Game Control Problem for a Phase Field Equation

Abstract

A game control problem for a phase field equation is considered. This problem is investigated from the viewpoint of both the first player (the partner) and of the second player (the opponent). For both players, their own procedures for forming feedback controls are specified.

1 Introduction

A system modeling the solidification process and governed by so-called phase field equations is considered. The state variables are the order parameter (also called the phase function) and the temperature. In contrast to the classical Stefan problem, which models the solidification process with a sharp solid–liquid interface, the phase field equations are applicable to fuzzy domains. For the aforementioned system, we discuss a game control problem, which consists in the following. Some quality criterion, depending on the trajectory of the system, is given. At discrete time instants, the phase function is inaccurately measured. There are two antagonistic players. The problem undertaken by the first player (the partner) is to construct (using measurements of the phase functions) a law of forming a feedback control that minimizes some quality criterion. The goal of the second player (the opponent) is opposite.

One of the approaches to solving the problems of guaranteed control (they are also called positional differential games) for dynamical systems, described by ordinary differential equations, was developed in [1]. The fundamental theory of guaranteed control for some systems with distributed parameters within the framework of the formalization from [1] was presented in [2, 3]. In all the works cited above, the cases when the full phase state of a system is inaccurately measured at frequent enough time instants were considered. In the present work, from the position of the approach [1–3], the problems of guaranteed control undertaken by the partner and opponent are investigated under measuring a “part” of system’s phase state.

2 Problem Statement and Solution Method

Consider the system (introduced in [4])

$$\begin{aligned}&\displaystyle \frac{\partial }{\partial t}\psi + l\displaystyle \frac{\partial }{\partial t}\varphi =k{\varDelta }_L \psi +Bu-Cv\quad \text{ in }\quad {\varOmega }\times {]}t_0,\vartheta ],\quad \vartheta =\mathop {{\mathrm {const}}}<+\infty , \end{aligned}$$

(1)

$$\begin{aligned}&\tau \displaystyle \frac{\partial }{\partial t}\varphi =\xi ^2{\varDelta }_L \varphi + g(\varphi )+\psi \end{aligned}$$

(2)

with the boundary condition $\frac{\partial }{\partial n} \psi =\frac{\partial }{\partial n}\varphi =0$ on $\partial {\varOmega }\times {]}t_0,\vartheta ]$ and the initial condition $\psi (t_0)=\psi _0, \varphi (t_0)=\varphi _0$ in ${\varOmega }$. Here, $\psi$ is the temperature, $\varphi$ is the phase function, ${\varOmega }\subset {\mathbb {R}}^n$ is a bounded domain with the sufficiently smooth boundary $\partial {\varOmega }, {\varDelta }_L$ is the Laplace operator, ${\partial }/{\partial n}$ is the outward normal derivative, $(U,|\cdot |_U)$ and $(V,|\cdot |_V)$ are Banach spaces, $B\in {\fancyscript{L}}(U;H)$ and $C\in {\fancyscript{L}}(V;H)$ are linear continuous operators, $H=L_2({\varOmega })$, and the function $g(z)$ is the derivative of a so-called potential $G(z)$. Following [4], we assume that $g(z)=az+bz^2-cz^3$.

System (1), (2) (we call it the system $S$) has been investigated by many authors. A rather detailed analysis of the previous results is presented in [5–7]. Among more recent works, we note [8]. Therefore, we do not dwell on this aspect. In what follows, for the sake of simplicity, we assume that $k=\xi =\tau =c=1$. Furthermore, we assume that the following conditions are fulfilled: (A1) The domain ${\varOmega }\subset {\mathbb {R}}^n, n=2,3$, has the boundary of $C^2$-class and (A2) the coefficients $a$ and $b$ are elements of $L_{\infty }([t_0,\vartheta ]\times {\varOmega })$, and $\mathop {{\mathrm {vrai}}}\sup c(t,\eta )>0$ for $(t,\eta )\in [t_0,\vartheta ]\times {\varOmega }$; (A3)  $\{\psi _0,\varphi _o\}\in {\fancyscript{R}}$, where ${\fancyscript{R}} := \{\psi ,\varphi \in W_{\infty }^2({\varOmega }){:}\ \frac{\partial }{\partial n} \psi =\frac{\partial }{\partial n}\varphi =0\ \text{ on }\ \partial {\varOmega }\}$.

Introduce the notation: $Q={\varOmega }\times {]}t_0,\vartheta {[}$;

$$\begin{aligned} W_p^{2,1}(Q)=\left\{ u \mid u, \frac{\partial u}{\partial \eta _i}, \frac{\partial ^2 u}{\partial \eta _i \partial \eta _j}, \frac{\partial u}{\partial t}\in L^p(Q)\right\} \quad \text{ for }\quad p\in [1,\infty {[} \end{aligned}$$

is the standard Sobolev space with the norm

$$\begin{aligned} \Vert u\Vert _{W_p^{2,1}(Q)} =\Bigg ( \int \limits _{{\varOmega }} |u|^p+ \sum _{i=1}^n \left| \frac{\partial u}{\partial \eta _i}\right| ^p+ \sum _{i,j=1}^n \left| \frac{\partial ^2 u}{\partial \eta _i \partial \eta _j}\right| ^p+ \left| \frac{\partial u}{\partial t}\right| ^p\,{\mathrm {d}}\eta {\mathrm {d}}t\Bigg )^{1/p}; \end{aligned}$$

$(\cdot ,\cdot )_H$ and $|\cdot |_H$ are the scalar product and the norm in $H$. A solution of the system $S\, x(\cdot ;t_0,x_0,u(\cdot ),v(\cdot ))=\{\psi (\cdot ;t_0,\psi _0, u(\cdot ),v(\cdot )), \varphi (\cdot ;t_0,\varphi _0,u(\cdot ),v(\cdot ))\}$ is a unique function $x(\cdot )= x(\cdot ; t_0,x_0,u(\cdot ),v(\cdot ))\in V_T^{(1)}=V_1\times V_1, V_1=W_2^{2,1}(Q)$, satisfying relations (1) and (2). As is known (see [7, p. 25, Assertion 5]), under our conditions, there exists a unique solution of $S$ for any $u(\cdot )\in L_{\infty }(T;U)$ and $v(\cdot )\in L_{\infty }(T;V)$.

Let the cost functional

$$\begin{aligned} I(x(\cdot ;t_0,x_0,u_T(\cdot ),v_T(\cdot )))=\int \limits _{t_0}^{\vartheta } \! \int \limits _{{\varOmega }} f(t,\eta ,x(t,\eta ),\nabla x(t,\eta )) \, {\mathrm {d}}\eta \, {\mathrm {d}}t \end{aligned}$$

be given. Here, the symbol $\nabla x$ stands for the gradient of the function $x$; the function $f(t,\eta ,x,y)$ satisfies the Carathéodory condition, i.e., $f(t,\eta ,x,y)$ is measurable (in the Lebesgue sense) in $(t,\eta ) \in T \times {\varOmega }$ for all $x \in {\mathbb {R}}, y \in {\mathbb {R}}^n$, and Lipschitz in $x \in {\mathbb {R}}, y \in {\mathbb {R}}^n$ for almost all $t, \eta \in T \times {\varOmega }$. In addition, $|f(t,\eta ,0,\dots ,0)|\le c_0(t,\eta )$ for almost all $t, \eta \in T \times {\varOmega }$, and $c_0(t,\eta )\in L_{\infty }(T\times {\varOmega })$. At discrete time instants $\tau _{i}\in {\varDelta }=\{\tau _i\}_{i=0}^m, \tau _0=t_0, \tau _{i+1}=\tau _i+\delta , \tau _m=\vartheta$, the phase function $\varphi$ is measured. The results of these measurements are functions $\xi _i^h\in H$ satisfying the inequalities

$$\begin{aligned} |\varphi (\tau _i)-\xi _i^h|_H\le h. \end{aligned}$$

(3)

Here, $h\in {]}0,1{[}$ stands for the level of informational noise. There are two antagonistic players controlling the system $S$ by means of various input actions. One of the players is called a partner, the other one is called an opponent. Let $P\subset U$ and $E \subset V$ be given convex, bounded, and closed sets. The problem undertaken by the partner is as follows. It is necessary to construct a law (a strategy) for forming the control $u$ (with values from $P$) by the feedback principle (on the base of measuring the state $\varphi (\tau _i)$) in such a way that this control minimizes the quality criterion under any possible actions of opponent, whose goals are opposite. Thus, the partner solves the minimax control problem. The problem undertaken by the opponent is “inverse”: it consists in the choice of a law (a strategy) for forming the control $v$ (with values from $E$) also by the feedback principle [on the base of measuring the state $\varphi (\tau _i)$] in such a way that this control maximizes the quality criterion under any possible actions of the partner, whose goals, as mentioned above, are opposite. Consequently, the opponent solves the maximin control problem. This is the description of the problem considered in the paper. The minimax game control problem for systems with distributed parameters has been investigated by many authors (see, for example, [9–12]). In the present work, to solve this problem, we use the approach from [1–3, 13–16].

We denote the function $u(t), t \in [a,b]$, by $u_{a,b}(\cdot )$. The sets of all controls of the partner and the opponent are denoted by the symbols $P_T(\cdot )$ and $E_T(\cdot )$: $P_T(\cdot ):=\{u(\cdot )\in L_2(T;U){:}\ u(t)\in P\ \text{ a.e. }\ t\in T\},\, E_T(\cdot ):=\{v(\cdot )\in L_2(T;V){:}\ v(t)\in E\ \text{ a.e. }\ t\in T\}.$ Any function (perhaps, multifunction) ${\fancyscript{U}}: T\times {\fancyscript{H}}\rightarrow P, {\fancyscript{H}} := H\times H\times H\times H$, is said to be a positional strategy of the partner. A positional strategy of the opponent is defined by analogy: ${\fancyscript{V}}: T\times {\fancyscript{H}}\rightarrow E.$ Positional strategies adjust controls at discrete time moments, given by some partition of the interval $T$. Any function ${\fancyscript{V}}_1:T\times H\times H\rightarrow H$ is said to be a reconstruction strategy. The strategy ${\fancyscript{V}}_1$ is destined to reconstruct the unknown component $\psi (\cdot )$.

Let us present the exact statement of the problems under consideration. Let the partition of $T$ be any finite family ${\varDelta }=\{\tau _i\}_{i=0}^m$, where $\tau _0=t_0, \tau _m=\vartheta , \tau _{i+1}=\tau _i+\delta$; $\delta =\delta ({\varDelta })$ is the diameter of ${\varDelta }$. Auxiliary systems $M_1$ and $M_2$ (models) are introduced. The system $M_1$ has an input $u^*(\cdot )$ and an output $w(\cdot )$; the system $M_2$ has an input $p^h(\cdot )$ and an output $w_1(\cdot )$, respectively. In the process, $p^h(\cdot )$ is formed in such a way that $p^h(\cdot )$ approximates the unknown coordinate $\psi (\cdot )$ of the system $S$. A solution $x(\cdot )$ of the system $S$, starting from an initial state $(t_*,x_*)$ and corresponding to piecewise constant controls $u^h(\cdot )$ and $p^h(\cdot )$ (formed by the feedback principle) and to a control $v_{t_*,\vartheta }(\cdot )\in E_{t_*,\vartheta }(\cdot )$, is called an $(h,{\varDelta },w,{\fancyscript{U}},{\fancyscript{V}}_1)$-motion $x_{{\varDelta },w}^h(\cdot )=x_{{\varDelta },w}^h(\cdot ;t_*,x_*,{\fancyscript{U}},{\fancyscript{V}}_1,v_{t_*,\vartheta }(\cdot ))$, generated by the positional strategies ${\fancyscript{U}}$ and ${\fancyscript{V}}_1$ on the partition ${\varDelta }$. The process of forming the motions $x_{{\varDelta },w}^h(\cdot ), w(\cdot )$, and $w_1(\cdot )$ is realized simultaneously. These three trajectories are all formed by the feedback principle, i.e., it is assumed that $x_{{\varDelta },w}^h(t)=x(t;\tau _i,x_{{\varDelta },w}^h(\tau _i),u_{\tau _i,\tau _{i+1}}^h(\cdot ),v_{\tau _i,\tau _{i+1}}(\cdot )), \, w(t)=w(t;\tau _i,w(\tau _i),u^*_{\tau _i,\tau _{i+1}}(\cdot ))$, $w_1(t)=w_1(t;\tau _i,w_1(\tau _i),p_{\tau _i,\tau _{i+1}}^h(\cdot )) , t\in [\tau _i,\tau _{i+1}{[}$, where

$$\begin{aligned} \begin{array}{c} u^h(t)=u_i^h\in {\fancyscript{U}}(\tau _i,\xi ^h_i,p_i^h,w(\tau _i)),\ \ p^h(t)=p_i^h\in {\fancyscript{V}}_1(\tau _i,\xi ^h_i,w_1(\tau _i)) \\ \text{ for }\ \ t\in [\tau _i,\tau _{i+1}{[},\ \ i\in [i(t_*):m-1],\ \ |\xi _i^h-\varphi (\tau _i)|_H\le h, \\ u^h(t)=u_*^h\in P,\ \ p^h(t)=p_*^h\in H \ \ \text{ for }\ \ t\in [t_*,\tau _{i(t_*)}{[}, \ \ i(t_*)=\min \{i:\tau _i>t_*\}. \end{array} \end{aligned}$$

The set of all $(h,{\varDelta },w,{\fancyscript{U}},{\fancyscript{V}}_1)$-motions is denoted by $X_h(t_*,x_*,{\fancyscript{U}},{\fancyscript{V}}_1,{\varDelta },w)$.

Problem 1

(Problem of the partner) It is necessary to find models $M_1$ and $M_2$, a control $u^*(\cdot )$ for the model $M_1$, as well as a positional strategy of the partner ${\fancyscript{U}}: T\times {\fancyscript{H}} \rightarrow P$ and a positional reconstruction strategy ${\fancyscript{V}}_1: T\times H \rightarrow H$, and a number $c_1$ with the following properties: whatever the value $\varepsilon >0$ may be, one can specify (explicitly) numbers $h_*>0$ and $\delta _*>0$ such that the inequality $I(x_{{\varDelta },w}^h(\cdot )) \le c_1+\varepsilon , \forall x_{{\varDelta },w}^h(\cdot )\in X_h(t_0,x_0,{\fancyscript{U}},{\fancyscript{V}}_1,{\varDelta },w)$ is fulfilled uniformly with respect to all measurements $\xi _i^h$ with properties (3) if $h \le h_*$ and $\delta =\delta ({\varDelta }) \le \delta _*$.

By analogy with the motion $x_{{\varDelta },w}^h(\cdot )=x_{{\varDelta },w}^h(\cdot ;t_*,x_*,{\fancyscript{U}},{\fancyscript{V}}_1,v_{t_*,\vartheta }(\cdot ))$, we define the motion $x_{{\varDelta },z}^h(\cdot ):=x_{{\varDelta },z}^h(\cdot ;t_*,x_*,{\fancyscript{V}},{\fancyscript{V}}_1,{\fancyscript{V}}_2,u_{t_*,\vartheta }(\cdot ))$ corresponding to piecewise constant controls $v^h(\cdot ), v^*(\cdot )$, and $p^h(\cdot )$ (formed by the feedback principle) and to a control $u_{t_*,\vartheta }(\cdot )\in P_{t_*,\vartheta }(\cdot )$. This motion is called an $(h,{\varDelta },{\fancyscript{V}},{\fancyscript{V}}_1,{\fancyscript{V}}_2)$-motion, generated by the positional strategies ${\fancyscript{V}}, {\fancyscript{V}}_1$, and ${\fancyscript{V}}_2$ on the partition ${\varDelta }$. The set of all $(h,{\varDelta },{\fancyscript{V}},{\fancyscript{V}}_1,{\fancyscript{V}}_2)$-motions is denoted by $X_h(t_*,x_*,{\fancyscript{V}},{\fancyscript{V}}_1,{\fancyscript{V}}_2,{\varDelta })$. Note that the trajectory $x_{{\varDelta },z}^h(\cdot ;t_*,x_*,{\fancyscript{V}},{\fancyscript{V}}_1,{\fancyscript{V}}_2,u_{t_*,\vartheta }(\cdot ))$ is formed simultaneously with the other two trajectories, $z(\cdot )$ and $w_1(\cdot )$. Here, $z(\cdot )$ is the trajectory of some auxiliary system $M_3$ (a model), whereas $w_1(\cdot )$ is the trajectory of the system $M_2$. These three trajectories are all formed by the feedback principle, i.e., it is assumed that $x_{{\varDelta },z}^h(t)=x_{{\varDelta },z}^h(t;\tau _i,x_{{\varDelta },z}^h(\tau _i),u_{\tau _i,\tau _{i+1}}(\cdot ),v^h_{\tau _i,\tau _{i+1}}(\cdot )), z(t)=z(t;\tau _i,z(\tau _i),v_{\tau _i,\tau _{i+1}}^*(\cdot )), w_1(t)=w_1(t;\tau _i,w_1(\tau _i),p_{\tau _i,\tau _{i+1}}^h(\cdot )), t\in [\tau _i,\tau _{i+1}{[}$, where

$$\begin{aligned} \begin{array}{c} v^h(t)=v_i^h\in {\fancyscript{V}}(\tau _i,\xi ^h_i,p_i^h,z(\tau _i)),\ \ v^*(t)=v_i^*\in {\fancyscript{V}}_2(\tau _i,\xi ^h_i,p_i^h,z(\tau _i)),\\ [1ex] p^h(t)=p_i^h\in {\fancyscript{V}}_1(\tau _i,\xi ^h_i,w_1(\tau _i))\ \ \text{ for }\ \ t\in [\tau _i,\tau _{i+1}{[},\ \ i\in [i(t_*):m-1],\\ [1ex] |\xi _i^h-\varphi (\tau _i)|_H\le h,\ \ i(t_*)=\min \{i:\tau _i>t_*\}, \\ [1ex] p^h(t)=p_*^h\in H, \ \ v^*(t)=v^*\in E,\ \ v^h(t)=v_*^h\in E \ \ \text{ for }\ \ t\in [t_*,\tau _{i(t_*)}{[}. \end{array} \end{aligned}$$

Problem 2

(Problem of the opponent) It is necessary to find models $M_3$ and $M_2$, as well as a positional strategy of the opponent ${\fancyscript{V}}: T\times {\fancyscript{H}} \rightarrow E$, a positional strategy ${\fancyscript{V}}_2: T\times {\fancyscript{H}} \rightarrow E$, a positional reconstruction strategy ${\fancyscript{V}}_1: T\times H \rightarrow H$ with the following properties: whatever the value $\varepsilon >0$ may be, one can specify (explicitly) numbers $h_*>0$ and $\delta _*>0$ such that the inequality $I(x_{{\varDelta },z}^h(\cdot )) \ge c_1-\varepsilon , \forall x_{{\varDelta },z}^h(\cdot )\in X_h(t_0,x_0,{\fancyscript{V}},{\fancyscript{V}}_1,{\fancyscript{V}}_2,{\varDelta })$, is fulfilled uniformly with respect to all measurements $\xi _i^h$ with properties (3) if $h \le h_*$ and $\delta =\delta ({\varDelta }) \le \delta _*$. Here, the number $c_1$ is the same as in Problem 1.

Remark 2.1

The fact that the value $c_1$ in Problem 2 is the same as in Problem 1 means that $c_1$ is the value of the game, in which the partner aims to minimize a maximally possible value of the quality criterion. At the same time, the goal of the opponent is opposite: He aims to maximize a minimally possible value of the criterion. In this case, a strategy of the partner ${\fancyscript{U}}$ solving Problem 1 is called an $\varepsilon$-optimal minimax strategy, whereas a strategy of the opponent ${\fancyscript{V}}$ solving Problem 2 is called an $\varepsilon$-optimal maximin strategy. Moreover, the pair $({\fancyscript{U}},{\fancyscript{V}})$ constitutes an $\varepsilon$-saddle point.

3 Algorithm for Solving Problem 1

Let the following condition be fulfilled.

Condition 3.1

There exists a convex and closed set $D\subset H$ such that $BP=CE+D$.

Here, we use the notation $BP:=\{ Bu: u\in P\}, CE:=\{ Cv: v\in E\}, CE+D:=\{ u: u=u_1+u_2, u_1 \in CE, u_2 \in D \}$. Let $u^*(\cdot )$ be an optimal control solving

Problem 3

It is necessary to minimize $I(w(\cdot ;t_0,x_0,u(\cdot )))$ over the set $D_T(\cdot )=\{ u(\cdot )\in L_2(T;H): u(t)\in D\ \text{ for } \text{ a. } \text{ a. }\ t\in T\}$. Here, the symbol $w(\cdot )=\{w^{(1)}(\cdot ),w^{(2)}(\cdot )\}=w(\cdot ;t_0,x_0,u(\cdot )), u(\cdot )\in D_T(\cdot )$, denotes the solution of the system

$$\begin{aligned} \begin{array}{l} \displaystyle \frac{\partial }{\partial t}w^{(1)}+ l\displaystyle \frac{\partial }{\partial t}w^{(2)}={\varDelta }_L w^{(1)}+u\quad \text{ in }\quad {\varOmega }\times {]}t_0,\vartheta ],\\ [2ex] \displaystyle \frac{\partial }{\partial t}w^{(2)}={\varDelta }_L w^{(2)}+ g(w^{(2)})+w^{(1)} \end{array} \end{aligned}$$

(4)

with the boundary condition $\frac{\partial }{\partial n} w^{(1)}=\frac{\partial }{\partial n}w^{(2)}=0$ on $\partial {\varOmega }\times {]}t_0,\vartheta ]$ and the initial condition $w^{(1)}(t_0)=\psi _0, w^{(2)}(t_0)=\varphi _0$ in ${\varOmega }$.

Note that Problem 3 was investigated in a number of papers (see, for example, [6, 7, 17], where some optimality conditions were stated). Let $w(\cdot )=w(\cdot ;t_0,x_0,u^*(\cdot ))$ be an optimal trajectory in Problem 3 and $C_{{\mathrm {opt}}}=\inf \{I(w(\cdot ;t_0,x_0,u(\cdot ))){:}\,u(\cdot )\!\in \! D_T(\cdot )\}$ be the optimal value of the quality criterion. As the model $M_1$, we take system (4) with the control $u(\cdot )=u^*(\cdot )$; as the model $M_2$, we take the equation

$$\begin{aligned} \frac{\partial w_1(t,\eta )}{\partial t} = {\varDelta }_L w_1(t,\eta )+ p^h(t,\eta ) + g(w_1(t,\eta ))\quad \text{ in }\quad {\varOmega }\times {]}t_0,\vartheta ] \end{aligned}$$

(5)

with the boundary condition $\frac{\partial w_1}{\partial n} = 0$ on $\partial {\varOmega }\times {]}t_0,\vartheta ]$ and the initial condition $w_1(t_0)=\varphi _0$ in ${\varOmega }$. The strategies ${\fancyscript{U}}$ and ${\fancyscript{V}}_1$ are defined in such a way that:

$$\begin{aligned} {\fancyscript{U}}(t,\xi ,p,w)&= \arg \max \{L(u,y):u\in P\}, \end{aligned}$$

(6)

$$\begin{aligned} {\fancyscript{V}}_1(t,\xi ,w_1)&= \arg \min \{l(t,\alpha ,u,s):u\in U_d\}, \end{aligned}$$

(7)

where $w=\{w^{(1)},w^{(2)}\}, L(u,y)=(y,Bu)_H, y=w^{(1)}-p+l(w^{(2)}-\xi ), l(t,\alpha ,u,s)=\exp ({-}2bt)(s,u)_H+\alpha |u|^2_H, s=w_1-\xi , U_d:=\{u\in H:|u|_H\le d\}$.

Let us describe the algorithm for solving Problem 1. Before the algorithm starts, we fix a value $h \in {]}0,1{[}$, a partition

$$\begin{aligned} {\varDelta }_h = \{ \tau _{h,i} \}_{i=0}^{m_h},\quad \tau _{h,i}=\tau _{h,i-1}+\delta ,\quad \delta =\delta (h), \quad \tau _{h,0}=t_0,\quad \tau _{h,m_h}=\vartheta , \end{aligned}$$

with the diameter $\delta (h)=\tau _{h,i+1}-\tau _{h,i}$, and a function $\alpha =\alpha (h)$: ${]}0,1{[} \rightarrow {\mathbb {R}}^+$,

$$\begin{aligned} \alpha (h)\rightarrow h, \quad (h+\delta (h))\alpha ^{-1}(h)\rightarrow 0 \quad \text{ as }\ h\rightarrow 0. \end{aligned}$$

(8)

The work of the algorithm is decomposed into $m-1, m=m_h$, identical steps. We assume that $u^h(t)=u_0^h\in {\fancyscript{U}}(t_0,\xi _0^h,p_0^h,w(t_0)), p^h(t)= p_0^h\in {\fancyscript{V}}_1(t_0,\xi _0^h,\varphi _0), |\xi _0^h-\varphi _0|_H\le h$ on the interval $[t_0,\tau _1{[}$. Under the action of these piecewise constant controls, as well as of an unknown disturbance $v_{t_0,\tau _1}(\cdot )$, the $(h,{\varDelta },w,{\fancyscript{U}},{\fancyscript{V}}_1)$-motion $\{x_{{\varDelta },w}^h(\cdot )\}_{t_0,\tau _1}=\{ x_{{\varDelta },w}^h (\cdot ;t_0,x_0,u^h_{t_0,\tau _1}(\cdot ),v_{t_0,\tau _1}(\cdot )) \}_{t_0,\tau _1}$ of the system $S$, the trajectory $\{w_1(\cdot )\}_{t_0,\tau _1}=\{w_1(\cdot ;t_0,w_1(t_0),p^h_{t_0,\tau _1}(\cdot ))\}_{t_0,\tau _1}$ of the model $M_2$, and the trajectory $\{w(\cdot )\}_{t_0,\tau _1}=\{w(\cdot ;t_0,x_0,u^*_{t_0,\tau _1}(\cdot ))\}_{t_0,\tau _1}$ of the model $M_1$ are realized. At the moment $t=\tau _1$, we determine $u_1^h$ and $p_1^h$ from the condition

$$\begin{aligned} u_1^h \in {\fancyscript{U}}(\tau _1,\xi _1^h,p_1^h, w(\tau _1)), \quad |\xi _1^h - \varphi _{{\varDelta },w}^h(\tau _1)|_H \le h, \quad p_1^h\in {\fancyscript{V}}_1(\tau _1,\xi _1^h,w_1(\tau _1)); \end{aligned}$$

i.e., we assume that $u^h(t)=u_1^h$ and $p^h(t)=p_1^h$ for $t\in [\tau _1,\tau _2{[}.$ Then, we calculate the $(h,{\varDelta },w,{\fancyscript{U}},{\fancyscript{V}}_1)$-motion $\{x_{{\varDelta },w}^h(\cdot )\}_{\tau _1,\tau _2} =\{x_{{\varDelta },w}^h(\cdot ;\tau _1,x_{{\varDelta },w}^h(\tau _1), u^h_{\tau _1,\tau _2}(\cdot ) , v_{\tau _1,\tau _2}(\cdot ))\}_{\tau _1,\tau _2},$ the trajectory $\{w_1(\cdot )\}_{\tau _1,\tau _2}=\{w_1(\cdot ;\tau _1,w_1(\tau _1),p^h_{\tau _1,\tau _2}(\cdot ))\}_{\tau _1,\tau _2}$ of the model $M_2$, and the trajectory $\{w(\cdot )\}_{\tau _1,\tau _2}=\{w(\cdot ;\tau _1,w(\tau _1),u^*_{\tau _1,\tau _2}(\cdot )\}_{\tau _1,\tau _2}$ of the model $M_1$. Let the $(h,{\varDelta },w,{\fancyscript{U}},{\fancyscript{V}}_1)$-motion $x_{{\varDelta },w}^h(\cdot )$, the trajectory $w_1(\cdot )$ of the model $M_2$, and the trajectory $w(\cdot )$ of the model $M_1$ be defined on the interval $[t_0,\tau _i]$. At the moment $t=\tau _i$, we assume that

$$\begin{aligned} u_i^h \in {\fancyscript{U}}(\tau _i,\xi _i^h,p_i^h,w(\tau _i)), \quad |\xi _i^h - \varphi _{{\varDelta },w}^h(\tau _i)|_H \le h, \quad p_i^h\in {\fancyscript{V}}_1(\tau _i,\xi _i^h,w_1(\tau _i)); \end{aligned}$$

(9)

i.e., we set $u^h(t)=u_i^h$ and $p^h(t)=p_i^h$ for $t\in [\tau _i,\tau _{i+1}{[}$. As the result of the action of these controls and of an unknown disturbance $v_{\tau _i,\tau _{i+1}}(\cdot )$, the $(h,{\varDelta },w,{\fancyscript{U}},{\fancyscript{V}}_1)$-motion $\{x_{{\varDelta },w}^h(\cdot )\}_{\tau _i,\tau _{i+1}} =\{x_{{\varDelta },w}^h(\cdot ;\tau _i,x_{{\varDelta },w}^h(\tau _i),u^h_{\tau _i,\tau _{i+1}}(\cdot ),v_{\tau _i,\tau _{i+1}}(\cdot ) )\}_{\tau _i,\tau _{i+1}},$ the trajectory $\{w_1(\cdot )\}_{\tau _i,\tau _{i+1}}=\{w_1(\cdot ;\tau _i,w_1(\tau _i),p^h_{\tau _i,\tau _{i+1}}(\cdot ))\}_{\tau _i,\tau _{i+1}}$ of the model $M_2$, and the trajectory $\{w(\cdot )\}_{\tau _i,\tau _{i+1}}=\{w(\cdot ;\tau _i,w(\tau _i),u^*_{\tau _i,\tau _{i+1}}(\cdot ))\}_{\tau _i,\tau _{i+1}}$ of the model $M_1$ are realized on the interval $[\tau _i,\tau _{i+1}]$. The procedure of forming the $(h,{\varDelta },w,{\fancyscript{U}},{\fancyscript{V}}_1)$-motion and the trajectories of models $M_2$ and $M_1$ stops at the moment $\vartheta$.

Theorem 3.1

Let $c_1=C_{{\mathrm {opt}}}$ and let the models $M_1$ and $M_2$ be specified by relations (4) and (5). Then, the strategies ${\fancyscript{U}}$ and ${\fancyscript{V}}_1$ of form (6), (7) solve Problem 1.

Proof

To prove the theorem, we estimate the variation in the functional

$$\begin{aligned} \begin{array}{c} {\varLambda }(t,x_{{\varDelta },w}^h(\cdot ),w(\cdot ))= {\varLambda }^0(t,x_{{\varDelta },w}^h(\cdot ),w(\cdot )) \\ {} + 0.5\displaystyle {\int \limits _0^t\Big \{\int \limits _{\varOmega } |\nabla \pi ^h(\rho ,\eta )|^2\,{\mathrm {d}}\eta +l^2\int \limits _{\varOmega } |\nabla \mu ^h(\rho ,\eta )|^2\,{\mathrm {d}}\eta \Big \} {\mathrm {d}}\rho ,} \end{array} \end{aligned}$$

where ${\varLambda }^0(t,x_{{\varDelta },w}^h(\cdot ),w(\cdot ))=0.5|g^h(t)|_H^2 + 0.5l^2|\mu ^h(t)|_H^2, \pi ^h(t) = w^{(1)}(t) - \psi _{{\varDelta },w}^h(t), \mu ^h(t)=w^{(2)}(t)-\varphi _{{\varDelta },w}^h(t), g^h(t)=\pi ^h(t)+l\mu ^h(t)$. It is easily seen that the functions $\pi ^h(\cdot )$ and $\mu ^h(\cdot )$ are solutions of the system

$$\begin{aligned} \begin{array}{c} \displaystyle \frac{\partial \pi ^h(t,\eta )}{\partial t}+ l \displaystyle \frac{\partial \mu ^h(t,\eta )}{\partial t} = {\varDelta }_L \pi ^h(t,\eta )+u^*(t,\eta )-(Bu^h)(t,\eta )+(Cv)(t,\eta )\\ \text{ in }\ {\varOmega }\times {]}t_0,\vartheta ], \quad \displaystyle \frac{\partial \mu ^h(t,\eta )}{\partial t} = {\varDelta }_L\mu ^h(t,\eta )+ R^h(t,\eta )\mu ^h(t,\eta )+\pi ^h(t,\eta ) \end{array} \end{aligned}$$

(10)

with the initial condition $\pi ^h(t_0)=\mu ^h(t_0)=0\ \text{ in }\ {\varOmega }$ and with the boundary condition $\displaystyle {\frac{\partial \pi ^h}{\partial n}=\frac{\partial \mu ^h}{\partial n}=0\ \text{ on }\ \partial {\varOmega }\times {]}t_0,\vartheta ].}$ Here, $R^h(t,\eta )=a(t,\eta )+ b(t,\eta )(w^{(1)}(t,\eta )+ \varphi ^h_{{\varDelta },w}(t,\eta ))- (({w^{(1)}}(t,\eta ))^2+w^{(1)}(t,\eta )\varphi ^h_{{\varDelta },w}(t,\eta )+(\varphi ^h_{{\varDelta },w})^2(t,\eta )).$ Multiplying scalarly the first equation of (10) by $g^h(t)$, and the second one by $\mu ^h(t)$, we obtain

$$\begin{aligned}&(g^h(t),g_t^h(t))_H+\int \limits _{{\varOmega }}\{ |\nabla \pi ^h(t,\eta )|^2+ l\nabla \pi ^h(t,\eta )\nabla \mu ^h(t,\eta )\}\,{\mathrm {d}}\eta \nonumber \\&\quad =(g^h(t),u^*(t)-Bu^h(t)+Cv(t))_H, \\&(\mu ^h(t),\mu _t^h(t))_H+\int \limits _{{\varOmega }} |\nabla \mu ^h(t,\eta )|^2\,{\mathrm {d}}\eta \le (\pi ^h(t),\mu ^h(t))_H+b|\mu ^h(t)|_H^2\quad \text{ for } \text{ a.a. }\ t\in T. \nonumber \end{aligned}$$

(11)

Here, we use the inequality $\mathop {\mathop {{\mathrm {vrai}}}\max }\limits _{(t,\eta )\in T\times {\varOmega }} \{a(t,\eta )+b(t,\eta )(v_1+v_2)- (v_1^2+v_1v_2+v_2^2)\}\le b$, which is valid for any $v_1, v_2\in {\mathbb {R}}$. It is evident that the inequality

$$\begin{aligned} \int \limits _{{\varOmega }} l(\nabla \pi ^h(t,\eta ),\nabla \mu ^h(t,\eta ))\,{\mathrm {d}}\eta \ge -0.5\int \limits _{{\varOmega }} \{ |\nabla \pi ^h(t,\eta )|^2+ l^2|\nabla \mu ^h(t,\eta )|^2\}\,{\mathrm {d}}\eta , \end{aligned}$$

(12)

for a.a. $t\in T$, is fulfilled. Let us multiply the first inequality of (11) by $l^2$ and add to the second one. Taking into account (12), we have for a.a. $t\in T$

$$\begin{aligned}&(g^h(t),g_t^h(t))_H+l^2(\mu ^h(t),\mu _t^h(t))_H+0.5 \int \limits _{\varOmega }\{|\nabla \pi ^h(t,\eta )|^2+l^2 |\nabla \mu ^h(t,\eta )|^2\}\,{\mathrm {d}}\eta \nonumber \\&\quad \le (g^h(t), u^*(t)-Bu^h(t)+Cv(t))_H{} + l^2(\pi ^h(t),\mu ^h(t))_H+ bl^2|\mu ^h(t)|_H^2. \end{aligned}$$

(13)

Note that $\pi ^h(t)=g^h(t)-l\mu ^h(t)$. In this case, for a.a. $t\in T$

$$\begin{aligned} (\pi ^h(t),\mu ^h(t))_H+b|\mu ^h(t)|^2_H&= (g^h(t)-l\mu ^h(t),\mu ^h(t))_H {}+b|\mu ^h(t)|^2_H \nonumber \\&= (g^h(t),\mu ^h(t))_H+(b-l)|\mu ^h(t)|^2_H \nonumber \\&\le 0.5 (|g^h(t)|^2_H+ (0.5+|b-l|)|\mu ^h(t)|^2_H. \end{aligned}$$

(14)

Combining (13) and (14), we obtain for a.a. $t\in T$

$$\begin{aligned}&\frac{{\mathrm {d}}}{{\mathrm {d}}t}{\varLambda }^0(t,x_{{\varDelta },w}^h(\cdot ),w(\cdot ))+0.5\int \limits _{\varOmega }\{|\nabla \pi ^h(t,\eta )|^2+ l^2|\nabla \mu ^h(t,\eta )|^2\}\,{\mathrm {d}}\eta \nonumber \\&\quad \le 2l^2\lambda ^2{\varLambda }^0(t,x_{{\varDelta },w}^h(\cdot ),w(\cdot )) +(g^h(t),u^*(t)-Bu^h(t)+Cv(t))_H. \end{aligned}$$

(15)

Estimate the last term in the right-hand side of inequality (15). For $t\in [\tau _i,\tau _{i+1}{[}$,

$$\begin{aligned} |g^h(t)-y_i^h|_H=|\pi ^h(t)+l \mu ^h(t)-y_i^h|_H\le \lambda _{1,i}(t)+\lambda _{2,i}(t), \end{aligned}$$

(16)

where $y_i^h=w^{(1)}(\tau _i)-p_i^h-l(w^{(2)}(\tau _i)-\xi _i^h), \lambda _{1,i}(t)=|w^{(1)}(t)-\psi ^h_{{\varDelta },w}(t)-w^{(1)}(\tau _i)+p_i^h|_H,\, \lambda _{2,i}(t)=l|w^{(2)}(t)-\varphi ^h_{{\varDelta },w}(t)-w^{(2)}(\tau _i)+\xi _i^h|_H.$ By virtue of (3), we have

$$\begin{aligned} \begin{array}{c} \displaystyle {\lambda _{1,i}(t)\le |p_i^h-\psi ^h_{{\varDelta },w}(t)|_H+ \int \limits _{\tau _i}^{t}|\dot{w}^{(1)}(\tau )|_H\,{\mathrm {d}}\tau }, \\ \displaystyle {\lambda _{2,i}(t)\le lh+ l\int \limits _{\tau _i}^{t}\{|\dot{\varphi }^h_{{\varDelta },w}(\tau )|_H+|\dot{w}^{(2)}(\tau )|_H\}\,{\mathrm {d}}\tau , \quad t\in \delta _i= [\tau _i,\tau _{i+1}{[}.} \end{array} \end{aligned}$$

(17)

From (16) and (17), for $t\in \delta _i$, it follows that

$$\begin{aligned} |g^h(t)-y_i^h|_H\le lh+ \int \limits _{\tau _i}^t \{ l|\dot{\varphi }^h_{{\varDelta },w}(\tau )|_H+|\dot{w}^{(1)}(\tau )|_H+|\dot{w}^{(2)}(\tau )|_H\}\,{\mathrm {d}}\tau + |p_i^h-\psi ^h_{{\varDelta },w}(t)|_H. \end{aligned}$$

(18)

It follows from [16] that $|p^h(\cdot )-\psi (\cdot )|^2_{L_2(T;H)}\le K\mu (h),\, K={\mathrm {const}} > 0, \mu (h)=(h+\delta (h)+\alpha (h))^{1/2}+(h+\delta (h)))\alpha ^{-1}(h))$, i.e., the strategy ${\fancyscript{V}}_1$ of form (7) is a reconstruction strategy. By virtue of (3), taking into account the latter inequality, from estimate (18), we derive

$$\begin{aligned} \sum _{i=0}^{m-1}\int \limits _{\tau _i}^{\tau _{i+1}} M(t;\tau _i)\,{\mathrm {d}}t\le k_1(h+\delta )+k_2\int \limits _{t_0}^{\vartheta } |p^h(\tau )-\psi ^h_{{\varDelta },w}(\tau )|_H\,{\mathrm {d}}\tau \le k_3\mu ^{1/2}(h), \end{aligned}$$

(19)

where $M(t;\tau _i)=|g^h(t)-y_i^h|_H\{ |Bu_i^{h}|_H+|Cv(t)|_H+|u^*(t)|_H\}$ for a.a. $t\in \delta _i$. Then, it follows from (15) that for a.a. $t\in \delta _i$,

$$\begin{aligned}&(g^h(t),g_t^h(t))_H+l^2(\mu ^h(t),\mu _t^h(t))_H+0.5\!\displaystyle \int \limits _{\varOmega }\{|\nabla \pi ^h(t,\eta )|^2 + l^2|\nabla \mu ^h(t,\eta )|^2\}\, {\mathrm {d}}\eta \nonumber \\&\quad \le 2l^2\lambda ^2{\varLambda }^0(t,x^h_{{\varDelta },w}(\cdot ),w(\cdot ))+ (y_i^h,u^*(t)-Bu_i^h+Cv(t))_H+M(t;\tau _i). \end{aligned}$$

(20)

Here (see (9)), $u_i^h \in {\fancyscript{U}}(\tau _i,\xi _i^h,p_i^h,w^{(1)}(\tau _i),w^{(2)}(\tau _i))$; $\xi _i^h$ is an inaccurate measurement of the phase state $\varphi ^h_{{\varDelta },w}(\tau _i)$; $v_{\tau _i,\tau _{i+1}}(\cdot )$ is an unknown realization of the control of the opponent; the strategy ${\fancyscript{U}}$ is determined from formula (6). By virtue of Condition 3.1, there exists a control $u_{\tau _i,\tau _{i+1}}^{(1)}(\cdot )\in P_{\tau _i,\tau _{i+1}}(\cdot )$ such that

$$\begin{aligned} Bu^{(1)}(t)=Cv(t)+u^*(t)\quad \text{ for } \text{ a. } \text{ a. }\quad t\in [\tau _i,\tau _{i+1}]. \end{aligned}$$

(21)

From (21), we have that for a. a. $t\in [\tau _i,\tau _{i+1}]$

$$\begin{aligned} (y_i^h,Cv(t)+u^*(t)-Bu_i^h)_H = (B(u^{(1)}(t)-u_i^h),y^h_i)_H\le 0. \end{aligned}$$

(22)

We deduce from (20) and (22) that

$$\begin{aligned} \frac{{\mathrm {d}}{\varLambda }(t,x^h_{{\varDelta },w} (\cdot ),w(\cdot ))}{{\mathrm {d}}t}&= (g^h(t),g_t^h(t))_H+l^2(\mu ^h(t),\mu _t^h(t))_H {} +0.5\int \limits _{\varOmega }\{|\nabla \pi ^h(t,\eta )|^2\nonumber \\&+\,\, l^2|\nabla \mu ^h(t,\eta )|^2 \}{\mathrm {d}}\eta \le l^2\lambda ^2{\varLambda }^0(t,x^h_{{\varDelta },w}(\cdot ),w(\cdot ))\nonumber \\&+ \,M(t;\tau _i)\quad \text{ for } \text{ a.a. }\ t\in \delta _i. \end{aligned}$$

(23)

Using (23) and (19), by virtue of the Gronwall Lemma, we obtain

$$\begin{aligned} {\varLambda }^0(t,x_{{\varDelta },w}^h(\cdot ),w(\cdot ))\le k_4\sum _{i=0}^{m-1} \int \limits _{\tau _i}^{\tau _{i+1}} M(t;\tau _i)\,{\mathrm {d}}t\le k_5\mu ^{1/2}(h),\quad \forall t\in T. \end{aligned}$$

Hence and from (23), we derive ${\varLambda }(t,x_{{\varDelta },w}^h(\cdot ),w(\cdot ))\le k_6\mu ^{1/2}(h), \forall t\in T$. The statement of the theorem follows from the last inequality. The theorem is proved. $\square$

4 Algorithm for Solving Problem 2

We design an algorithm for solving Problem 2. Assume that, as everywhere above, Condition 3.1 is fulfilled. As the model $M_3$, we take the system

$$\begin{aligned} \begin{array}{l} \displaystyle \frac{\partial }{\partial t}z^{(1)}+ l\displaystyle \frac{\partial }{\partial t}z^{(2)}={\varDelta }_L z^{(1)}+v^*\quad \text{ in }\quad {\varOmega }\times {]}t_0,\vartheta ],\\ [1ex] \displaystyle \frac{\partial }{\partial t}z^{(2)}={\varDelta }_L z^{(2)}+ g(z^{(2)})+z^{(1)} \end{array} \end{aligned}$$

(24)

with the boundary condition $\frac{\partial }{\partial n} z^{(1)}=\frac{\partial }{\partial n}z^{(2)}=0$ on $\partial {\varOmega }\times {]}t_0,\vartheta ]$ and the initial condition $z^{(1)}(t_0)=\psi _0, z^{(2)}(t_0)=\varphi _0$ in ${\varOmega }$. Its solution is denoted by the symbol $z(\cdot )=\{z^{(1)}(\cdot ),z^{(2)}(\cdot )\}=z(\cdot ;t_0,z_0,v^*(\cdot ))$, where $z_0=\{\psi _0,\varphi _0\}$. The model $M_2$ is described by relations (5). The strategies ${\fancyscript{V}}, {\fancyscript{V}}_1$, and ${\fancyscript{V}}_2$ are defined as follows:

$$\begin{aligned} {\fancyscript{V}}(t,\xi ,p,z)&:= \arg \max \{L_1(v,\chi ){:}\ v\in E\},\end{aligned}$$

(25)

$$\begin{aligned} {\fancyscript{V}}_1(t,\xi ,w_1)&:= \arg \min \{l(t,\alpha ,u,s){:}\ u\in U_d\}, \end{aligned}$$

(26)

$$\begin{aligned} {\fancyscript{V}}_2(t,\xi ,p,z)&:= B\tilde{u}-C\tilde{v}, \end{aligned}$$

(27)

where $\tilde{u}\in \arg \min \{L(u,\chi ):u\in P\}, L_1(v,\chi )=(\chi ,Cv)_H, \chi =z^{(1)}-p+l(z^{(2)}-\xi ), z=\{z^{(1)},z^{(2)}\}, L(u,\chi )=(\chi ,Bu)_H, \tilde{v}=\tilde{v}(\tilde{u})$ is an arbitrary element from the set $E$ such that $B\tilde{u}-C\tilde{v}\in D$.

Let us pass to the description of the algorithm for solving Problem 2. Before the algorithm starts, we fix a value $h \in {]}0,1{[}$, a function $\alpha =\alpha (h)$: ${]}0,1{[}\rightarrow {\mathbb {R}}^+$ with properties (8), and a partition ${\varDelta }_h= \{\tau _{h,i}\}_{i=0}^{m_h}$ with the diameter $\delta (h)$. The work of the algorithm is decomposed into $m-1, m=m_h$, identical steps. We assume that

$$\begin{aligned} \begin{array}{c} v^h(t)=v_0^h\in {\fancyscript{V}}(t_0,\xi _0^h,p_0^h,z(t_0)),\quad |\xi _0^h-\varphi _0|_H\le h, \\ [1ex] p^h(t)=p_0^h\in {\fancyscript{V}}_1(t_0,\xi _0^h,w_1(t_0)),\quad v^*(t)=v^*_0\in {\fancyscript{V}}_2(t_0,\xi _0^h,p_0^h,z(t_0)) \end{array} \end{aligned}$$

(28)

on the interval $[t_0,\tau _1{[}$. Under the action of these piecewise constant controls, as well as of an unknown control $u_{t_0,\tau _1}(\cdot )$, the $(h,{\varDelta },{\fancyscript{V}},{\fancyscript{V}}_1,{\fancyscript{V}}_2)$-motion $\{x_{{\varDelta },z}^h(\cdot )\}_{t_0,\tau _1}= \{x_{{\varDelta },z}^h (\cdot ;t_0,x_0, u_{t_0,\tau _1}(\cdot ), v^h_{t_0,\tau _1}(\cdot ))\}_{t_0,\tau _1}$ of the system $S$, the trajectory $\{w_1(\cdot )\}_{t_0,\tau _1}= \{w_1(\cdot ;t_0,w_1(t_0),p_{t_0,\tau _2}^h(\cdot ))\}_{t_0,\tau _1}$ of the model $M_2$, and the trajectory $\{z(\cdot )\}_{t_0,\tau _1}=\{z(\cdot ; t_0, z_0, v^*_{t_0,\tau _1}(\cdot ))\}_{t_0,\tau _1}$ of the model $M_3$ are realized. At the moment $t=\tau _1$, we determine $v_1^h, p_1^h$, and $v^*_1$ from the condition

$$\begin{aligned} v_1^h \in {\fancyscript{V}}(\tau _1,\xi _1^h,p_1^h,z(\tau _1)), \ \ p_1^h\in {\fancyscript{V}}_1(\tau _1,\xi _1^h,w_1(\tau _1)),\ \ v_1^*\in {\fancyscript{V}}_2(\tau _1,\xi _1^h,p_1^h,z(\tau _1)), \end{aligned}$$

(29)

$|\xi _1^h - \varphi _{{\varDelta },z}^h(\tau _1)|_H \le h$, where $v_1^*=B\tilde{u}^h_{1}-C\tilde{v}^h_{1}, \tilde{u}^h_{1}=\arg \min \{L(u,\chi _1^h){:}\ u\in P\}, \chi _1^h=z^{(1)}(\tau _1)-p_1^h+l(z^{(2)}(\tau _1)-\xi ^h_1),\,\tilde{v}^h_{1}=\tilde{v}^h_{1}(\tilde{u}^h_{1})$ is an arbitrary element from the set $E$ such that $B\tilde{u}^h_{1}-C\tilde{v}^h_{1}\in D$. We assume that

$$\begin{aligned} v^h(t)=v_1^h,\quad p^h(t)=p_1^h,\quad v^*(t)=v_1^*\quad \text{ for }\quad t\in [\tau _1,\tau _2{[}. \end{aligned}$$

Then, we calculate the realization of the $(h,{\varDelta },{\fancyscript{V}},{\fancyscript{V}}_1,{\fancyscript{V}}_2)$-motion $\{x_{{\varDelta },z}^h(\cdot )\}_{\tau _1,\tau _2} \!=\! \{x_{{\varDelta },z}^h(\cdot ;\tau _1,x_{{\varDelta },z}^h(\tau _1), u_{\tau _1,\tau _2}(\cdot ),v^h_{\tau _1,\tau _2}(\cdot ))\}_{\tau _1,\tau _2}$, the trajectory $\{w_1(\cdot )\}_{\tau _1,\tau _2}\!=\! \{w_1(\cdot ;\tau _1,w_1(\tau _1),p_{\tau _1,\tau _2}^h(\cdot )))\}_{\tau _1,\tau _2}$ of the model $M_2$, and the trajectory $z_{\tau _1,\tau _2}(\cdot )= \{z(\cdot ;\tau _1,z(\tau _1),v^*_{\tau _1,\tau _2}(\cdot )\}_{\tau _1,\tau _2}$ of the model $M_3$. Let the $(h,{\varDelta },{\fancyscript{V}},{\fancyscript{V}}_1,{\fancyscript{V}}_2)$-motion $x_{{\varDelta },z}^h(\cdot )$, the trajectory $w_1(\cdot )$ of the model $M_2$, and the trajectory $z(\cdot )$ of the model $M_3$ be defined on the interval $[t_0,\tau _i]$. At the moment $t=\tau _i$, we assume that

$$\begin{aligned} v_i^h \in {\fancyscript{V}}(\tau _i,\xi _i^h,p_i^h,z(\tau _i)), \quad p_i^h\in {\fancyscript{V}}_1(\tau _i,\xi _i^h,w_1(\tau _i)),\quad v_i^*\in {\fancyscript{V}}_2(\tau _i,\xi _i^h,p_i^h,z(\tau _i)), \end{aligned}$$

(30)

$|\xi _i^h -\varphi _{{\varDelta },z}^h(\tau _i)|_H \le h$, where $v^*_i=B\tilde{u}^h_{i}-C\tilde{v}^h_{i}, \tilde{u}^h_{i}=\arg \min \{L(u,\chi _i^h){:}\ u\in P\}, \chi _i^h=z^{(1)}(\tau _i)-p_i^h+l(z^{(2)}(\tau _i)-\xi _i^h)$, $\tilde{v}^h_{i}=\tilde{v}^h_{i}(\tilde{u}^h_{i})$ is an arbitrary element from the set $E$ such that $B\tilde{u}^h_{i}-C\tilde{v}^h_{i}\in D$. We assume that

$$\begin{aligned} v^h(t)=v_i^h, \quad p^h(t)=p_i^h,\quad v^*(t)=v_i^*\quad \text{ for }\quad t\in [\tau _i,\tau _{i+1}{[}. \end{aligned}$$

The trajectory $\{w_1(\cdot )\}_{\tau _i,\tau _{i+1}}= \{w_1(\cdot ;\tau _i,w_1(\tau _i),p_{\tau _i,\tau _{i+1}}^h(\cdot ))\}_{\tau _i,\tau _{i+1}}$ of the model $M_2$, the trajectory $\{z(\cdot )\}_{\tau _i,\tau _{i+1}}=\{z(\cdot ;\tau _i,z(\tau _i),v^*_{\tau _i,\tau _{i+1}}(\cdot ))\}_{\tau _i,\tau _{i+1}}$ of the model $M_3$, and the $(h,{\varDelta },{\fancyscript{V}},{\fancyscript{V}}_1,{\fancyscript{V}}_2)$-motion

$$\begin{aligned} \{x_{{\varDelta },z}^h(\cdot )\}_{\tau _i,\tau _{i+1}} = \{x_{{\varDelta },z}^h(\cdot ;\tau _i,x_{{\varDelta },z}^h(\tau _i), u_{\tau _i,\tau _{i+1}}(\cdot ),v^h_{\tau _i,\tau _{i+1}}(\cdot ) )\}_{\tau _i,\tau _{i+1}} \end{aligned}$$

are realized on the interval $[\tau _i,\tau _{i+1}]$ as the result of the action of these controls and an unknown control $u_{\tau _i,\tau _{i+1}}(\cdot )$. The above procedure of forming the $(h,{\varDelta },{\fancyscript{V}},{\fancyscript{V}}_1,{\fancyscript{V}}_2)$-motion and the trajectories of the models $M_2$ and $M_3$ stops at the moment $\vartheta$.

Theorem 4.1

Let $c_1=C_{{\mathrm {opt}}}$ and let the models $M_3$ and $M_2$ be specified by relations (24) and (5). Then, the strategies ${\fancyscript{V}}, {\fancyscript{V}}_1$, and ${\fancyscript{V}}_2$ of form (25)–(27) solve Problem 2.

Proof

Note that $I(w(\cdot ))\ge c_1, \forall w(\cdot )\in W_T(\cdot ).$ Here and below, the symbol $W_T(\cdot )=W_T(\cdot ;t_0,x_0)$ stands for the bundle of solutions of system (4), i.e., $W_T(\cdot ;t_0,x_0)=\{w(\cdot ;t_0,x_0,u(\cdot )),\ u(\cdot )\in D_T(\cdot )\},$ whereas the symbol $W_T(t)$ denotes the section of this bundle at the moment $t$. Therefore, by virtue of the Lipschitz property of the function $f$, for any $\varepsilon >0$, it is sufficient to find $h_1>0$ and $\delta _1>0$ such that the inequality

$$\begin{aligned} \lambda (\vartheta ,x^h_{{\varDelta },z}(\cdot ),W_{T}(\cdot ))= \inf \{{\varLambda }(\vartheta ,x^h_{{\varDelta },z}(\cdot ),z(\cdot )):z(\cdot )\in W_{T}(\cdot )\}\le \varepsilon \end{aligned}$$

is fulfilled for $h\in {]}0,h_1{[}$ and $\delta \in {]}0,\delta _1{[}$. Here,

$$\begin{aligned}&{\varLambda }(t,x^h_{{\varDelta },z}(\cdot ),z(\cdot )) = {\varLambda }^0(t,x_{{\varDelta },z}^h(\cdot ),z(\cdot )) {}+ 0.5\int \limits _0^t\Big \{\int \limits _{\varOmega } |\nabla \tilde{\pi }^h(\rho ,\eta )|^2\,{\mathrm {d}}\eta \\&{}+l^2\int \limits _{\varOmega } |\nabla \tilde{\mu }^h(\rho ,\eta )|^2\,{\mathrm {d}}\eta \Big \} {\mathrm {d}}\rho , \quad {\varLambda }^0(t,x^h_{{\varDelta },z}(\cdot ),z(\cdot ))=0.5|\tilde{g}^h(t)|_H^2 + 0.5l^2|\tilde{\mu }^h(t)|_H^2, \\&\tilde{\pi }^h(t) = z^{(1)}(t) {-} \psi _{{\varDelta },z}^h(t), \quad \tilde{\mu }^h(t)=z^{(2)}(t)-\varphi _{{\varDelta },z}^h(t), \quad \tilde{g}^h(t)=\tilde{\pi }^h(t)+l\tilde{\mu }^h(t). \end{aligned}$$

It is easily seen that the inequality

$$\begin{aligned} \lambda (\tau _{i+1},x^h_{{\varDelta },z}(\cdot ),W_{T}(\cdot ))\le {\varLambda }(\tau _{i+1},x^h_{{\varDelta },z}(\cdot ),z(\cdot )) \end{aligned}$$

is valid, since $z(\tau _{i+1};\tau _{i},z(\tau _{i}),v^*_{\tau _{i},\tau _{i+1}}(\cdot ))\in W_{T}(\tau _{i+1})$ by the choice of the control $v^*(t), t\in [\tau _i,\tau _{i+1}{[}$. Thus, to prove the theorem, it is sufficient to estimate the variation in the value ${\varLambda }(t,x^h_{{\varDelta },z}(\cdot ),z(\cdot ))$. Note that the functions $\tilde{\pi }^h(\cdot )$ and $\tilde{\mu }^h(\cdot )$ are solutions of the system

$$\begin{aligned} \begin{array}{c} \displaystyle \frac{\partial \tilde{\pi }^h(t,\eta )}{\partial t}+ l \displaystyle \frac{\partial \tilde{\mu }^h(t,\eta )}{\partial t} = {\varDelta }_L \tilde{\pi }^h(t,\eta )+v^*(t,\eta )-(Bu)(t,\eta )+(Cv^h)(t,\eta ) \\ \text{ in }\ \ {\varOmega }\times {]}t_0,\vartheta ], \quad \displaystyle \frac{\partial \tilde{\mu }^h(t,\eta )}{\partial t} = {\varDelta }_L\tilde{\mu }^h(t,\eta )+ \tilde{R}^h(t,\eta )\tilde{\mu }^h(t,\eta )+\tilde{\pi }^h(t,\eta ) \end{array} \end{aligned}$$

(31)

with the initial condition $\tilde{\pi }^h(t_0)=\tilde{\mu }^h(t_0)=0$ in ${\varOmega }$ and the boundary condition $\frac{\partial \tilde{\pi }^h}{\partial n}=\frac{\partial \tilde{\mu }^h}{\partial n}=0$ on $\partial {\varOmega }\times {]}t_0,\vartheta ]$. Here, $\tilde{R}^h(t,\eta ) =a(t,\eta )+b(t,\eta )(z^{(1)}(t,\eta )+ \varphi ^h_{{\varDelta },z}(t,\eta ))- (({z^{(1)}}(t,\eta ))^2+ z^{(1)}(t,\eta )\varphi ^h_{{\varDelta },z}(t,\eta )+ (\varphi ^h_{{\varDelta },z})^2(t,\eta )).$ Multiplying scalarly the first equation of (31) by $\tilde{g}^h(t)$, and the second one by $\tilde{\mu }^h(t)$, we have

$$\begin{aligned}&(\tilde{g}^h(t),\tilde{g}_t^h(t))_H+\int \limits _{{\varOmega }}\{ |\nabla \tilde{\pi }^h(t,\eta )|^2+ l\nabla \tilde{\pi }^h(t,\eta )\nabla \tilde{\mu }^h(t,\eta )\}\,{\mathrm {d}}\eta \nonumber \\&\quad =(\tilde{g}^h(t),v^*(t)-Bu(t)+Cv^h(t))_H, \\&(\tilde{\mu }^h(t),\tilde{\mu }_t^h(t))_H+\int \limits _{{\varOmega }} |\nabla \tilde{\mu }^h(t,\eta )|^2\,{\mathrm {d}} \eta \le (\tilde{\pi }^h(t),\tilde{\mu }^h(t))_H+b|\tilde{\mu }^h(t)|_H^2\quad \text{ for } \text{ a.a. }\ t\in T. \nonumber \end{aligned}$$

(32)

Now, multiply the second inequality of (32) by $l^2$ and add to the first one. We obtain for a.a. $t\in T$

$$\begin{aligned}&(\tilde{g}^h(t),\tilde{g}_t^h(t))_H+l^2(\tilde{\mu }^h(t),\tilde{\mu }_t^h(t))_H+0.5\!\int \limits _{\varOmega }\{|\nabla \tilde{\pi }^h(t,\eta )|^2+l^2 |\nabla \tilde{\mu }^h(t,\eta )|^2\}\,{\mathrm {d}}\eta \nonumber \\&\le (\tilde{g}^h(t), v^*(t)-Bu(t)+Cv^h(t))_H{} + l^2(\tilde{\pi }^h(t),\tilde{\mu }^h(t))_H+ bl^2|\tilde{\mu }^h(t)|_H^2. \end{aligned}$$

(33)

Then, we have (see (19))

$$\begin{aligned} \sum _{i=0}^{m-1}\int \limits _{\tau _i}^{\tau _{i+1}} M_*(t;\tau _i)\,{\mathrm {d}}t\le c_1\mu ^{1/2}(h), \end{aligned}$$

(34)

where

$$\begin{aligned} M_*(t;\tau _i)=|\tilde{g}^h(t)-\chi _i^h|_H\{ |Bu(t)|_H+|Cv^h_i|_H+|v^*(t)|_H\} \ \ \text{ for } \text{ a.a. }\ t\in \delta _i. \end{aligned}$$

By analogy with (20), from (33), we deduce that for a.a. $t\in \delta _i$,

$$\begin{aligned}&(\tilde{g}^h(t),\tilde{g}_t^h(t))_H+l^2(\tilde{\mu }^h(t),\tilde{\mu }_t^h(t))_H+0.5\!\int \limits _{\varOmega }\{|\nabla \tilde{\pi }^h(t,\eta )|^2 {}+ l^2|\nabla \tilde{\mu }^h(t,\eta )|^2\}\, {\mathrm {d}}\eta \nonumber \\&\quad \le 2l^2\lambda ^2{\varLambda }^0(t,x^h_{{\varDelta },z}(\cdot ),z(\cdot ))+ (\chi _i^h,v^*(t)-Bu(t)+Cv_i^h)_H+M_*(t;\tau _i). \end{aligned}$$

(35)

For $t\in [\tau _i,\tau _{i+1}{[}$, by virtue of (28)–(30), we get

$$\begin{aligned} (\chi _i^h,v^*(t)-Bu(t)+Cv_i^h)_H=(\chi _i^h,B(\tilde{u}_i^h-u(t)))_H+(\chi _i^h,C(\tilde{v}_i^h-v_i^h))_H. \end{aligned}$$

Taking into account the rules for forming the controls $\tilde{u}_i^h$ and $v_i^h$, we conclude that

$$\begin{aligned} (\chi _i^h,v^*(t)-Bu(t)+Cv_i^h)_H\le 0 \quad \text{ for }\ t\in [\tau _i,\tau _{i+1}{[}. \end{aligned}$$

(36)

We deduce from (35) and (36) that

$$\begin{aligned}&\frac{{\mathrm {d}}{\varLambda }(t,x^h_{{\varDelta },z}(\cdot ), z(\cdot ))}{{\mathrm {d}}t} =(\tilde{g}^h(t),\tilde{g}_t^h(t))_H+l^2(\tilde{\mu }^h(t),\tilde{\mu }_t^h(t))_H+0.5\!\int \limits _{\varOmega }\{|\nabla \tilde{\pi }^h(t,\eta )|^2 \nonumber \\&\quad {}+ l^2|\nabla \tilde{\mu }^h(t,\eta )|^2 \}\,{\mathrm {d}}\eta \le l^2\lambda ^2{\varLambda }^0(t,x^h_{{\varDelta },z}(\cdot ),z(\cdot ))+ M_*(t;\tau _i) \quad \text{ for } \text{ a.a. }\ t\in \delta _i.\nonumber \\ \end{aligned}$$

(37)

Thus,

$$\begin{aligned} \displaystyle \frac{{\mathrm {d}}{\varLambda }^0(t,x^h_{{\varDelta },z}(\cdot ), z(\cdot ))}{{\mathrm {d}}t}\le 2l^2\lambda ^2{\varLambda }^0(t,x^h_{{\varDelta },z}(\cdot ),z(\cdot ))+M_*(t;\tau _i) \quad \text{ for } \text{ a.a. }\ t\in \delta _i. \end{aligned}$$

Using (34), by virtue of the Gronwall Lemma, we derive

$$\begin{aligned} {\varLambda }^0(t,x^h_{{\varDelta },z}(\cdot ),z(\cdot ))\le c_2\sum _{i=0}^{m-1} \int \limits _{\tau _i}^{\tau _{i+1}} M_*(t;\tau _i)\,{\mathrm {d}}t\le c_3\mu ^{1/2}(h),\quad \forall t\in T. \end{aligned}$$

Hence and from (37), we get ${\varLambda }(t,x^h_{{\varDelta },z}(\cdot ),z(\cdot ))\le c_4\mu ^{1/2}(h), \forall t\in T$. The theorem is proved. $\square$

Remark 2.1 and Theorems 3.1 and 4.1 imply the main result of the paper.

Theorem 4.2

The strategy ${\fancyscript{U}}$, defined by (6), is an $\varepsilon$-optimal minimax strategy, and the strategy ${\fancyscript{V}}$, defined by (25), is an $\varepsilon$-optimal maximin strategy. Thus, the pair $({\fancyscript{U}},{\fancyscript{V}})$ constitutes an ${\varepsilon }$-saddle point in the game, and $c_1=C_{{\mathrm {opt}}}= \inf \{I(w(\cdot ;t_0,x_0,u(\cdot ))):u(\cdot )\in D_T(\cdot )\}$ (see Problem 3) is the value of the game.

5 Conclusions

In this paper, we studied the game control problem for the nonlinear distributed system described by the phase field equations. The work was aimed at building stable algorithms solving the problem. The suggested algorithms are based on constructions from the dynamical reconstruction theory and on the method of extremal shift, which is known in the theory of positional differential games.