ is valid for a.e. $\xi \in [\tau ,t]$ then the equality $Q^v_\tau [u(\cdot )](\xi ) = Q^v_\tau [u'(\cdot )](\xi )$ holds for a.e. $\xi \in [\tau ,t]$. Such a non-anticipative strategy together with $u(\cdot ) \in \mathcal {U}_\tau$ define the realization $v(\cdot ) = Q^v_\tau [u(\cdot )](\cdot )$, the motion $x(\cdot ) = x(\cdot \,\vert \,\tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$, and the quality index $\gamma = \gamma (\tau ,z,w(\cdot ),u(\cdot ),Q^v_\tau )$. The upper value of differential game (2)–(4) is

$$\begin{aligned} \rho ^v(\tau ,z,w(\cdot )) = \sup \limits _{Q^v_\tau } \inf \limits _{u(\cdot ) \in \mathcal {U}_\tau } \gamma (\tau ,z,w(\cdot ),u(\cdot ),Q^v_\tau ). \end{aligned}$$

(8)

The functional $\mathbb G \ni (\tau ,z,w(\cdot )) \mapsto \rho ^v = \rho ^v(\tau ,z,w(\cdot )) \in \mathbb R$ is the upper value functional of differential game (2)–(4).

Note that the functionals $\rho ^u$ and $\rho ^v$ satisfy the following conditions:

$(\rho _0)$:

The equality $\rho (\vartheta ,z,w(\cdot )) = \sigma (z,w(\cdot ))$, $(z,w(\cdot )) \in \mathbb R^n \times \textrm{PLip}$ holds.

$(\rho _1)$:

For each pair $(\tau ,w(\cdot )) \in [0,\vartheta ] \times \textrm{Lip}([-h,0),\mathbb R^n)$, the function $\hat{\rho }(t) = \rho (t,w(-0),w(\cdot ))$ is continuous on $[\tau ,\vartheta ]$.

$(\rho _2)$:

For every $\alpha > 0$, there exists $\lambda _\rho = \lambda _\rho (\alpha ) > 0$ such that

$$\begin{aligned} \big \vert \rho (\tau ,z,w(\cdot )) - \rho (\tau ,z',w'(\cdot )) \big \vert \le \lambda _\rho \upsilon (\tau ,z-z',w(\cdot ) - w'(\cdot )) \end{aligned}$$

(9)

for any $\tau \in [0,\vartheta ]$ and $(z,w(\cdot )), (z',w'(\cdot )) \in P(\alpha )$, where

$$\begin{aligned} \upsilon (\tau ,z,w(\cdot )) = \Vert z\Vert + \Vert w(\cdot )\Vert _1 + \Vert w(-h)\Vert + \Vert w(j h - \tau )\Vert \end{aligned}$$

(10)

in which $j \in \overline{-1,J-1}$ is such that $\tau \in (j h, (j+1) h]$.

$(\rho _3)$:

For every $(\tau ,z,w(\cdot )) \in \mathbb G$, $\tau < \vartheta$, $\zeta > 0$, $t_* \in (\tau ,\vartheta ]$, and $v(\cdot ) \in \mathcal {V}_\tau$, there exists $u(\cdot ) \in \mathcal {U}_\tau$ such that the motion $x(\cdot ) = x(\cdot \,\vert \,\tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$ satisfies the estimate

$$\begin{aligned} \rho (t_*,x(t_*),x_{t_*}(\cdot )) + \int _{\tau }^{t_*} f^0(t,x(t),x(t-h),u(t),v(t)) d t \le \rho (\tau ,z,w(\cdot )) + \zeta . \end{aligned}$$

$(\rho _4)$:

For every $(\tau ,z,w(\cdot )) \in \mathbb G$, $\tau < \vartheta$, $\zeta > 0$, $t_* \in (\tau ,\vartheta ]$, and $u(\cdot ) \in \mathcal {U}_\tau$, there exists $v(\cdot ) \in \mathcal {V}_\tau$ such that the motion $x(\cdot ) = x(\cdot \,\vert \,\tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$ satisfies the estimate

$$\begin{aligned} \rho (t_*,x(t_*),x_{t_*}(\cdot )) + \int _{\tau }^{t_*} f^0(t,x(t),x(t-h),u(t),v(t)) d t \ge \rho (\tau ,z,w(\cdot )) - \zeta . \end{aligned}$$

The fulfillment of terminal condition $(\rho _0)$ follows directly from definitions (7) and (8) of value functionals $\rho ^u$ and $\rho ^v$. Conditions $(\rho _1)$, $(\rho _2)$ naturally generalize the continuity properties of value functionals to the case of neural-type systems considered on a set $\mathbb G$ (see [24, Remark 2]) and are proved in Proposition 10. Conditions $(\rho _3)$, $(\rho _4)$ describe the dynamic programming principle for the functionals $\rho ^u$ and $\rho ^v$ and can be proved following the scheme from [2, Chapter VIII, Theorem 1.9] if we take into account that any non-anticipative strategy $Q^u_\tau$ defines a rule—for every $v(\cdot ) \in \mathcal {V}_\tau$ there exists $u(\cdot ) \in \mathcal {U}_\tau$ and similarly for $Q^v_\tau$. Note also the fact that system (2) has a delay does not in any way affect the proof scheme.

2.3 Hamilton–Jacobi Equation

In this section, we consider the corresponding to differential game (2)–(4) Hamilton-Jacobi equation with coinvariant derivatives to prove the existence of the value and other auxiliary properties.

Following [24] (see also [12, 15]), a functional $\rho :\mathbb G \mapsto \mathbb R$ is called coinvariantly (ci-) differentiable at a point $(\tau ,z,w(\cdot )) \in \mathbb G$, $\tau < \vartheta$ if there exist $\partial ^{ci}_{\tau ,w}\rho (\tau ,z,w(\cdot )) \in \mathbb R$ and $\nabla _z\rho (\tau ,z,w(\cdot )) \in \mathbb R^n$ such that, for every $t \in [\tau ,\vartheta ]$, $y \in \mathbb R^n$, and $x(\cdot ) \in \Lambda (\tau ,z,w(\cdot ))$, the relation below holds

$$\begin{aligned} \begin{array}{c} \rho (t,y,x_t(\cdot )) - \rho (\tau ,z,w(\cdot )) = (t - \tau ) \partial ^{ci}_{\tau ,w}\rho (\tau ,z,w(\cdot )) \\ + \langle y - z, \nabla _z \rho (\tau ,z,w(\cdot )) \rangle + o(\vert t - \tau \vert + \Vert y - z\Vert ), \end{array} \end{aligned}$$

(11)

where the value $o(\delta )$ can depend on $x(\cdot )$ and $o(\delta )/\delta \rightarrow 0$ as $\delta \downarrow 0$. Then $\partial ^{ci}_{\tau ,w}\rho (\tau ,z,w(\cdot ))$ is called the ci-derivative of $\rho$ with respect to $\{\tau ,w(\cdot )\}$ and $\nabla _z \rho (\tau ,z,w(\cdot ))$ is the gradient of $\rho$ with respect to z.

Denote

$$\begin{aligned} \begin{array}{c} G(t,x,y) = \partial g(t,x) / \partial t + \nabla _x g(t,x) y \\ H(t,x,y,s) = \min \limits _{u \in \mathbb U}\max \limits _{v \in \mathbb V} \chi (t,x,y,u,v,s) \end{array} \quad t \in [0,\vartheta ],\quad x,y,s \in \mathbb R^n, \end{aligned}$$

(12)

and consider the Cauchy problem for the Hamilton-Jacobi equation

$$\begin{aligned} \begin{array}{l} \partial ^{ci}_{\tau ,w} \rho (\tau ,z,w(\cdot )) + \langle G(\tau ,w(-h),d^+ w(-h) / d \xi ), \nabla _z \rho (\tau ,z,w(\cdot )) \rangle \\ \hspace{2cm}+ H(\tau ,z,w(-h),\nabla _z \rho (\tau ,z,w(\cdot ))) = 0,\quad (\tau ,z,w(\cdot )) \in \mathbb G_*, \end{array} \end{aligned}$$

(13)

and the terminal condition

$$\begin{aligned} \rho (\vartheta ,z,w(\cdot )) =\sigma (z,w(\cdot )),\quad (z,w(\cdot )) \in \mathbb R^n \times \textrm{PLip}, \end{aligned}$$

(14)

where $d^+ w(-h) / d \xi$ is the right derivative of the function $w(\xi )$, $\xi \in [-h,0)$ at the point $\xi =-h$. The properties and singularities of such Cauchy problems were studied in [24]. In particular, this paper proves the existence and uniqueness of the generalized (minimax or viscosity) solution of such a problem. Thus, showing in Proposition 11 that the both functionals $\rho ^u$ and $\rho ^v$ are the viscosity solution of problem (13), (14), we obtain the following statement.

Theorem 1

Differential game (2)–(4) has the value functional

$$\begin{aligned} \rho ^\circ (\tau ,z,w(\cdot )) = \rho ^u(\tau ,z,w(\cdot )) = \rho ^v(\tau ,z,w(\cdot )),\quad (\tau ,z,w(\cdot )) \in \mathbb G. \end{aligned}$$

(15)

Note also that from this theorem and [24, Theorem 5], we get the following auxiliary property which directly connects $\rho ^\circ$ and equation (13):

$(\rho _5)$:

Let the value functional $\rho ^\circ$ be ci-differentiable on $\mathbb G_*$. Then $\rho ^\circ$ satisfies Hamilton-Jacobi equation (13) for any $(\tau ,z,w(\cdot )) \in \mathbb G_*$.

2.4 Optimal Positional Strategies

In this section, we introduce the concept of positional (feedback) players’ strategies following the scheme from [14] (see also [16] for time-delay systems). Note that the positional strategies are a particular case of non-anticipative strategies, and therefore, their guaranteed results cannot be better than values (7) and (8). Below, we present positional strategies capable of providing precisely these values (i.e. optimal strategies), which is the main result of this paper. Namely, following the differential game theory [25], firstly, we describe optimal strategies based on the value functional gradient if it satisfies certain smoothness conditions, are secondly, we present strategies utilizing quasi-gradients for the general case.

By a positional strategy of the first player, we mean an arbitrary function $U:\mathbb G \mapsto \mathbb U$. Let us fix $(\tau ,z,w(\cdot )) \in \mathbb G$ and a partition of the interval $[\tau ,\vartheta ]$:

$$\begin{aligned} \Delta _\delta = \{t_j :t_1 = \tau ,\, 0 < t_{j+1} - t_{j} \le \delta ,\, j = \overline{1,k},\, t_{k+1} = \vartheta \}. \end{aligned}$$

(16)

The pair $\{U,\Delta _\delta \}$ defines a control law that forms a piecewise constant function $u(\cdot ) \in \mathcal {U}_\tau$ according to the following step-by-step rule:

$$\begin{aligned} u(t) = U(t_j,x(t_j),x_{t_j}(\cdot )),\quad t \in [t_j, t_{j+1}),\quad j=\overline{1,k}. \end{aligned}$$

(17)

This control law together with any function $v(\cdot ) \in \mathcal {V}_\tau$ uniquely determine the motion $x(\cdot ) = x(\cdot \,\vert \,\tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$ and the quality index $\gamma = \gamma (t,z,w(\cdot ), U, \Delta _\delta , v(\cdot ))$ of quality index (4). The guaranteed result of the strategy U is defined by

$$\begin{aligned} \rho _u(\tau ,z,w(\cdot ),U) = \displaystyle \lim _{\delta \downarrow 0} \sup _{\Delta _\delta } \sup _{v(\cdot )} \gamma (\tau ,z,w(\cdot ),U,\Delta _\delta ,v(\cdot )). \end{aligned}$$

(18)

Similarly, with the corresponding changes, for the second player, we define a positional control strategy $V:\mathbb G \mapsto \mathbb V$, control law $\{V,\Delta _\delta \}$ that forms a function $v(\cdot ) \in \mathcal {V}_\tau$ by

$$\begin{aligned} v(t) = V(t_j,x(t_j),x_{t_j}(\cdot )),\quad t\in [t_j, t_{j+1}),\quad j=\overline{1,k}, \end{aligned}$$

the guaranteed result of the strategy V

$$\begin{aligned} \rho _v(\tau ,z,w(\cdot ),V) = \displaystyle \lim _{\delta \downarrow 0} \, \inf _{\Delta _\delta } \, \inf _{u(\cdot )} \gamma (\tau ,z,w(\cdot ), u(\cdot ), V, \Delta _\delta ). \end{aligned}$$

(19)

Theorem 2

Let the value functional $\rho ^\circ = \rho ^\circ (t,z,w(\cdot ))$ is differentiable by z on $\mathbb G$ and ci-differentiable on $\mathbb G_*$ (see (1)). Let the functional $\nabla _z \rho ^\circ = \nabla _z \rho ^\circ (\tau ,z,w(\cdot ))$ satisfy the condition

$(\rho ^*_1)$:

For each $(\tau ,w(\cdot )) \in [0,\vartheta ] \times \textrm{Lip}$, the function $\hat{\rho }(t) = \nabla _z \rho ^\circ (t,w(-0),w(\cdot ))$ is continuous on $[\tau ,\vartheta ] \cap (j h, (j + 1) h)$ for each $j \in \overline{0,J-1}$.

and condition $(\rho _2)$. Then, for every $(\tau ,z,w(\cdot )) \in \mathbb G$, the players’ strategies

$$\begin{aligned} \begin{array}{c} U(t,x,r(\cdot )) \in \mathop {\textrm{argmin}}\limits _{u \in \mathbb U} \max \limits _{v \in \mathbb V} \chi \big (t,x,r(-h),u,v, \nabla _z\rho ^\circ (t,x,r(\cdot ))\big ) \\ V(t,x,r(\cdot )) \in \mathop {\textrm{argmax}}\limits _{v \in \mathbb V} \min \limits _{u \in \mathbb U} \chi \big (t,x,r(-h),u,v, \nabla _z\rho ^\circ (t,x,r(\cdot ))\big ) \end{array} \end{aligned}$$

provide the equalities

$$\begin{aligned} \rho _u(\tau ,z,w(\cdot ),U) = \rho ^\circ (\tau ,z,w(\cdot )) = \rho _v(\tau ,z,w(\cdot ),V). \end{aligned}$$

(20)

Thus, the positional strategies U and V described above are optimal.

The simplest example of a differential game in which the conditions of Theorem 2 are satisfied can be found in [8]. This example also forces us to use a weaker condition $(\rho _1^*)$ than $(\rho _1)$ in this theorem because we can see that even in such straightforward cases, condition $(\rho _1)$ for the value functional gradient does not hold.

Next, determine the optimal strategies for the general case following [25]. For every $\lambda ,\varepsilon > 0$, denote

$$\begin{aligned} \begin{array}{c} \eta ^{\lambda ,\varepsilon }(t,x) = \theta ^{\lambda ,\varepsilon }(t) \mu ^{\varepsilon }(x),\quad \theta ^{\lambda ,\varepsilon }(t) = (e^{- \lambda t} - \varepsilon ) / \varepsilon ,\\ \mu ^{\varepsilon }(x) = \sqrt{\varepsilon ^4 + \Vert x\Vert ^2}, \end{array} \quad (t,x) \in [0,\vartheta ] \times \mathbb R^n \end{aligned}$$

(21)

and consider the functionals

$$\begin{aligned} \begin{array}{c} p^{\lambda ,\varepsilon }(t,x,r(\cdot )) = \mathop {\textrm{argmin}}\limits _{p \in \mathbb R^n} \big (\rho ^\circ (t,p,r(\cdot )) + \eta ^{\lambda ,\varepsilon }(t,x - p)\big ), \\ q^{\lambda ,\varepsilon }(t,x,r(\cdot )) = \mathop {\textrm{argmax}}\limits _{q \in \mathbb R^n} \big (\rho ^\circ (t,q,r(\cdot )) - \eta ^{\lambda ,\varepsilon }(t,x - q)\big ), \end{array} \end{aligned}$$

(22)

where $(t,x,r(\cdot )) \in \mathbb {G}$. Proposition 14 proves the ${{\,\textrm{argmin}\,}}$ and ${{\,\textrm{argmax}\,}}$ values are archived for sufficiently small $\varepsilon$ and therefore, these functionals are well defined. Let us consider the quasi-gradients

$$\begin{aligned} \begin{array}{c} \nabla ^{\lambda , \varepsilon , +}_z\rho ^\circ (t,x,r(\cdot )) = \nabla _z\eta ^{\lambda ,\varepsilon }(t,x - p^{\lambda ,\varepsilon }(t,x,r(\cdot ))), \\ \nabla ^{\lambda , \varepsilon , -}_z \rho ^\circ (t,x,r(\cdot )) = \nabla _z\eta ^{\lambda ,\varepsilon }(t,x - q^{\lambda ,\varepsilon }(t,x,r(\cdot ))), \end{array} \end{aligned}$$

(23)

and describe the optimal strategies that does not require additional smoothness conditions for the value functional.

Theorem 3

For every $(\tau ,z,w(\cdot )) \in \mathbb G$ and $\zeta > 0$, there exist $\lambda > 0$ such that the players’ strategies

$$\begin{aligned} \begin{array}{c} U^{\lambda , \varepsilon }(t,x,r(\cdot )) \in \mathop {\textrm{argmin}}\limits _{u \in \mathbb U} \max \limits _{v \in \mathbb V} \chi \big (t,x,r(-h),u,v, \nabla ^{\lambda , \varepsilon , +}_z \rho ^\circ (t,x,r(\cdot ))\big ) \\ V^{\lambda , \varepsilon }(t,x,r(\cdot )) \in \mathop {\textrm{argmax}}\limits _{v \in \mathbb V} \min \limits _{u \in \mathbb U} \chi \big (t,x,r(-h),u,v, \nabla ^{\lambda ,\varepsilon ,-}_z\rho ^\circ (t,x,r(\cdot ))\big ) \end{array} \end{aligned}$$

(24)

provide the equities

$$\begin{aligned} \begin{array}{c} \underset{\varepsilon \downarrow 0}{\overline{\lim }} \ \rho _u(\tau ,z,w(\cdot ),U^{\lambda , \varepsilon }) = \rho ^\circ (\tau ,z,w(\cdot )) = \underset{\varepsilon \downarrow 0}{\underline{\lim }} \ \rho _v(\tau ,z,w(\cdot ),V^{\lambda , \varepsilon }). \end{array} \end{aligned}$$

(25)

Note that, in contrast to Theorem 2, Theorem 3 describes strategies providing $\rho ^\circ (\tau ,z,w(\cdot ))$ only in the limit. Nonetheless, this result is typical for the theory of differential games (see, e.g., [25, Section 12.2]) since, even for ordinary differential equations, universal positional strategies (i.e. positional strategies independent of a particular initial condition) that provide the value without $\varepsilon$ may not exist [26].

3 Proof

3.1 Properties of the Dynamical System

In this section, we give some properties of dynamical system (2). Proposition 4 follows directly from condition $(g_1)$. Proposition 6 was proved in [24, Lemma 1]. The proofs of the remaining propositions are given below.

Proposition 4

For every $\alpha > 0$, there exists $\lambda _g = \lambda _g(\alpha ) > 0$ such that

$$\begin{aligned} \big \vert g(t,x) - g(t',x') \big \vert \le \lambda _g \big (\vert t - t' \vert + \Vert x - x'\Vert \big ) \end{aligned}$$

for any $t, t' \in [0,\vartheta ]$ and $x, x' \in B(\alpha ):= \{x \in \mathbb R^n :\Vert x\Vert \le \alpha \}$.

Proposition 5

There exists $c_X > 0$ such that, for every $(\tau ,z,w(\cdot )) \in \mathbb G$, $u(\cdot ) \in \mathcal {U}_\tau$, and $v(\cdot ) \in \mathcal {V}_\tau$, the motion $x(\cdot ) = x(\cdot \,\vert \,\tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$ satisfies the estimate

$$\begin{aligned} \Vert x(t)\Vert \le c_X\big (1 + \Vert z\Vert + \Vert w(\cdot )\Vert _\infty \big ),\quad t \in [\tau ,\vartheta ]. \end{aligned}$$

(26)

Proof

Define $c_g, c_f > 0$ according to conditions $(g_2)$, $(f_2)$. Denote $c_* = 1 + 2 c_g + 2 \vartheta$ and put $c_X = c_*^{\,J+1} e^{c_f \vartheta }$, where J is from (1). Then, due to (2), we have

$$\begin{aligned} \Vert x(t)\Vert \le c_g (1 + \Vert x(t-h)\Vert ) + \Vert z\Vert + c_g (1 + \Vert w(-h)\Vert ) + c_f \int \limits _\tau ^t (1 + \Vert x(\xi )\Vert + \Vert x(\xi -h)\Vert \big ) d \xi \end{aligned}$$

for any $t \in [\tau , \vartheta ]$. Define the function $\kappa (t) = 1 + \max \{\Vert x(\xi )\Vert \,\vert \, \xi \in [\tau -h, t]\}$, $t \in [\tau ,\vartheta ]$. Denote $t_j = \min \{\tau + j h,\vartheta \}$, $j \in \overline{0,J}$. Then, we derive

$$\begin{aligned} \kappa (t) \le c_* \kappa (t_j) + c_f \int \limits _{t_j}^{t} \kappa (\xi ) d \xi ,\quad t \in [t_j, t_{j+1}],\quad j \in \overline{0,J-1}. \end{aligned}$$

From this estimate, applying the method of mathematical induction and Gronwall–Bellman Lemma (see, e.g., [3, p. 31]), we obtain the estimate

$$\begin{aligned} \kappa (t) \le c_*^{\,j+1} \kappa (\tau ) e^{c_f (t - \tau )},\quad t \in [t_j, t_{j+1}],\quad j \in \overline{0,J-1}. \end{aligned}$$

(27)

which implies (26). $\square$

Proposition 6

For every $\alpha > 0$, there exist $\alpha _X = \alpha _X(\alpha ) > 0$, $\alpha ^g_X = \alpha ^g_X(\alpha ) > 0$, and $\lambda ^g_X = \lambda ^g_X(\alpha ) > 0$ such that, for each $\tau \in [0,\vartheta ]$, $(z,w(\cdot )) \in P(\alpha )$ (see (6)), $u(\cdot ) \in \mathcal {U}_\tau$, and $v(\cdot ) \in \mathcal {V}_\tau$, the motion $x(\cdot ) = x(\cdot \,\vert \,\tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$ and the function $x^g(t) = x(t) - g(t,x(t-h))$, $t \in [\tau ,\vartheta ]$, satisfy the relations

$$\begin{aligned} (x(t),x_t(\cdot )) \in P(\alpha _X),\ \ \Vert x^g(t)\Vert \le \alpha ^g_X,\ \ \Vert x^g(t) - x^g(t')\Vert \le \lambda ^g_X \vert t - t' \vert ,\ \ t,t' \in [\tau ,\vartheta ]. \end{aligned}$$

Proposition 7

For every $\alpha > 0$, there exist $\lambda _{XX} = \lambda _{XX}(\alpha ) > 0$ such that, for every $\tau \in [0,\vartheta ]$, $(z,w(\cdot )), (p,r(\cdot )) \in P(\alpha )$, $u(\cdot ) \in \mathcal {U}_\tau$, and $v(\cdot ) \in \mathcal {V}_\tau$, the motions $x(\cdot ) = x(\cdot \vert \tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$ and $y(\cdot ) = x(\cdot \vert \tau ,p,r(\cdot ),u(\cdot ),v(\cdot ))$ satisfy the inequality

$$\begin{aligned} \Vert x(t) - y(t)\Vert + \int _{\tau -h}^t \Vert x(\xi ) - y(\xi )\Vert d \xi \le \lambda _{XX} \upsilon (\tau ,z-p,w(\cdot )-r(\cdot )), \quad t \in [\tau ,\vartheta ]. \end{aligned}$$

Proof

Let $\alpha > 0$. According to Propositions 4, 6 and condition $(f_3)$, define the numbers $\alpha _X = \alpha _X(\alpha ) > 0$, $\lambda _g = \lambda _g(\alpha _X) > 1$, and $\lambda _f = \lambda _f(\alpha _X) > 0$. Define also the numbers $\lambda ^g_0 = 1$, $\lambda ^s_0 = 1$, and

$$\begin{aligned} \lambda ^g_j = (1 + \lambda _f (2 + \lambda _g) \lambda ^s_{j-1}) e^{\lambda _f h},\quad \lambda ^s_j = 1 + j h \lambda _j^g + \lambda _g \lambda ^s_{j-1},\quad j \in \overline{1,J}. \end{aligned}$$

(28)

Put $\lambda _{XX} = (J \lambda ^J_g \lambda _J^g +\lambda _J^s) \lambda _g$.

Let $\tau \in [0,\vartheta ]$, $(z,w(\cdot )), (p,r(\cdot )) \in P(\alpha )$, $u(\cdot ) \in \mathcal {U}_\tau$, and $v(\cdot ) \in \mathcal {V}_\tau$. Define the motions $x(\cdot ) = x(\cdot \vert \tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$ and $y(\cdot ) = x(\cdot \vert \tau ,p,r(\cdot ),u(\cdot ),v(\cdot ))$. Denote

$$\begin{aligned} s(t) = x(t) - y(t),\quad t \in [\tau -h,\vartheta ],\ \ s^g(t) = s(t) - g(t,x(t-h) + g(t,y(t-h)),\ \ t \in [\tau ,\vartheta ]. \end{aligned}$$

Let us prove the estimates

$$\begin{aligned} \Vert s^g(t)\Vert \le \lambda ^g_j \big (\Vert s^g(\tau )\Vert + \Vert s_\tau (\cdot )\Vert _1\big ),\quad \int _{\tau - h}^{t} \Vert s(\xi )\Vert d \xi \le \lambda ^s_j \big (\Vert s^g(\tau )\Vert + \Vert s_\tau (\cdot )\Vert _1\big ) \end{aligned}$$

(29)

for any $t \in [\tau , \tau + j h] \cap [\tau ,\vartheta ]$ and $j \in \overline{0,J}$. Note that these estimates hold for $j=0$. Following the method of mathematical induction, we assume that these estimates are proved for $j-1$ and prove them for j. Due to the definitions of $x(\cdot )$, $y(\cdot )$, $s(\cdot )$, and $\lambda _g$, $\lambda _f$, using the second estimate in (29) for $j-1$, we derive

$$\begin{aligned} \begin{array}{c} \Vert s^g(t)\Vert \le \Vert s^g(\tau )\Vert + \lambda _f \displaystyle \int \limits _{\tau +(j-1)h}^{t} \Vert s^g(\xi )\Vert d \xi + \lambda _f (2 + \lambda _g) \displaystyle \int \limits _{\tau -h}^{\tau +(j-1)h} \Vert s(\xi )\Vert d \xi \\ \le (1 + \lambda _f (2 + \lambda _g) \lambda ^s_{j-1}) \big (\Vert s^g(\tau )\Vert + \Vert s_\tau (\cdot )\Vert _1\big ) + \lambda _f \displaystyle \int \limits _{\tau +(j-1)h}^{t} \Vert s^g(\xi )\Vert d \xi . \end{array} \end{aligned}$$

Then, applying Gronwall–Bellman Lemma (see, e.g., [3, p. 31]), we get the first estimate in (29) on $[\tau +(j-1)h, \tau +jh]$. Since $\lambda ^g_j > \lambda ^g_i$ for any $i \in \overline{0,j-1}$, this estimate also holds on $[\tau , \tau + jh]$. The second estimate for j follows from the relations

$$\begin{aligned} \begin{array}{c} \displaystyle \int _{\tau -h}^{t} \Vert s(\xi )\Vert d \xi \le \Vert s_\tau (\cdot )\Vert _1 + \int _\tau ^{t} \Vert s^g(\xi )\Vert d \xi + \lambda _g \int _{\tau -h}^{t-h} \Vert s(\xi )\Vert d \xi \\ \le \Vert s_\tau (\cdot )\Vert _1 + j h \lambda ^g_j \big (\Vert s^g(\tau )\Vert + \Vert s_\tau (\cdot )\Vert _1\big ) + \lambda _g \lambda ^s_{j-1} \big (\Vert s^g(\tau )\Vert + \Vert s_\tau (\cdot )\Vert _1\big ), \end{array} \end{aligned}$$

in which, we use the choice of $\lambda _g$, the first estimate in (29) for j, and the second estimate in (29) for $j-1$. Thus, we have proved (29) for any $j \in \overline{0,J}$.

Let $t \in [\tau ,\vartheta ]$. Let $j_t \in \overline{0,J}$ be such that $t - (j_t+1) h \in [\tau - h, \tau )$. Then, applying (29) for $i \in \overline{0,j_t}$, taking into account the choice of $\lambda _g$ and $\lambda _{XX}$, we conclude

$$\begin{aligned} \Vert s(t)\Vert + \int _{\tau -h}^t \Vert s(\xi )\Vert d \xi \le \sum _{i=0}^{j_t} \lambda _g^i\Vert s^g(t - i h)\Vert + \lambda _g^{j_t+1}\Vert s^g(t - (j_*+1) h)\Vert + \int _{\tau -h}^t \Vert s(\xi )\Vert d \xi \end{aligned}$$

$$\begin{aligned} \le (J \lambda ^J_g \lambda _J^g +\lambda _J^s) \big (\Vert s^g(\tau )\Vert + \Vert s_\tau (\cdot )\Vert _1\big ) + \lambda _g^J\Vert s^g(\vartheta - (j+1) h)\Vert \le \lambda _{XX} \upsilon (\tau ,s(\tau ),s_\tau (\cdot )) \end{aligned}$$

and, hence, prove the proposition. $\square$

Let $(\tau ,w(\cdot )) \in [0,\vartheta ] \times \textrm{PLip}$. Let $-h = \xi _0< \xi _1< \ldots < \xi _{I+1} = 0$ be such that the function $w(\cdot )$ is Lipschitz continuous on $[\xi _i,\xi _{i+1})$ for each $i \in \overline{0,I}$. Denote $j_\tau \in \overline{0,J}$ such that $j_\tau h - \tau \in [-h,0)$. Then, without loss of generality, we can assume that $\xi _i = j_\tau h - \tau$ for some $i \in \overline{0,I}$. Denote

$$\begin{aligned} \Theta _\nu (\tau ,w(\cdot )) = \big \{[\tau + \xi _i +j h, \tau + \xi _{i+1} + j h - \nu ) \cap [\tau ,\vartheta ) :i \in \overline{0,I},\, j \in \overline{1,J}\big \}, \end{aligned}$$

(30)

where $\nu \in [0,\nu _*)$ and $\nu _* = \min \{(\xi _{i+1} - \xi _{i}) \vert i \in \overline{0,I}\}/2$.

Proposition 8

For each $w(\cdot ) \in \textrm{PLip}$, there exists $\lambda _X = \lambda _X(w(\cdot )) > 0$ such that, for every $\tau \in [0,\vartheta ]$, $z \in \mathbb R^n$, $u(\cdot ) \in \mathcal {U}_\tau$, and $v(\cdot ) \in \mathcal {V}_\tau$, the motion $x(\cdot ) = x(\cdot \,\vert \,\tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$ satisfies the inequality

$$\begin{aligned} \Vert x(t) - x(t')\Vert + \Vert x(t-h) - x(t'-h)\Vert \le \lambda _X \vert t - t' \vert ,\quad t,t' \in \theta ,\quad \theta \in \Theta _0(\tau ,w(\cdot )). \end{aligned}$$

Proof

Due to the inclusion $x_\tau (\cdot ) = w(\cdot ) \in \textrm{PLip}$, there exists $\lambda _w > 0$ such that

$$\begin{aligned} \Vert x(t-h) - x(t'-h)\Vert = \Vert w(t-\tau -h) - w(t'-\tau -h)\Vert \le \lambda _w \vert t - t'\vert \end{aligned}$$

for any $t,t' \in [\xi _{i},\xi _{i+1})$ and $i \in \overline{0,I}$. Let $\alpha > 0$. Taking $\alpha _X = \alpha _X(\alpha ) > 0$, $\lambda ^g_X = \lambda ^g_X(\alpha ) > 0$, and $\lambda _g = \lambda _g(\alpha _X) > 0$ from Propositions 6 and 4, respectively, define

$$\begin{aligned} \lambda _0 = \lambda _w,\quad \lambda _{j + 1} = \lambda _X^g + (1 + \lambda _g) \lambda _{j},\quad j \in \overline{0,J},\quad \lambda _X = \lambda _J. \end{aligned}$$

Then, denoting $x^g(t) = x(t) - g(t,x(t-h))$, we derive

$$\begin{aligned} \begin{array}{c} \Vert x(t) - x(t')\Vert + \Vert x(t-h) - x(t'-h)\Vert \\ \le \Vert x^g(t) + x^g(t)\Vert + (1 + \lambda _g) \Vert x(t-h) - x(t'-h)\Vert \le \lambda ^g_X + (1 + \lambda _g) \lambda _j = \lambda _{j+1} \end{array} \end{aligned}$$

for any $t, t' \in [\tau + \xi _i +j h, \tau + \xi _{i+1} + j h)$, $i \in \overline{0,I}$, and $j \in \overline{1,J}$. Since $\lambda _X = \lambda _J > \lambda _j$, $j \in \overline{0,J}$, from this inequalty, we get the statement of the proposition. $\square$

3.2 Property of the Value Functional

Proposition 9

Let $\rho$ satisfy $(\rho ^*_1)$ and $(\rho _2)$. Let $(\tau ,z,w(\cdot )) \in \mathbb G$ and $\nu \in (0, \nu _*)$. Then, for every $\zeta > 0$, there exists $\delta > 0$ such that, for every $u(\cdot ) \in \mathcal {U}_\tau$ and $v(\cdot ) \in \mathcal {V}_\tau$, the motion $x(\cdot ) = x(\cdot \vert \tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$ satisfies the inequality

$$\begin{aligned} \big \vert \rho (t,x(t),x_{t}(\cdot )) - \rho (t',x(t'),x_{t'}(\cdot )) \big \vert \le \zeta \end{aligned}$$

for any $t,t' \in \theta$: $\vert t - t' \vert \le \delta$, and $\theta \in \Theta _\nu (\tau ,w(\cdot ))$.

Proof

For the sake of contradiction, we assume the existence of $\zeta > 0$, $u^m(\cdot ) \in \mathcal {U}_\tau$, $v^m(\cdot ) \in \mathcal {V}_\tau$, $\theta _m \in \Theta$, and $t_m, t'_m \in \theta _m$ such that $\vert t_m - t'_m \vert \le 1 / m$ and the motion $x^m(\cdot ) = x(\cdot \vert \tau ,z,w(\cdot ),u^m(\cdot ),v^m(\cdot ))$ satisfies

$$\begin{aligned} \big \vert \rho (t_m,x^m(t_m),x^m_{t_m}(\cdot )) - \rho (t'_m,x^m(t'_m),x^m_{t'_m}(\cdot )) \big \vert > \zeta . \end{aligned}$$

(31)

Without loss of generality, taking into account definition (30) of $\Theta _\nu (\tau ,w(\cdot ))$, we can assume the existence of $t_* \in [\tau ,\vartheta ]$ such that $\vert t_* - t_m \vert + \vert t_* - t'_m \vert \le 1/m$, $m \in \mathbb N$ and $t_m, t'_m, t_* \in \theta _*$ for some $\theta _* \in \Theta _\nu (\tau ,w(\cdot ))$. Moreover, in accordance with (10) and Proposition 8, we can assume the existence of $x^*(\cdot ) \in \textrm{Lip}([\tau -h,T], \mathbb R^n)$ such that

$$\begin{aligned} \upsilon (t,x^m(t) - x^*(t),x^m_t(\cdot )- x^*_t(\cdot )) \le 1/m,\quad t \in \theta _*,\quad m \ge \in N. \end{aligned}$$

Let $\lambda _*$ be a Lipschitz constant of $x^*(\cdot )$. Then, due to (10) and Proposition 8, we have

$$\begin{aligned} \begin{array}{l} \upsilon (t_m,x^*(t_m) - x^*(t_*),x^*_{t_m}(\cdot )- x^*_{t_*}(\cdot )) = \Vert x^*(t_m) - x^*(t_*)\Vert \\ \quad + \Vert x^*(t_m-h) - x^*(t_*-h)\Vert + \Vert x^*_{t_m}(\cdot )- x^*_{t_*}(\cdot )\Vert _1 \le (2 + h) \lambda _* / m, \end{array} \quad m \in \mathbb N. \end{aligned}$$

According to $(\rho ^*_1)$, there exists $M_* > 0$ such that

$$\begin{aligned} \Vert \rho (t_m,x^*(t_*),x^*_{t_*}(\cdot )) - \rho (t_*,x^*(t_*),x^*_{t_*}(\cdot ))\Vert \le \zeta /4,\quad m \ge M_*. \end{aligned}$$

Thus, defining $\alpha = \max \{\Vert z\Vert ,\Vert w(\cdot )\Vert _\infty \}$ and taking $\alpha _X = \alpha _X(\alpha )$ and $\lambda _\rho = \lambda _\rho (\alpha _X)$ in accordance with Propositions 6 and condition $(\rho _2)$, we derive

$$\begin{aligned} \big \Vert \rho (t_m,x^m(t_m),x^m_{t_m}(\cdot )) - \rho (t_*,x^*(t_*),x^*_{t_*}(\cdot )) \big \Vert \le \zeta / 4 + \lambda _\rho (1 + (2 + h) \lambda _*) / m \le \zeta /2 \end{aligned}$$

for any $m > M_1:= \max \{M_*, 4 \lambda _\rho (1 + (2 + h) \lambda _*) / \zeta \}$.

By the same way, we can find $M_2 > 0$ such that

$$\begin{aligned} \big \Vert \rho (t'_m,x^m(t'_m),x^m_{t'_m}(\cdot )) - \rho (t_*,x^*(t_*),x^*_{t_*}(\cdot )) \big \Vert \le \zeta /2,\quad m > M_2, \end{aligned}$$

and, thus, obtain the contradiction with (31). $\square$

Proposition 10

The functionals $\rho ^u$ and $\rho ^v$ satisfy conditions $(\rho _1)$ and $(\rho _2)$.

Proof

We prove the statement only for $\rho ^u$ since it is proved similarly for the $\rho ^v$.

First, we prove that $\rho ^u$ satisfies condition $(\rho _2)$. Let $\alpha > 0$. According to Propositions 6, 7 and conditions $(f_3)$, $(\sigma _1)$, define $\alpha _X = \alpha _X(\alpha ) > 0$, $\lambda _{XX} = \lambda _{XX}(\alpha _X) > 0$, $\lambda _f = \lambda _f(\alpha _X) > 0$, and $\lambda _\sigma = \lambda _\sigma (\alpha _X) > 0$. Put $\lambda _\rho = (\lambda _\sigma + 2 \lambda _f) \lambda _{XX}$. To prove that $\rho ^u$ satisfies condition $(\rho _2)$, it suffices to show the inequality

$$\begin{aligned} \rho ^u(\tau ,p,r(\cdot )) - \rho ^u(\tau ,z,w(\cdot )) \le \lambda _\rho \upsilon (\tau ,z-p,w(\cdot )-r(\cdot )) + \zeta , \end{aligned}$$

for any $\tau \in [0,\vartheta ]$, $(z,w(\cdot )), (p,r(\cdot )) \in P(\alpha )$, and $\zeta > 0$.

Let us take $\tau \in [0,\vartheta ]$, $(z,w(\cdot )), (p,r(\cdot )) \in P(\alpha )$, and $\zeta > 0$. According to definition (18) of $\rho ^u$, there exists $\hat{Q}^u_\tau$ such that

$$\begin{aligned} \sup \limits _{v(\cdot ) \in \mathcal {V}_\tau } \gamma (\tau ,z,w(\cdot ),\hat{Q}^u_\tau , v(\cdot )) \le \rho ^u(\tau ,z,w(\cdot )) + \zeta / 2, \end{aligned}$$

there exists $\hat{v}(\cdot ) \in \mathcal {V}_\tau$ such that

$$\begin{aligned} \sup \limits _{v(\cdot ) \in \mathcal {V}_\tau } \gamma (\tau ,p,r(\cdot ), \hat{Q}^u_\tau , v(\cdot )) \le \gamma (\tau ,p,r(\cdot ), \hat{Q}^u_\tau , \hat{v}(\cdot )) + \zeta / 2, \end{aligned}$$

and therefore we have

$$\begin{aligned} \rho ^u(\tau ,p,r(\cdot )) - \rho ^u(\tau ,z,w(\cdot )) \le \gamma (\tau ,p,r(\cdot ), \hat{Q}^u_\tau , \hat{v}(\cdot )) - \gamma (\tau ,z,w(\cdot ), \hat{Q}^u_\tau , \hat{v}(\cdot )) + \zeta . \end{aligned}$$

Define $\hat{u}(\cdot ) = \hat{Q}^u_\tau [\hat{v}(\cdot )](\cdot )$, the motions $x(\cdot ) = x(\cdot \,\vert \,\tau ,z,w(\cdot ),\hat{u}(\cdot ),\hat{v}(\cdot ))$ and $y(\cdot ) = x(\cdot \,\vert \,\tau ,p,r(\cdot ),\hat{u}(\cdot ),\hat{v}(\cdot ))$, and the function $s(t) = x(t) - y(t)$, $t \in [\tau -h,\vartheta ]$. Then, due to the definition (4) of $\gamma$ and the choice of the numbers $\lambda _\sigma$, $\lambda _f$, and $\lambda _{XX}$, we derive

$$\begin{aligned} \gamma (\tau ,p,r(\cdot ), \hat{Q}^u_\tau , \hat{v}(\cdot )) - \gamma (\tau ,z,w(\cdot ), \hat{Q}^u_\tau , \hat{v}(\cdot ))\le & {} \lambda _\sigma \Vert s(\vartheta )\Vert + (\lambda _\sigma + 2 \lambda _f) \displaystyle \int _{\tau -h}^\vartheta \Vert s(t)\Vert d t \\\le & {} \lambda _\rho \upsilon (\tau ,z-p,w(\cdot ) - r(\cdot )). \end{aligned}$$

Thus, we has shown that $\rho ^u$ satisfies $(\rho _2)$.

Now, let us prove that the functional $\rho ^u$ satisfies condition $(\rho _1)$. Let $(\tau ,w(\cdot )) \in [0,\vartheta ] \times \textrm{Lip}$. Let us show the function $\hat{\rho }^u(t) = \rho ^u(t,w(-0),w(\cdot ))$ is uniformly continuous on $[\tau ,\vartheta ]$. Let $\zeta > 0$. According to Proposition 6 and conditions $(f_2)$ and $(\rho _2)$ proved above, define $\alpha _X(\alpha _0) > 0$, $c_f > 0$, and $\lambda _\rho (\alpha _X) > 0$, where $\alpha _0 = \Vert w(\cdot )\Vert _\infty$. Note also that, since $w(\cdot ) \in \textrm{Lip}$ due to [19, Lemma 3], there exists $\lambda _X = \lambda _X(w(\cdot )) > 0$ such that, for every $t \in [\tau ,\vartheta ]$, $u(\cdot ) \in \mathcal {U}_t$, and $v(\cdot ) \in \mathcal {V}_t$, the motion $x(\cdot ) = x(\cdot \,\vert \,t,w(-0),w(\cdot ),u(\cdot ),v(\cdot ))$ satisfies

$$\begin{aligned} \big \Vert x(\xi ') - x(\xi )\big \Vert \le \lambda _X \vert \xi ' - \xi \vert ,\quad \xi ,\xi ' \in [t-h,\vartheta ]. \end{aligned}$$

Put $\delta = \zeta / 3 \max \{\lambda _\rho (2 + h) \lambda _X, c_f (1 + 2 \alpha _X)\}$.

Let $t,t_* \in [\tau ,\vartheta ]$ be such that $\vert t - t_* \vert \le \delta$. Without loss of generality, suppose that $t \le t_*$. Let $v(\cdot ) \in \mathcal {V}_\tau$. According to $(\rho _3)$, there exists $u(\cdot ) \in \mathcal {U}_\tau$ such that, for the motion $x(\cdot ) = x(\cdot \,\vert \,t,w(-0),w(\cdot ),u(\cdot ),v(\cdot ))$, we have

$$\begin{aligned} \rho ^u(t_*,x(t_*),x_{t_*}(\cdot )) + \int \limits _t^{t_*} f^0(\xi ,x(\xi ),x_\xi (\cdot ),u(\xi ),v(\xi )) d \xi \le \rho ^u(t,w(-0),w(\cdot )) + \zeta / 3. \end{aligned}$$

Due to the choice of $\lambda _\rho$, $\lambda _X$, and $\delta$, we derive

$$\begin{aligned}{} & {} \vert \rho ^u(t_*,x(t_*),x_{t_*}(\cdot )) - \rho ^u(t_*,w(-0),w(\cdot )) \vert \\{} & {} \quad \le \lambda _\rho \upsilon \big (t_*, x(t_*) - w(-0), x_{t_*}(\cdot ) - w(\cdot )\big ) \le \lambda _\rho (2 + h) \lambda _X \delta \le \zeta / 3. \end{aligned}$$

According to the choice of $\alpha _X$, $c_f$, and $\delta$, we get

$$\begin{aligned} \int \limits _{t}^{t_*} \big \vert f^0(\xi ,x(\xi ),x_\xi (\cdot ),u(\xi ),v(\xi )) \big \vert d \xi \le c_f(1 + 2 \alpha _X)(t_* - t) \le \zeta / 3. \end{aligned}$$

Thus, we obtain $\rho ^u(t_*,w(-0),w(\cdot )) - \rho ^u(t,w(-0),w(\cdot )) \le \zeta$. The inequality $\rho ^u(t_*,w(-0),w(\cdot )) - \rho ^u(t,w(-0),w(\cdot )) \ge - \zeta$ can be proved in the similar way, using $(\rho _4)$ instead of $(\rho _3)$. $\square$

The subdifferential of a functional $\rho :\mathbb G \mapsto \mathbb R$ at a point $(\tau ,z,w(\cdot )) \in \mathbb G$, $\tau < \vartheta$ is a set, denoted by $D^-(\tau ,z,w(\cdot ))$, of pairs $(p_0,p) \in \mathbb R \times \mathbb R^n$ such that

$$\begin{aligned} \lim \limits _{\delta \rightarrow 0} \inf \limits _{(t,x) \in O^+_\delta (\tau ,z)} \!\!\frac{\varphi (t,x,\kappa _t(\cdot )) - \varphi (\tau ,z,w(\cdot )) - (t - \tau ) p_0 - \langle x - z, p \rangle }{\vert t - \tau \vert + \Vert x - z\Vert } \ge 0, \end{aligned}$$

(32)

where $\kappa (t) = w(t - \tau )$, $t \in [\tau -h,\tau )$, $\kappa (t) = z$, $t \in [\tau ,\vartheta ]$ and $O^+_\delta (\tau ,z) = \{(t,x) \in [\tau ,\vartheta ] \times \mathbb R^n :t \in [\tau , \tau + \delta ], \Vert x - z\Vert \le \delta \}$. The superdifferential of a functional $\varphi :\mathbb G \mapsto \mathbb R$ at a point $(\tau ,z,w(\cdot )) \in \mathbb G$, $\tau < \vartheta$ is a set, denoted by $D^+(\tau ,z,w(\cdot ))$, of pairs $(q_0,q) \in \mathbb R \times \mathbb R^n$ such that

$$\begin{aligned} \lim \limits _{\delta \rightarrow 0} \sup \limits _{(t,x) \in O^+_\delta (\tau ,z)} \!\!\frac{\varphi (t,x,\kappa _t(\cdot )) - \varphi (\tau ,z,w(\cdot )) - (t - \tau ) q_0 - \langle x - z, q \rangle }{\vert t - \tau \vert + \Vert x - z\Vert } \le 0. \end{aligned}$$

(33)

Proposition 11

Let a functional $\rho :\mathbb G \mapsto \mathbb R$ satisfy conditions $(\rho _2)$–$(\rho _4)$. Then, for every $(\tau ,z,w(\cdot )) \in \mathbb G_*$, the following inequalities holds:

$$\begin{aligned} \begin{array}{c} p_0 + \langle G(\tau ,w(-h),d^+ w(-h) / d \xi ), p \rangle \\ \quad \hspace{3cm}+ H(\tau ,z,w(-h),p) \le 0,\quad (p_0,p) \in D^-\rho (\tau ,z,w(\cdot )) \\ q_0 + \langle G(\tau ,w(-h),d^+ w(-h) / d \xi ), q \rangle \\ \quad \hspace{3cm}+ H(\tau ,z,w(-h),q) \ge 0,\quad (q_0,q) \in D^+\rho (\tau ,z,w(\cdot )) \end{array} \end{aligned}$$

(34)

Proof

We prove only the first inequality from (34) since the second one can be shown similarly. Let $(\tau , z, w(\cdot )) \in \mathbb G_*$ and $(p_0, p) \in D^-\rho (\tau , z, w(\cdot ))$. Note that the estimate

$$\begin{aligned} p_0 + \langle G(\tau ,w(-h),\textrm{d}^+ w(-h) / \textrm{d} \xi ), p \rangle + H(\tau ,z,w(-h),p) \le \zeta \end{aligned}$$

(35)

for any $\zeta > 0$ implies the first inequality from (34).

Let $\zeta > 0$. According to Proposition 8, define $\lambda _X = \lambda _X(w(\cdot )) > 0$. Due to definition (32) of $D^-\rho (\tau , z, w(\cdot ))$, there exists $\delta > 0$ such that

$$\begin{aligned} \frac{\rho (t,x,\kappa _t(\cdot )) - \rho (\tau ,z,w(\cdot )) - (t - \tau ) p_0 - \langle x - z, p \rangle }{\vert t - \tau \vert + \Vert x - z\Vert } \ge - \frac{\zeta }{3(1 + \lambda _X)} \end{aligned}$$

(36)

for every $(t,x) \in O^+_\delta (\tau ,z)$. Since $w(\cdot ) \in \textrm{PLip}_*$ (see (1)), taking into account (5), (12), and $(\rho _2)$, there exists $t \in (\tau ,\tau + \delta / (1 + \lambda _X))$ such that, for every $u(\cdot ) \in \mathcal {U}_\tau$ and $v(\cdot ) \in \mathcal {V}_\tau$, the motion $x(\cdot ) = x(\cdot \,\vert \,\tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$ satisfies the estimates

$$\begin{aligned} \begin{array}{l} \big \vert \chi (\tau ,z,w(-h),u,v) - \chi (\xi ,x(\xi ),x(\xi -h),u,v) \big \vert \le \zeta / 6, \\ \quad \big \vert G(\tau ,w(-h),d^+ w(-h) / d \xi ) - G(\xi ,x(\xi -h),d^+ x(\xi -h) / d \xi ) \big \vert \le \zeta / 6, \\ \quad \,\big \vert \rho (\xi ,x(\xi ),\kappa _\xi (\cdot )) - \rho (\xi ,x(\xi ),x_\xi (\cdot ))\big \vert \le (\xi - \tau ) \zeta / 3 \end{array} \end{aligned}$$

(37)

for any $\xi \in [\tau ,t]$, $u \in \mathbb U$, and $v \in \mathbb V$. Define

$$\begin{aligned} v_* \in \mathop {\textrm{argmax}}\limits _{v \in \mathbb V} \min \limits _{u \in \mathbb U} \chi \big (\tau ,z,w(-h),u,v,p\big ). \end{aligned}$$

(38)

Due to condition $(\rho _3)$, there exists $u(\cdot ) \in \mathcal {U}_\tau$ such that the motion $x(\cdot ) = x(\cdot \,\vert \,\tau ,z,w(\cdot ),u(\cdot ),v(\cdot )=v_*)$ satisfies

$$\begin{aligned} \rho (t,x(t),x_t(\cdot )) + \int _\tau ^t f^0(\xi ,x(\xi ),x(\xi ),u(\xi ),v_*) d \xi \le \rho (\tau ,z,w(\cdot )) + (t - \tau ) \zeta / 2. \end{aligned}$$

(39)

Thus, due to (2), (5), (12), (37), and (38), we derive

$$\begin{aligned} \begin{array}{l} \langle G(\tau ,w(-h),\textrm{d}^+ w(-h) / \textrm{d} \xi ), p \rangle + H(\tau ,z,w(-h),p) \\ \quad \le \displaystyle \frac{1}{t - \tau }\int _\tau ^t \bigg ( \frac{d}{d \xi } \Big (g(\xi ,x(\xi -h)\Big ) + \chi (\xi ,x(\xi ),x(\xi -h),u(\xi ),v_*,p)\bigg ) d \xi + \zeta / 3 \\ \quad = \displaystyle \frac{\langle x(t) - z, p \rangle }{t - \tau } + \frac{1}{t - \tau } \int _\tau ^t f^0(\xi ,x(\xi ),x(\xi ),u(\xi ),v_*) d \xi + \zeta / 3. \end{array} \end{aligned}$$

From this inequality, using (36), (37), and (39), we obtain (35).

Proposition 12

There exists $c_\rho > 0$ such that $\rho ^\circ$ satisfies the equality

$$\begin{aligned} \big \vert \rho ^\circ (\tau ,z,w(\cdot )) \big \vert \le c_\rho \big (1 + \Vert z\Vert + \Vert w(\cdot )\Vert _\infty \big ),\quad (\tau ,z,w(\cdot )) \in \mathbb G. \end{aligned}$$

(40)

Proof

Let us take $c_f$, $c_\sigma$, and $c_X$ from conditions $(f_2)$, $(\sigma _2)$ and the Proposition 6. Then, putting $c_\rho = c_\sigma (1 + c_X (1 + h)) + c_f \vartheta (1 + 2 c_X)$, for every $(\tau ,z,w(\cdot )) \in \mathbb G$, $u(\cdot ) \in \mathcal {U}_\tau$, and $v(\cdot ) \in \mathcal {V}_\tau$, the motion $x(\cdot ) = x(\cdot \,\vert \,\tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$ satisfies

$$\begin{aligned} \begin{array}{l} \sigma (x(\vartheta ),x_\vartheta (\cdot )) + \displaystyle \int _\tau ^\vartheta f^0(t,x(t),x(t-h),u(t),v(t)) d t \\ \quad \le c_\sigma \big (1 + \Vert x(\vartheta )\Vert + \Vert x_\vartheta (\cdot )\Vert _\infty \big ) + c_f \displaystyle \int _\tau ^\vartheta \big (1 + \Vert x(t)\Vert + \Vert x(t-h)\Vert \big ) d t \\ \quad \le c_\sigma (1 + c_X (1 + h)) + c_f \vartheta (1 + 2 c_X) + c_X (c_\sigma (1 + h) + 2 c_f \vartheta ) \alpha _0 \le c_\rho \big (1 + \alpha _0\big ) \end{array} \end{aligned}$$

where $\alpha _0 = \max \{\Vert z\Vert ,\Vert w(\cdot )\Vert _\infty \}$. Thus, from (4), (7), and Theorem 1, we obtain (40). $\square$

Proposition 13

Let the value functional $\rho ^\circ = \rho ^\circ (t,z,w(\cdot ))$ be differentiable by z. Then, for every $\alpha > 0$, there exists $\lambda _\rho = \lambda _\rho (\alpha ) > 0$ such that

$$\begin{aligned} \big \Vert \nabla _z \rho ^\circ (\tau ,z,w(\cdot ))\big \Vert \le \lambda _\rho ,\quad \tau \in [0,\vartheta ],\quad (z,w(\cdot )) \in P(\alpha ). \end{aligned}$$

(41)

Proof

Due to Theorem 1 and Proposition 10, there exists $\lambda _\rho = \lambda _\rho (\alpha ) > 0$ such that

$$\begin{aligned} \big \vert \rho ^\circ (\tau ,z,w(\cdot )) - \rho ^\circ (\tau ,p,w(\cdot )) \big \vert \le \lambda _\rho \Vert z - p\Vert ,\quad \tau \in [0,\vartheta ],\quad (z,w(\cdot )) \in P(\alpha ). \end{aligned}$$

Since $\rho ^\circ$ is differentiable by z, from this estimate, we obtain (41). $\square$

Proposition 14

Let $\lambda > 0$. Let $\varepsilon _* = \varepsilon _*(\lambda ) > 0$ be such that $\theta ^{\lambda , \varepsilon }(t) > 2 c_\rho$ for any $t \in [0,\vartheta ]$ and $\varepsilon \in (0,\varepsilon _*)$, where $\theta ^{\lambda ,\varepsilon }$ and $c_\rho$ are from (21) and Proposition 12, respectively. Then, the argmin and argmax values in (22) are achieved.

Proof

Let us prove the statement for the argmin value. Let $(t,x,r(\cdot )) \in \mathbb G$. Consider the function $\varphi (p) = \rho ^\circ (t,p,r(\cdot )) + \eta ^{\lambda ,\varepsilon }(t,x - p)$, $p \in \mathbb R^n$. According to Theorem 1 and Proposition 10, this function is continuous. Due to the choice of $\varepsilon _*$ and $c_\rho$, we derive

$$\begin{aligned} \varphi (p) \ge - c_\rho \big (1 + \Vert p\Vert + \Vert r(\cdot )\Vert _\infty \big ) + \theta ^{\lambda ,\varepsilon }(t) \Vert x - p\Vert \ge c_\rho \big (1 + \Vert p\Vert - 2 \Vert x\Vert + \Vert r(\cdot )\Vert _\infty \big ). \end{aligned}$$

Hence, the function $\varphi (p)$ is bounded below and $\varphi (p) \rightarrow +\infty$ as $p \rightarrow \infty$. Thus, the minimum of $\varphi (p)$ is achieved. $\square$

3.3 Proof of Theorem 2

Proof

The proof is carried out by the scheme from [16] (see also [8]).

Let us fix $(\tau ,z,w(\cdot )) \in \mathbb G$. According to Proposition 6, define $\alpha _* = \alpha _X(\alpha _X(\alpha _0))$ and $\lambda ^g_* = \lambda ^g_X(\alpha _X(\alpha _0))$, where $\alpha _0 = \max \{\Vert z\Vert ,\Vert w(\cdot )\Vert _\infty \}$. Then, for every $u(\cdot ) \in \mathcal {U}_\tau$, $v(\cdot ) \in \mathcal {V}_\tau$, $t_* \in [\tau ,\vartheta ]$, and for every $r(\cdot ) \in \textrm{PLip}$ such that $\Vert r(\cdot )\Vert _\infty \le \Vert x_{t_*}(\cdot )\Vert _\infty$, where $x(\cdot ) = x(\cdot \,\vert \,\tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$, the motion $y(\cdot ) = x(\cdot \,\vert \,t_*,x(t_*),r(\cdot ),u(\cdot ),v(\cdot ))$ and the function $y^g(t) = y(t) - g(t,y(t-h))$, $t \in [t_*,\vartheta ]$ satisfy the relations

$$\begin{aligned} (y(t),y_t(\cdot )) \in P(\alpha _*),\quad \big \vert y^g(t) - y^g(t')\big \vert \le \lambda ^g_* \vert t - t' \vert ,\quad t,t' \in [t_*,\vartheta ]. \end{aligned}$$

(42)

Moreover, due to condition $(f_2)$ and Proposition 13, there exist $\beta _f, \beta _\nabla > 0$ such that

$$\begin{aligned} \begin{array}{l} \big \Vert f(t,y(t),y(t-h),u(t),v(t)) \big \Vert + \big \vert f^0(t,y(t),y(t-h),u(t),v(t)) \big \vert \le \beta _f, \\ \quad \big \Vert \nabla _z \rho ^\circ (t,y(t),y_t(\cdot ))\big \Vert \le \beta _\nabla ,\ \ t \in [t_*,\vartheta ]. \end{array} \end{aligned}$$

(43)

Note that (42) and (43) are also valid for $x(\cdot )$ if we take $t_* = \tau$ and $r(\cdot )=w(\cdot )$.

Since both equalities in (20) are proved similarly, we present only the proof of the first equality which, according to (18), will be proved if we show that, for every $\zeta > 0$, there exists $\delta > 0$ such that, for every partition $\Delta _\delta$ (see (16)) and every $v(\cdot ) \in \mathcal {V}_\tau$, if $u(\cdot ) \in \mathcal {U}_\tau$ satisfies the relation

$$\begin{aligned} u(t) = u_j \in \mathop {\textrm{argmin}}\limits _{u \in \mathbb U} \max \limits _{v \in \mathbb V} \chi \big (t_j,x(x_j),x(t_j-h),u,v, \nabla _z\rho ^\circ (t_j,x(t_j),x_{t_j}(\cdot ))\big ) \end{aligned}$$

(44)

for any $t \in [t_j, t_{j+1})$ and $j \in \overline{1,k}$, then the motion $x(\cdot ) = x(\cdot \,\vert \,\tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$ satisfies the estimate

$$\begin{aligned} \sigma (x(\vartheta ),x_\vartheta (\cdot )) + \int _\tau ^\vartheta f^0(t,x(t),x(t-h),u(t),v(t)) d t \le \rho ^\circ (\tau ,z,w(\cdot )) + \zeta . \end{aligned}$$

(45)

Let $\zeta > 0$. According to the definition (30) of the set $\Theta _\nu (\tau ,w(\cdot ))$, the number of intervals in the set $[\tau ,\vartheta ]\, \backslash \, \Theta _\nu (\tau ,w(\cdot ))$ does not depend on $\nu \in (0,\nu _*)$. Denote this number as $l_*$. Due to Proposition 10, define $\lambda _\rho = \lambda _\rho (\alpha _*)$. Set

$$\begin{aligned} \nu = \min \Big \{\frac{\zeta }{6 \beta _* l_*}, \frac{\nu _*}{6}\Big \}, \quad \beta _* = \max \big \{4 \beta _f, 4 \lambda _\rho \big (4 \lambda ^g_* + 2 h \alpha _*\big )\big \},\quad \zeta _* = \frac{\zeta }{2(\vartheta - \tau )}. \end{aligned}$$

(46)

Due to condition $(f_1)$, Propositions 8, 9, and (43), there exists $\delta \in (0,\min \{\nu ,h\})$ such that, for every $u(\cdot ) \in \mathcal {U}_\tau$, $v(\cdot ) \in \mathcal {V}_\tau$, $\theta \in \Theta _\nu (\tau ,w(\cdot ))$, $t,t' \in \theta$: $\vert t - t' \vert \le \delta$, and $u \in \mathbb U$, $v \in \mathbb V$, the motion $x(\cdot ) = x(\cdot \,\vert \,\tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$ satisfies the inequality

$$\begin{aligned} \begin{array}{l} \big \vert \chi (t,x(t),x(t-h),u,v,\nabla _z \rho ^\circ (t,x(t),x_t(\cdot ))) \\ \quad - \chi (t',x(t'),x(t'-h),u,v,\nabla _z \rho ^\circ (t',x(t'),x_{t'}(\cdot ))) \big \vert \le \zeta _* / 16. \\ \end{array} \end{aligned}$$

(47)

and, taking into account (12), as a consequence, the estimate

$$\begin{aligned} \begin{array}{l} \big \vert H(t,x(t),x(t-h),\nabla _z \rho ^\circ (t,x(t),x_t(\cdot ))) \\ \quad \hspace{1cm} - H(t',x(t'),x(t'-h),\nabla _z \rho ^\circ (t',x(t'),x_{t'}(\cdot ))) \big \vert \le \zeta _* / 16. \\ \end{array} \end{aligned}$$

(48)

Let us take a partition $\Delta _\delta$ and a realization $v(\cdot ) \in \mathcal {V}_\tau$. Define the index sets

$$\begin{aligned} K_1 = \big \{j \in \overline{0,k} \, \big \vert \, \exists \theta \in \Theta _\nu (\tau ,w(\cdot )) :[t_j,t_{j+1}] \subset \theta \big \},\quad K_2 = \big \{j \in \overline{0,k} \, \big \vert \, j \notin K_1\big \}. \end{aligned}$$

(49)

Then, according to the choice of the numbers $\nu$, $l_*$ and $\delta$, we have

$$\begin{aligned} \sum \limits _{j \in K_1} (t_{j+1} - t_j) \le \vartheta - \tau ,\quad \sum \limits _{j \in K_2} (t_{j+1} - t_j) \le 3 \nu l_* \le \zeta / (2 \beta _*). \end{aligned}$$

Thus, to show (45), we need to prove the inequality

$$\begin{aligned} \begin{array}{l} \rho ^\circ (t_{j+1},x(t_{j+1}),x_{t_{j+1}}(\cdot )) + \displaystyle \int _{t_j}^{t_{j+1}} f^0(\xi ,x(\xi ),x(\xi -h),u(\xi ),v(\xi )) d \xi \\ \quad \le \rho ^\circ (t_{j},x(t_{j}),x_{t_{j}}(\cdot )) + (t_{j+1} - t_j) \zeta _*,\quad j \in K_1, \end{array} \end{aligned}$$

(50)

and the inequality

$$\begin{aligned} \begin{array}{l} \rho ^\circ (t_{j+1},x(t_{j+1}),x_{t_{j+1}}(\cdot )) + \displaystyle \int _{t_j}^{t_{j+1}} f^0(\xi ,x(\xi ),x(\xi -h),u(\xi ),v(\xi )) d \xi \\ \quad \le \rho ^\circ (t_{j},x(t_{j}),x_{t_{j}}(\cdot )) + (t_{j+1} - t_j) \beta _*, \quad j \in K_2. \end{array} \end{aligned}$$

(51)

Let us prove (50). Let $j \in K_1$. Then, due to Proposition 8, the function $x(\cdot )$ is Lipschitz continuous on $[t_j-h,t_{j+1}-h]$. One can take a sequence of continuously differentiable on $[-h,-h+t_{j+1} - t_j]$ functions $r^m(\cdot ) \in \textrm{PLip}$, $m \in \mathbb N$ such that

$$\begin{aligned} \Vert r^m(\cdot )\Vert _\infty \le \Vert x_{t_j}(\cdot )\Vert _\infty ,\quad \Vert x_{t_j}(\cdot ) - r^m(\cdot )\Vert _\infty \le 1/m,\quad m \in \mathbb N. \end{aligned}$$

Define the sequence of motions $y^m(\cdot ) = x(\cdot \vert t_j,x(t_j),r^m(\cdot ),u(\cdot ),v(\cdot ))$. Note that, according to Proposition 7 and (10), there exists $\lambda _{XX} = \lambda _{XX}(\alpha _*) > 0$ such that

$$\begin{aligned} \Vert x(t) - y^m(t)\Vert \le \lambda _{XX} (2 + h) / m,\quad m \in \mathbb N. \end{aligned}$$

Then, due to condition $(f_3)$, Proposition 10, and relations (42), (43), we can take $y(\cdot ) \in \{y^m(\cdot )\,\vert \,m \in \mathbb N\}$ such that

$$\begin{aligned} \begin{array}{l} \big \vert \rho ^\circ (t,x(t),x_t(\cdot )) - \rho ^\circ (t,y(t),y_t(\cdot )) \big \vert \le (t_{j+1} - t_{j}) \zeta _* / 4, \\ \quad \big \Vert f^0(t,x(t),x(t-h),u(t),v(t)) - f^0(t,y(t),y(t-h),u(t),v(t))\big \Vert \le \zeta _* / 4, \\ \quad \big \vert \chi (t,x(t),x(t-h),u(t),v(t),\nabla _z\rho ^\circ (t,x(t),x_t(\cdot ))) \\ \quad - \chi (t,y(t),y(t-h),u(t),v(t),\nabla _z\rho ^\circ (t,y(t),y_t(\cdot ))) \big \vert \le \zeta _* / 16 \end{array} \end{aligned}$$

(52)

for any $t \in [t_{j}, t_{j+1}]$, and, taking into account (12), as a consequence

$$\begin{aligned} \begin{array}{l} \big \vert H(t,x(t),x(t-h),\nabla _z\rho ^\circ (t,x(t),x_t(\cdot ))) \\ \quad - H(t,y(t),y(t-h),\nabla _z\rho ^\circ (t,y(t),y_t(\cdot ))) \big \vert \le \zeta _* / 16 \end{array} \end{aligned}$$

(53)

for any $t \in [t_{j}, t_{j+1}]$. Thus, in order to conclude (50), it is necessary to prove

$$\begin{aligned} \begin{array}{l} \rho ^\circ (t_{j+1},y(t_{j+1}),y_{t_{j+1}}(\cdot )) + \displaystyle \int _{t_j}^{t_{j+1}} f^0(\xi ,y(\xi ),y(\xi -h),u(\xi ),v(\xi )) d \xi \\ \quad \le \rho ^\circ (t_{j},y(t_{j}),y_{t_{j}}(\cdot )) + (t_{j+1} - t_j) \zeta _* / 4. \end{array} \end{aligned}$$

(54)

Let us consider the function

$$\begin{aligned} \omega (t) = \rho ^\circ (t,y(t),y_t(\cdot )) + \int _{t_j}^{t} f^0(\xi ,y(\xi ),y(\xi -h),u(\xi ),v(\xi )) d t,\quad t \in [t_j, t_{j+1}]. \end{aligned}$$

(55)

Since $y(\cdot )$ is continuously differentiable on $[t_{j}-h,t_{j+1}-h]$, we have $y_t(\cdot ) \in \textrm{PLip}_*$ for any $t \in [t_{j},t_{j+1})$. Then, taking into account definition (1) of the set $\mathbb G_*$ we derive $(t,y(t),y_t(\cdot )) \in \mathbb G_*$ for almost every $t \in [t_{j},t_{j+1})$. Since $y(\cdot )$ is Lipschitz continuous on $[t_{j},t_{j+1}]$, there exists dy(t) /dt for almost every $[t_{j},t_{j+1}]$. Then, from the coinvariant differentiability of $\rho ^\circ$ on $\mathbb G_*$, we obtain

$$\begin{aligned} \displaystyle \frac{d}{d t} \omega (t){} & {} = \partial _{\tau ,w}^{ci} \rho ^\circ (t,y(t),y_t(\cdot )) + \langle \frac{d}{d t} y(t), \nabla _z \rho ^\circ (t,y(t),y_t(\cdot )) \rangle \\{} & {} \quad + f^0(t,y(t),y(t-h),u(t),v(t)) \end{aligned}$$

for almost every $t \in [t_{j}, t_{j+1}]$. Then, due to property $(\rho _5)$, taking into account (5) and (12), we have

$$\begin{aligned} \begin{array}{c} \displaystyle \frac{d}{d t} \omega (t) = \chi (t,y(t),y(t-h),u(t),v(t),\nabla _z \rho ^\circ (t,y(t),y_t(\cdot ))) \\ \quad - H(t,y(t),y(t-h),\nabla _z \rho ^\circ (t,y(t),y_t(\cdot ))) \end{array} \end{aligned}$$

(56)

for almost every $t \in [t_{j}, t_{j+1}]$. Next, in accordance with (44), (47), and (52), we derive

$$\begin{aligned} \begin{array}{l} \chi (t,y(t),y(t-h),u(t),v(t),\nabla _z \rho ^\circ (t,y(t),y_t(\cdot ))) \\ \quad \le \chi (t,x(t),x(t-h),u(t),v(t),\nabla _z \rho ^\circ (t,x(t),x_t(\cdot ))) + \zeta _* / 16 \\ \quad \le \chi (t_j,x(t_j),x(t_j-h),u(t),v(t),\nabla _z \rho ^\circ (t_j,x(t_j),x_{t_j}(\cdot ))) + \zeta _* / 8 \\ \quad \le \max \limits _{v \in \mathbb V} \chi (t_j,x(t_j),x(t_j-h),u_j,v,\nabla _z \rho ^\circ (t_j,x(t_j),x_{t_j}(\cdot ))) + \zeta _* / 8 \\ \quad = \min \limits _{u \in \mathbb U} \max \limits _{v \in \mathbb V} \chi (t_j,x(t_j),x(t_j-h),u,v,\nabla _z \rho ^\circ (t_j,x(t_j),x_{t_j}(\cdot ))) + \zeta _* / 8. \end{array} \end{aligned}$$

Finally, according to (12), (48), and (53), we get

$$\begin{aligned} \begin{array}{c} \min \limits _{u \in \mathbb U} \max \limits _{v \in \mathbb V} \chi (t_j,x(t_j),x(t_j-h),u,v,\nabla _z \rho (t_j,x(t_j),x_{t_j}(\cdot ))) \\ \le H(t,x(t),x(t-h),\nabla _z \rho ^\circ (t,x(t),x_t(\cdot ))) + \zeta _* / 16 \\ \le H(t,y(t),y(t-h),\nabla _z \rho ^\circ (t,y(t),y_t(\cdot ))) + \zeta _* / 8. \end{array} \end{aligned}$$

Thus, we obtain $d \omega (t) / d t \le \zeta _* / 4$ and conclude (54) which proves (50).

Let us prove (51). Let $j \in K_2$. According to condition $(\rho _3)$, there exists $u_j(\cdot ) \in \mathcal {U}_{t_j}$ such that the motion $y(\cdot ) = x(\cdot \,\vert \,t_j,x(t_j),x_{t_j}(\cdot ),u_j(\cdot ),v(\cdot ))$ satisfies the estimate

$$\begin{aligned} \begin{array}{l} \rho ^\circ (t_{j+1},y(t_{j+1}),y_{t_{j+1}}(\cdot )) + \displaystyle \int _{t_j}^{t_{j+1}} f^0(\xi ,y(\xi ),y(\xi -h),u(\xi ),v(\xi )) d \xi \\ \le \rho ^\circ (t_{j},x(t_{j}),x_{t_{j}}(\cdot )) + (t_{j+1} - t_j) \beta _* / 4. \end{array} \end{aligned}$$

(57)

Define the function $s(t) = x(t) - y(t)$, $t \in [t_{j}-h, \vartheta ]$. Then, since $t_{j+1} - t_{j} \le \delta < h$ and according to (42), we have

$$\begin{aligned} \begin{array}{c} s(t) = 0,\quad t \in [t_j-h, t_j],\quad \Vert s(t)\Vert \le 2\alpha _*,\\ \Vert s(t)\Vert = \Vert x(t) - g(t,x(t-h)) - y(t) + g(t,x(t-h))\Vert \le 2 \lambda ^g_* (t_{j+1} - t_{j}) \end{array} \end{aligned}$$

(58)

for any $t \in [t_{j}, t_{j+1}]$. Hence, according to the choice of $\lambda _\rho$ and $\beta _*$ (see (46)), we derive

$$\begin{aligned} \begin{array}{l} \big \vert \rho (t_{j+1},x(t_{j+1}),x_{t_{j+1}}(\cdot )) - \rho (t_{j+1},y(t_{j+1}),y_{t_{j+1}}(\cdot )) \big \vert \\ \quad \le \lambda _\rho \big (\Vert s(t_{j+1})\Vert + \Vert s(l h)\Vert + \Vert s_{j+1}(\cdot )\Vert _1\big ) \le (t_{j+1} - t_{j}) \beta _* / 4, \end{array} \end{aligned}$$

(59)

where $l \in \overline{-1,J-1}$ satisfies $l h \in [t_{j+1}-h, t_{j+1})$. Due to (43) and (46), we have

$$\begin{aligned} \begin{array}{c} \displaystyle \int _{t_j}^{t_{j+1}} f^0(\xi ,x(\xi ),x(\xi -h),u(\xi ),v(\xi )) d \xi \le (t_{j+1} - t_{j}) \beta _* / 4,\\ \displaystyle \int _{t_j}^{t_{j+1}} f^0(\xi ,y(\xi ),y(\xi -h),u_j(\xi ),v(\xi )) d \xi \le (t_{j+1} - t_{j}) \beta _* / 4. \end{array} \end{aligned}$$

(60)

From (57)–(60), we obtain (51). Thus, we have proved (57), (51), and the theorem. $\square$

3.4 Proof of Theorem 3

Define the functional

$$\begin{aligned} \varphi ^{\lambda ,\varepsilon }(t,x,r(\cdot )) = \min \limits _{p \in \mathbb R^n}\big (\rho ^\circ (t,p,r(\cdot )) - \eta ^{\lambda ,\varepsilon }(t,x-p)\big ), \quad (t,x,r(\cdot )) \in \mathbb G, \end{aligned}$$

(61)

where $\eta ^{\lambda ,\varepsilon }$ is from (21).

Proposition 15

Let $(\tau ,z,w(\cdot )) \in \mathbb G$ and $\lambda ,\zeta _1 > 0$. There exists $\varepsilon _1 > 0$ such that, for every $\varepsilon \in (0,\varepsilon _1]$, $u(\cdot ) \in \mathcal {U}_\tau$, and $v(\cdot ) \in \mathcal {V}_\tau$, the motion $x(\cdot ) = x(\cdot \,\vert \,\tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$ satisfies the estimates

$$\begin{aligned} \big \Vert p^{\lambda ,\varepsilon }(t,x(t),x_t(\cdot )) - x(t)\big \Vert \le \zeta _1,\quad t \in [\tau ,\vartheta ], \end{aligned}$$

(62)

where the functional $p^{\lambda ,\varepsilon }$ is defined in (22).

Proof

Define $\alpha _X = \alpha _X(\alpha _0)$, where $\alpha _0 = \max \{\Vert z\Vert ,\Vert w(\cdot )\Vert _\infty \}$, and $c_\rho$ according to Propositions 6 and 12. Due to definitions (21), we can take $\varepsilon _1 > 0$ so that

$$\begin{aligned} \theta ^{\lambda ,\varepsilon }(t) \ge c_\rho + 3 \beta / \zeta _1,\ \ \eta ^{\lambda ,\varepsilon }(t,0) \le \beta ,\ \ \beta = c_\rho (1 + (1 + h) \alpha _X),\ \ t \in [0,\vartheta ],\ \ \varepsilon \in (0,\varepsilon _1]. \end{aligned}$$

Let $u(\cdot ) \in \mathcal {U}_\tau$, $v(\cdot ) \in \mathcal {V}_\tau$, $\varepsilon \in (0,\varepsilon _1]$, and $t \in [\tau ,\vartheta ]$. Denote $p = p^{\lambda ,\varepsilon }(t,x(t),x_t(\cdot ))$. Then, we have

$$\begin{aligned} \begin{array}{l} 3 \beta / \zeta _1 \Vert x(t) - p\Vert - \beta \le \big (\theta ^{\lambda ,\varepsilon }(t) - c_\rho \big )\big \Vert x(t) - p\big \Vert - c_\rho \big (1 + \Vert x(t)\Vert + \Vert x_t(\cdot )\Vert _\infty \big )\\ \quad \le \theta ^{\lambda ,\varepsilon }(t) \Vert x(t) - p\Vert - c_\rho \big (1 + \Vert p\Vert + \Vert x_t(\cdot )\Vert _\infty \big ) \le \rho ^\circ (t,p,x_t(\cdot )) + \eta ^{\lambda ,\varepsilon }(t,x(t) - p) \\ \quad = \varphi ^{\lambda ,\varepsilon }(t,x(t),x_t(\cdot )) \le \rho ^\circ (t,x(t),x_t(\cdot )) + \eta ^{\lambda ,\varepsilon }(t,0) \le 2 \beta . \end{array} \end{aligned}$$

Thus, we obtain (62). $\square$

Proposition 16

Let $(\tau ,z,w(\cdot )) \in \mathbb G$ and $\lambda ,\zeta _2 > 0$. There exists $\varepsilon _2 > 0$ such that, for every $\varepsilon \in (0,\varepsilon _2]$, $u(\cdot ) \in \mathcal {U}_\tau$, and $v(\cdot ) \in \mathcal {V}_\tau$, the motion $x(\cdot \,\vert \,\tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$ satisfies the estimates

$$\begin{aligned} \big \Vert \varphi ^{\lambda ,\varepsilon }(t,x(t),x_t(\cdot )) - \rho ^{\circ }(t,x(t),x_t(\cdot ))\big \Vert \le \zeta _2,\quad t \in [\tau ,\vartheta ]. \end{aligned}$$

(63)

Proof

In accordance with Proposition 6, define $\alpha _X = \alpha _X(\alpha _0) > 0$, where $\alpha _0 = \max \{\Vert z\Vert ,\Vert w(\cdot )\Vert _\infty \}$. Due to Theorem 1 and Proposition 10, the value functional $\rho ^\circ$ satisfies condition $(\rho _2)$. According to this condition, determine $\lambda _\rho = \lambda _\rho (\alpha _X) > 0$. Due to Proposition 15, putting $\zeta _1 = \zeta _2 / \lambda _\rho$, define $\varepsilon _1 > 0$. Set $\varepsilon _2 = \min \{\varepsilon _1, \zeta _2\}$. Then, in particular, due to (21), we have $\eta ^{\lambda ,\varepsilon }(t,0) \le \zeta _2$, $t \in [0,\vartheta ]$, $\varepsilon \in (0,\varepsilon _2]$. Thus, taking into account (61), for every $\varepsilon \in (0,\varepsilon _2]$ and $t \in [\tau ,\vartheta ]$, we obtain

$$\begin{aligned} \begin{array}{l} - \zeta _2 \le - \eta ^{\lambda ,\varepsilon }(t,0) \le \rho ^{\circ }(t,x(t),x_t(\cdot )) - \varphi ^{\lambda ,\varepsilon }(t,x(t),x_t(\cdot )) \\ \quad = \rho ^{\circ }(t,x(t),x_t(\cdot )) - \rho ^\circ (t,p,r(\cdot )) - \eta ^{\lambda ,\varepsilon }(t,x-p) \le \lambda _\rho \Vert x(t) - p\Vert \le \zeta _2, \end{array} \end{aligned}$$

where we denote $p = p^{\lambda ,\varepsilon }(t,x(t),x_t(\cdot ))$. $\square$

Proposition 17

Let $(\tau ,z,w(\cdot )) \in \mathbb G$ and $\zeta _3 > 0$. There exists $\lambda , \varepsilon _3 > 0$ with the following properties. For every $\varepsilon \in (0,\varepsilon _3]$, there exists $\delta > 0$ such that, for every partition $\Delta _\delta$ and every $v(\cdot ) \in \mathcal {V}_\tau$, if $u(\cdot ) \in \mathcal {U}_\tau$ is defined according to the strategy $U^{\lambda ,\varepsilon }$ (see (24)), then the motion $x(\cdot ) = x(\cdot \,\vert \,\tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$ satisfies the estimate

$$\begin{aligned} \varphi ^{\lambda ,\varepsilon }(\vartheta ,x(\vartheta ),x_\vartheta (\cdot )) + \int _\tau ^\vartheta f^0(\xi ,x(\xi ),x(\xi -h),u(\xi ),v(\xi )) \le \varphi ^{\lambda ,\varepsilon }(\tau ,z,w(\cdot )) + \zeta _3. \end{aligned}$$

Proof

Let $(\tau ,z,w(\cdot )) \in \mathbb G$. According to Propositions 6 and 8, define $\alpha _* = \alpha _X(\alpha _X(\alpha _0) + 1)$, $\lambda ^g_* = \lambda ^g_X(\alpha _X(\alpha _0) + 1)$, and $\lambda _* = \lambda _X(\alpha _X(\alpha _0) + 1)$, where $\alpha _0 = \max \{\Vert z\Vert ,\Vert w(\cdot )\Vert _\infty \}$. Then, for every $u(\cdot ) \in \mathcal {U}_\tau$, $v(\cdot ) \in \mathcal {V}_\tau$, and $t_* \in [\tau ,\vartheta ]$, for every $p \in \mathbb R^n$ such that $\Vert p\Vert \le \Vert x(t_*)\Vert + 1$, where $x(\cdot ) = x(\cdot \,\vert \,\tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$, the motion $y(\cdot ) = x(\cdot \,\vert \,t_*,p,x_{t_*}(\cdot ),u(\cdot ),v(\cdot ))$ and the function $y^g(t) = y(t) - g(t,y(t-h))$, $t \in [t_*,\vartheta ]$ satisfy the relations

$$\begin{aligned} (y(t),y_t(\cdot )) \in P(\alpha _*),\quad \Vert y^g(t) - y^g(t')\Vert \le \lambda ^g_* \vert t - t' \vert ,\quad \Vert y(t) - y(t')\Vert \le \lambda _* \vert t - t' \vert \end{aligned}$$

(64)

for any $t \in [t_*,\vartheta ]$. Moreover, due to condition $(f_2)$, there exist $\beta _f > 0$ such that

$$\begin{aligned} \big \Vert f(t,y(t),y(t-h),u(t),v(t)) \big \Vert + \big \vert f^0(t,y(t),y(t-h),u(t),v(t)) \big \vert \le \beta _f \end{aligned}$$

(65)

for any $t \in [t_*,\vartheta ]$. Note that relations (64) and (65) are also valid for the motion $x(\cdot )$ and the function $x^g(t) = x(t) - g(t,x(t-h))$ if we take $t_* = \tau$ and $p=z$. In accordance with condition $(f_3)$, put

$$\begin{aligned} \lambda = \lambda _f(\alpha _*). \end{aligned}$$

(66)

Let $\zeta _3 > 0$. Denote

$$\begin{aligned} \zeta _* = \zeta _3 / (2(\vartheta - \tau )). \end{aligned}$$

(67)

In accordance with condition $(\rho _2)$, define $\lambda _\rho = \lambda _\rho (\alpha _*)$. Due to Propositions 15 and 16, there exists $\varepsilon _3 > 0$ such that, for every $\varepsilon \in (0,\varepsilon _3]$, $u(\cdot ) \in \mathcal {U}_\tau$, $v(\cdot ) \in \mathcal {V}_\tau$, and $t \in [\tau ,\vartheta ]$, the motion $x(\cdot ) = x(\cdot \,\vert \,\tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$ satisfies the estimates

$$\begin{aligned} \big \Vert p^{\lambda ,\varepsilon }(t,x(t),x_t(\cdot )) - x(t)\big \Vert \le \zeta _* / (12 \lambda _\rho ). \end{aligned}$$

(68)

Let us take $\varepsilon \in (0,\varepsilon _3]$. Recall that, according to (30), the number $l_*$ of intervals in the set $[\tau ,\vartheta ]\, \backslash \, \Theta _\nu (\tau ,w(\cdot ))$ does not depend on $\nu \in (0,\nu _*)$. Set

$$\begin{aligned} \nu = \zeta _3 / (6 l_* \beta _*),\quad \beta _* = 2 \lambda \alpha _* (1 + 1/\varepsilon ). \end{aligned}$$

(69)

Using condition $(f_1)$, Proposition 8, and (43), one can show the existence of

$$\begin{aligned} \delta \in (0,\delta _*],\quad \delta _* = \min \big \{\nu ,h,\zeta _* / (12 \lambda _\rho \lambda ^g_*)\big \} \end{aligned}$$

(70)

such that, for each $u(\cdot ) \in \mathcal {U}_\tau$ and $v(\cdot ) \in \mathcal {V}_\tau$, the motion $x(\cdot ) = x(\cdot \,\vert \,\tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$ satisfies the inequality

$$\begin{aligned} \begin{array}{l} \big \vert \chi (t,x(t),x(t-h),u,v, \nabla _x\eta ^{\lambda ,\varepsilon }(t,s(t))) \\ \quad - \chi (t',x(t'),x(t'-h),u,v,\nabla _x\eta ^{\lambda ,\varepsilon }(t',s(t'))) \big \vert \le \zeta _* / 6 \end{array} \end{aligned}$$

(71)

for any $t,t' \in \theta$: $\vert t - t' \vert \le \delta$, $\theta \in \Theta _\nu (\tau ,w(\cdot ))$, any $u \in \mathbb U$, $v \in \mathbb V$, and any function $s(\cdot )$, satisfying

$$\begin{aligned} \big \vert s(t) - s(t') \big \vert \le 2 \lambda ^g_* \vert t - t' \vert ,\quad t,t' \in \theta , \quad \theta \in \Theta _\nu (\tau ,w(\cdot )). \end{aligned}$$

(72)

Let us take $\Delta _\delta$ and $v(\cdot ) \in \mathcal {V}_\tau$. Let $x(\cdot ) = x(\cdot \,\vert \,\tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$ be the motion with $u(\cdot ) \in \mathcal {U}_\tau$ defined by the strategy $U^{\lambda ,\varepsilon }$. In accordance with (17), it means that

$$\begin{aligned} u(t) = u_j \in \mathop {\textrm{argmin}}\limits _{u \in \mathbb U} \max \limits _{v \in \mathbb V} \chi \big (t_j,x(t_j),x(t_j - h),u,v, \nabla _z\eta ^{\lambda ,\varepsilon }(t,x(t_j) - p_j)\big ) \end{aligned}$$

(73)

for any $t \in [t_j,t_{j+1})$ and $j \in \overline{1,k}$, where we denote $p_j = p^{\lambda ,\varepsilon }(t_j,x(t_j),x_{t_j}(\cdot ))$. Let us consider the index sets $K_1$ and $K_2$ from (49) and prove the inequalities

$$\begin{aligned} \begin{array}{l} \varphi ^{\lambda ,\varepsilon }(t_{j+1},x(t_{j+1}),x_{t_{j+1}}(\cdot )) + \displaystyle \int _{t_j}^{t_{j+1}} \!\! f^0(\xi ,x(\xi ),x(\xi -h),u(\xi ),v(\xi )) d \xi \\ \quad \le \varphi ^{\lambda ,\varepsilon }(t_{j},x(t_{j}),x_{t_{j}}(\cdot )) + (t_{j+1} - t_j) \zeta _*,\quad j \in K_1, \end{array} \end{aligned}$$

(74)

and

$$\begin{aligned} \begin{array}{l} \varphi ^{\lambda ,\varepsilon }(t_{j+1},x(t_{j+1}),x_{t_{j+1}}(\cdot )) + \displaystyle \int _{t_j}^{t_{j+1}} \!\! f^0(\xi ,x(\xi ),x(\xi -h),u(\xi ),v(\xi )) d \xi \\ \quad \le \varphi ^{\lambda ,\varepsilon }(t_{j},x(t_{j}),x_{t_{j}}(\cdot )) + (t_{j+1} - t_j) \beta _*,\quad j \in K_2. \end{array} \end{aligned}$$

(75)

Let $j \in \overline{1,k}$. From (22) and (61), we have

$$\begin{aligned} \varphi ^{\lambda ,\varepsilon }(t_j,x(t_j),x_{t_j}(\cdot )) = \rho ^\circ (t_j,p_j,x_{t_j}(\cdot )) + \eta ^{\lambda ,\varepsilon }(t_j,x(t_j)-p_j). \end{aligned}$$

(76)

Define

$$\begin{aligned} v_j(t) = v_j \in \mathop {\textrm{argmax}}\limits _{v \in \mathbb V}\min \limits _{u \in \mathbb U} \chi (t_j,x(t_j),x(t_j-h),u,v,\nabla _x\eta ^{\lambda ,\varepsilon }(t_j,x(t_j)-p_j)) \end{aligned}$$

(77)

for any $t \in [t_j,\vartheta ]$. Then, due to condition $(\rho _3)$, there exists $u_j(\cdot ) \in \mathcal {U}_{t_j}$ such that the motion $y(\cdot ) = x(\cdot \,\vert \,t_j,p_j,x_{t_j}(\cdot ),u_j(\cdot ),v_j(\cdot ))$ satisfies the inequality

$$\begin{aligned} \begin{array}{l} \rho ^\circ (t_{j+1},y(t_{j+1}),y_{t_{j+1}}(\cdot )) + \displaystyle \int _{t_j}^{t_{j+1}}\!\! f^0(\xi ,y(\xi ),y(\xi -h),u_j(\xi ),v_j(\xi )) d \xi \\ \quad \le \rho ^\circ (t_j,p_j,x_{t_j}(\cdot )) + (t_{j+1} - t_j) \zeta _*/3. \end{array} \end{aligned}$$

(78)

Denote $s(t) = x(t) - y(t)$, $t \in [t_j-h,\vartheta ]$. Due to the inequality $t_{j+1} - t_{j} \le \delta < h$ and (64), the function $s(\cdot )$ satisfies estimates (58). Then, due to the choice of $\lambda _\rho$, the inclusion $j \in K_1$, and the relations (70), we have

$$\begin{aligned} \begin{array}{l} \big \vert \rho ^\circ (t_{j+1},y(t_{j+1}),y_{t_{j+1}}(\cdot )) - \rho ^\circ (t_{j+1},y(t_{j+1}),x_{t_{j+1}}(\cdot )) \big \vert \le \lambda _\rho \Vert s_{t_{j+1}}(\cdot )\Vert _1 \\ \quad \le \lambda _\rho \displaystyle \int _{t_j}^{t_j+1} \!\Vert s(\xi ) - s(t_{j})\Vert d \xi + 2 \lambda _\rho (t_{j+1} - t_{j}) \Vert x(t_{j}) - p_j\Vert \le (t_{j+1} - t_{j}) \zeta _*/3. \end{array} \end{aligned}$$

(79)

Denote

$$\begin{aligned} \begin{array}{l} \kappa (t) = \eta ^{\lambda ,\varepsilon }(t,s(t)) + \displaystyle \int _{t_j}^t \Big (f^0(\xi ,x(\xi ),x(\xi -h),u(\xi ),v(\xi )) \\ \quad \quad \quad - f^0(\xi ,y(\xi ),y(\xi -h),u_j(\xi ),v_j(\xi ))\Big ) d \xi . \end{array} \end{aligned}$$

Note that, due to (21), we have

$$\begin{aligned} \partial \eta ^{\lambda ,\varepsilon }(t,x) / \partial t = - \lambda (\theta ^{\lambda ,\varepsilon }(t) + 1) \mu ^\varepsilon (x),\quad \nabla _x\eta ^{\lambda ,\varepsilon }(t,x) = (\theta ^{\lambda ,\varepsilon }(t)/\mu ^\varepsilon (x)) x \end{aligned}$$

(80)

Then, taking (58) into account, the function $s(\cdot )$ and, as a consequence the function $\kappa (\cdot )$ are Lipschitz continuous on $[t_{j},t_{j+1}]$, and, in accordance with (5), we obtain

$$\begin{aligned} \begin{array}{c} d \kappa (t) / d t = \partial \eta ^{\lambda ,\varepsilon }(t, s(t)) / \partial t + \chi (t,x(t),x(t-h),u(t),v(t),\nabla _x\eta ^{\lambda ,\varepsilon }(t,s(t))) \\ - \chi (t,y(t),y(t-h),u_j(t),v_j(t),\nabla _x\eta ^{\lambda ,\varepsilon }(t,s(t))). \end{array} \end{aligned}$$

(81)

Now, let us consider the case $j \in K_1$. Due to (58), estimate (72) holds for the function s(t). Then, using (12), (71), (73), and (77), we derive

$$\begin{aligned} \begin{array}{l} \chi \big (t,x(t),x(t-h),u(t),v(t),\nabla _x\eta ^{\lambda ,\varepsilon }(t,s(t))\big ) \\ \quad \le \max \limits _{v \in \mathbb V} \chi \big (t_j,x(t_j),x(t_j-h),u_j,v,\nabla _x\eta ^{\lambda ,\varepsilon }(t_j,s(t_j))\big ) + \zeta _* / 6 \\ \quad = H\big (t_j,x(t_j),x(t_j-h),\nabla _x\eta ^{\lambda ,\varepsilon }(t_j,s(t_j))\big ) + \zeta _* / 6 \\ \quad \le \min \limits _{u \in \mathbb U} \chi \big (t_j,x(t_j),x(t_j-h),u,v_j,\nabla _x\eta ^{\lambda ,\varepsilon }(t_j,s(t_j))\big ) + \zeta _* / 6 \\ \quad \le \chi \big (t,x(t),x(t-h),u_j(t),v_j(t), \nabla _x\eta ^{\lambda ,\varepsilon }(t,s(t))\big ) + \zeta _*/3 \end{array} \end{aligned}$$

and, hence, taking into account the inclusion in (64), choice (66) of $\lambda$, and equalities (58), (80), we obtain

$$\begin{aligned} \begin{array}{c} d \kappa (t) / d t \le \partial \eta ^{\lambda ,\varepsilon }(t, s(t)) / \partial t + \lambda _f \Vert s(t)\Vert \big (1 + \nabla _x\eta ^{\lambda ,\varepsilon }(t,s(t))\big ) + \zeta _*/3 \le \zeta _*/3. \end{array} \end{aligned}$$

(82)

Thus, due to (61), (76), (78), (79), and (82), we conclude (74).

In the case of $j \in K_2$, according to (21), (58), (64), (66), (69), (80), (81), we get

$$\begin{aligned} d \kappa (t) / dt \le \lambda \Vert s(t)\Vert \big (1 + \nabla _x\eta ^{\lambda ,\varepsilon }(t,s(t))\big ) \le 2 \lambda \alpha _* (1 + 1/\varepsilon ) \le \beta _*. \end{aligned}$$

and, taking into account (61), (76), and (78), we obtain (75).

From the inequalities (74), (75), and definitions in (69), we conclude the statement of the lemma. $\square$

Proof

Let us prove the first equality in Theorem 3. Let $(\tau ,z,w(\cdot )) \in \mathbb G$ and $\zeta > 0$. According to Proposition 17, taking $\zeta _3 = \zeta / 3$, define $\lambda _3, \varepsilon _3 > 0$. Due to Proposition 16, taking $\lambda = \lambda _3$ and $\zeta _2 = \zeta / 3$, define $\varepsilon _2 > 0$. Put $\varepsilon _* = \min \{\varepsilon _2, \varepsilon _3\}$. Then, for every $\varepsilon \in (0,\varepsilon _*]$, there exists $\delta > 0$ such that, for every partition $\Delta _\delta$ and every $v(\cdot ) \in \mathcal {V}_\tau$, if $u(\cdot ) \in \mathcal {U}_\tau$ is defined by $U^{\lambda ,\varepsilon }$ (see (24)), then the motion $x(\cdot ) = x(\cdot \,\vert \,\tau ,z,w(\cdot ),u(\cdot ),v(\cdot ))$ satisfies the estimate

$$\begin{aligned} \begin{array}{l} \rho ^\circ (\vartheta ,x(\vartheta ),x_\vartheta (\cdot )) + \displaystyle \int _\tau ^\vartheta f^0(\xi ,x(\xi ),x(\xi -h),u(\xi ),v(\xi ))) d \xi \\ \quad \le \varphi ^{\lambda ,\varepsilon }(\vartheta ,x(\vartheta ),x_\vartheta (\cdot )) + \displaystyle \int _\tau ^\vartheta f^0(\xi ,x(\xi ),x(\xi -h),u(\xi ),v(\xi ))) + \zeta / 3 \\ \quad \le \varphi ^{\lambda ,\varepsilon }(\tau ,z,w(\cdot )) + 2 \zeta / 3 \le \rho ^\circ (\tau ,z,w(\cdot )) + \zeta . \end{array} \end{aligned}$$

According to definition (18), the statement above means the first equality in Theorem 3 holds. The second equality can be proved by the similar way based on the statements symmetrical to Proposition 15–17. $\square$