SOME PROBLEMS OF GUARANTEED CONTROL OF THE SCHL¨OGL AND FITZHUGH—NAGUMO SYSTEMS

. A game control problems of the Schl¨ogl and FitzHugh—Nagumo equations are considered. The problems are investigated both from the view- point of the ﬁrst player (the partner) and of the second player (the opponent). For both players, their own procedures for forming feedback controls are spec- iﬁed.


1.
Introduction. In the recent years, a part of mathematical control theory, namely, the theory of control for distributed systems, has been intensively developed. To a considerable degree, this is stimulated by the fact that a rather wide set of applied problems is described by such systems. At present, there exists a number of monographs devoted to control problems for distributed systems [8,9,5,20,1,6]. In all these works, the emphasis is on the problems of open-loop or feedback control in the case when all system's parameters are precisely specified. But, the investigation of control problems for systems with uncontrollable disturbances (game or robust control problems) is also natural. Similar problems have been insufficiently investigated; in our opinion, this is connected with the fact that the well-known Pontryagin maximum principle is not really suitable for solving such problems. In the early 1970es, N.N. Krasovskii [7] suggested an effective approach to solving guaranteed control problems. This approach is based on the formalism of positional strategies.
In these paper, some problems of guaranteed (feedback) control for a parabolic equation with memory are investigated. Such an equation includes the Schlögl and FitzHugh-Nagumo equations. The fundamental theory of guaranteed control for some equations (variational inequalities) with distributed parameters within the framework of the formalization suggested in [7] was presented in [18,10,11,12,13]. In all the works cited above, the cases when equations (variational inequalities) do not contain a previous history were considered. In the present work, from the position of the approach [7,18,10,11,12,13], the problems of guaranteed control (for the partner and opponent) in the case of an equation containing the previous history of its phase state are investigated. To solve these problems, we use the known method of stable paths. Note that optimal control problems for the system in question were discussed in [3,4,19,17,2], see also their bibliography. For 560 VYACHESLAV MAKSIMOV example, in [3,17] different numerical and theoretical aspects of optimal control of the Schlögl and FitzHugh-Nagumo equations are discussed. The work [19] is devoted to the positional (feedback) control for such equations. In this paper three different approach are considered. First, an analytical solution is proposed. Second, an appropriate optimal control procedure is applied. The third approach extends the standard optimal control to the so-called spare optimal control; this results in very localized control signals and allows to analyze second order optimality conditions. The game control problem for systems with distributed parameters has been investigated by many authors (see, for example, [8,14,15,16]).
The game control problems under consideration in the paper consists in the following. Let a target set N or some quality criterion I = I(x(·), u(·), v(·))) depending on the solution x(·) of equation (1) and controls u(·) and v(·) and its prescribed value I * be given. At discrete times the phase state x(τ i ) is measured. The results of these measurements are functions Here, h ∈ (0, 1) stands for a level of informational noise. There are two playersantagonists controlling equation (1) by means of various input actions. One of them is called a partner, another one is called an opponent. Assume P ⊂ H 1 (Ω) and V ⊂ H 1 (Ω) are given 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 a certain feedback principle (on the base of measuring the state x(τ i )) in such a way that this control provides the attainment of the target set N by the solution (1) at the time t = ϑ or, in the case when the quality criterion is specified, guarantees a value of the criterion not exceeding I * for any admissible control v(·) chosen by the opponent. 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 V ) also by the feedback principle (on the base of measuring the state x(τ i )) in such a way that this control provides the evasion of the solution of equation (1) from the target set N at the time t = ϑ or guarantees a value of the quality criterion exceeding I * for any admissible control u(·) chosen by the partner. This is the description of the problems considered in the paper. The scheme of the algorithm for solving the problem undertaken by the partner, in the case when the target set is given, is as follows. In the beginning, an auxiliary equation M 1 is introduced. The equation M 1 has an input u * (·) and an output w(·) being the solution of this equation. Here, u * (·) is the solution of some auxiliary optimal control problem. Before the algorithm starts, a value h and a partition ∆ with diameter δ = δ(∆), as well as some open-loop control u * (·) serving as an input of M 1 are fixed. The process of synchronous feedback control of systems (1) and M 1 is organized on the interval T . This process is decomposed into m − 1 identical steps. At the ith step carried out during the time interval δ i = [τ i , τ i+1 ), the following actions are fulfilled. First, at the time τ i , according to some a priori chosen rule U, elements are calculated. Then (till the time τ i+1 ), the control u h (t) = u h i is fed onto the input of equation (1). Under the action of this control, as well as of the given control u * (t), τ i ≤ t < τ i+1 , and an unknown control of the opponent v(t), τ i ≤ t < τ i+1 , the states x(τ i+1 ) and w(τ i+1 ) are realized at the time τ i+1 . The procedure stops at the moment ϑ. Remark 1. Below, four problems are investigated. The first two problems are solved in the case when ξ h i , the results of measuring the states x(τ i ), satisfy inequalities (3) in the (H 1 (Ω)) * -metric (whereas the values ξ h i may be elements of the space L 2 (Ω)). The remaining two problems deal with the metric of the space L 2 (Ω). As is know, the first metric is weaker. Therefore, when solving the first two problems, we weaker the requirements on the measurement results in comparison with the requirements imposed for solving the later two problems.
The scheme of the algorithm for solving the problem undertaken by the opponent is analogous to the scheme of the algorithm for solving the problem by the partner. In the beginning, an auxiliary equation M 2 is introduced. The equation M 2 has an input v * (·) and an output z(·) being the solution of this equation. Before the algorithm starts, the value h and the partition ∆ with diameter δ are fixed. The process of synchronous feedback control of equations (1) and M 2 is organized on the interval T . This process is decomposed into m − 1 identical steps. At the ith step carried out during the time interval δ i = [τ i , τ i+1 ), the following actions are fulfilled. First, at the time τ i , according to some chosen rules V 1 and V, elements are calculated. Then (till the moment τ i+1 ), the control v In the case when the quality criterion is given, corresponding systems are used instead of equations M 1 or M 2 .
Let us give the strict formulation of the considered problems. Before this, we give some definitions. The sets of all open-loop controls of the partner and opponent are denoted by the symbols P T (·) and V T (·), respectively: The symbols P a,b (·) and u a,b (·) stand for the restrictions of the set P T (·) and function u(t), t ∈ T onto the segment is called a trajectory of equation (1) from the position (t * , x * ) corresponding to the controls u t * ,ϑ (·) ∈ P t * ,ϑ (·) and v t * ,ϑ (·) ∈ V t * ,ϑ (·). Any function (perhaps, multifunction) U : T × H → P is said to be a positional strategy of the partner. A positional strategy of the opponent is defined by analogy: Positional strategies correct controls at discrete times given by some partition of the interval T . Let the partition of T be any finite family Let an equation M 1 with a phase trajectory w(·) be fixed. The trajectory w(·) is generated by some openloop control u * (·). A trajectory x(·) of equation (1) starting from an initial state (t * , x * ) and corresponding to a piecewise constant control u h (·) (formed by the feedback principle) and to a control v t * , ) generated by the positional strategy U on the partition ∆. Thus, the process of forming the motions x h ∆,w (·) and w(·) is realized simultaneously. For t ∈ [τ i , τ i+1 ], these functions are specified as follows: we understand the solution of equation (1) constructed by the feedback principle described above; i.e., the control u h (·) is formed by formula (7) and x h ∆,w (·), by formula (6). The set of all (h, ∆, w, U)-motions is denoted by X h (t * , x * , U, ∆, w). By the results of [12], the set The task of the partner can be formulated in the following way. Let equation (1) be considered on the given time interval T . Its trajectory x(·) depends on two Problem 1. It is necessary to specify an equation M 1 , a control u * (·) for this equation, as well as a positional strategy of the partner U : T × H → P with the following properties: whatever a value ε > 0, one can specify (explicitly) numbers h * > 0 and δ * > 0 such that the inclusion In what follows, the symbol N ε stands for a closed ε-neighborhood of the set N in H.

Remark 2.
Taking into account the rule of constructing the set X h (0, x 0 , U, ∆, w), one can make the following conclusion. Whatever the open-loop control v T (·) ∈ V T (·) maybe, the solution x h ∆,w (·; 0, x 0 , U, v T (·)) belong to an ε-neighborhood of the set N at the time t = ϑ.
Remark 3. As is seen from the proof of Theorem 3.1 presented below, the rule for constructing the strategy U (see (4), (14)) is based on the following simple idea. Namely, the strategy U is chosen in such a way that the (h, ∆, w, U)-motion x h ∆,w (·) (see (6)-(8)), generated by this strategy slightly deviates from the solution w(·; 0, x 0 , u * (·)) for the whole time interval T . The last property should be valid for any open-loop control v(·)chosen by the opponent and unknown for the partner. In essence, the strategy U means the choice of a control according to the principle of extremal shift [7], which is well-known in the theory of guaranteed control. As to this principle applied to distributed systems, see, for example, [18,10,11,12,13].

By analogy with the motion
) corresponding to piecewise constant controls v h (·) and v * (·) (formed by the feedback principle) and to a control u t * ,ϑ (·) ∈ P t * ,ϑ (·). This motion is called an (h, ∆, V, V 1 )-motion generated by the positional strategies V and V 1 on the partition ∆. The set of all (h, ∆, Note that the trajectory x h ∆,z (·; t * , x * , V, V 1 , u t * ,ϑ (·)) is formed simultaneously with another trajectory z(·). All these two trajectories are formed by the feedback principle, i.e., for t ∈ [τ i , τ i+1 ], it is assumed that where we understand the trajectory of (1) constructed by the feedback principle described above; i.e., the controls v h (·) and v * (·) are formed by formula (11) and x h ∆,z (·), by formula (9). Here, the function z(·) (see (10)) is an auxiliary function, which is analogous to the function w(·) in the problem of the partner.
The problem undertaken by the opponent is inverse with respect to the problem undertaken by the partner. Its essence is as follows.
Problem 2. It is necessary to specify an equation M 2 , as well as a positional strategy of the opponent V : T × H → V , and a positional strategy V 1 : T × H → V with the following properties: whatever a value ε > 0, one can specify (explicitly) numbers h * > 0 and δ * > 0 such that the condition is fulfilled uniformly with respect to all measurements ξ h i with properties (12) if h ≤ h * and the diameter δ = δ(∆) ≤ δ * .
Along with Problems 1 and 2 formulated above, we consider another two problems (Problems 3 and 4). Let a cost functional and σ : H ∈ R are given functions satisfying the local Lipschitz condition with respect to the set of variables. For example, the cost functional may be a quadratic function, i.e.
To investigate Problems 3 and 4 formulated below, we need to extend the space of positions. Namely, instead of the space H, we take the space H 1 = H × R 2 as the space of positions. The reason of such a choice will be clear after the description of the algorithm for solving Problem 3.
So, let a prescribed value of the criterion, a number I * , be fixed.
is the solution of equation (1) generated by controls u h (·) and v(·) ∈ V T (·). In the process, the control u h (·) is formed by some strategy U. The set of triples φ h ∆,g (·) is denoted by the symbol Φ h (0, x 0 , U, ∆, g). Here, the symbol g stands for some additional function; its role is clarified below.
Problem 3. It is necessary to specify a system M 1 , a control u * (·) for this system, as well as a positional strategy of the partner U : T × H 1 → P with the following properties: whatever a value ε > 0, one can specify (explicitly) number h * > 0 such that the inequality Here C * 1 > 0 and µ 1 are fixed constants and δ is the diameter of the partition ∆. The problem undertaken by the opponent is inverse with respect to the problem undertaken by the partner. Its essence is as follows.
Introduce the set of functions is the solution of equation (1) generated by controls u(·) ∈ P T (·) and v h (·). In the process, the control v h (·) is formed by some strategy V. The set of triples ψ h ∆,g (·) is denoted by the symbol Ψ h (0, x 0 , ∆, V).

Problem 4.
It is necessary to specify a system M 2 , as well as a positional strategy of the opponent V : T × H 1 → V with the following properties: whatever a value ε > 0, one can specify (explicitly) number h * > 0 such that the inequality Here C * 2 > 0 and µ 2 are fixed constants. The solution of Problem 4 is omitted here, since it is analogous to the solution of Problem 3 (see the remark in the end of Section 5).
3. Algorithm for solving Problem 1. We specify an algorithm for solving Problem 1. In this paper, we assume that the following condition is fulfilled.

Condition 1 ([4]
). The parameter η in equation (1) satisfies the condition Here and below, |a| stands for the module of the number a. In this and the next sections, we assume that the following condition is also fulfilled.

Condition 2.
There exists a closed set D ⊂ H 1 (Ω) such that P = V + D.
Here, the symbol V +D stand for the algebraic sum of sets V and D, i.e. V +D = {u : u = u 1 + u 2 , u 1 ∈ V, u 2 ∈ D}.
The strategy U (see (4)) is defined in such a way: U(0, ξ, w) = P. Let us pass to the description of the algorithm for solving Problem 1. Namely, we describe the procedure of forming the (h, ∆, w, U)-motion x h ∆,w (·) corresponding to some fixed partition ∆ and the strategy U of form (4), (14).
Let the (h, ∆, w, U)-motion x h ∆,w (·) and the trajectory w(·) of the equation M 1 be defined on the interval [0, τ i ]. At the time t = τ i , we assume that i.e., we set u h (t) = u h i for t ∈ [τ i , τ i+1 ).

VYACHESLAV MAKSIMOV
As a result of the action of this control and of the unknown disturbance v τi,τi+1 (·), the (h, ∆, w,  Proof. Define R η (t, v) = e −ηt R(e ηt v) + η/3v. Then equations (1) and (13) take the following form: To prove the theorem, we estimate the variation of the function Subtracting (19) from (18) and multiplying scalarly the difference by x h ∆,w (t) − w(t) (in H), then integrating (for t ∈ [τ i , τ i+1 ], i = 0, . . . , m − 1) and taking into account the monotonicity of the mapping v → R η (t, v), we derive From the results of [4] we . Moreover, takes place. Therefore, by virtue of (22), we have In this case, for t ∈ [τ i , τ i+1 ], using (2) and (23), we obtain Then, using (8), we get In this case, the first term in the right-hand part of inequality (25), by virtue of the rule of finding the control u h (·) (see (15)- (17)), does not exceed the value It is easily seen that the inequality is fulfilled for t ∈ [τ i , τ i+1 ]. Consequently, using (24) and (26), we obtain (for t ∈ [0, ϑ]) From Theorem 2.1 [4], it follows that The latter inequality does not depend on ∆, u h (·), v(·), and u * (·) but depends on the parameters of equation (1), namely, on the initial state x 0 and the sets P and V . As well, ε(0) = 0. Therefore (for t ∈ T ), the estimate is fulfilled. By virtue of the Gronwall lemma, we obtain The statement of the theorem follows from the latter inequality; i.e. w(ϑ) = w(ϑ; 0, x 0 , u * (·)) ∈ N . The theorem is proved. (1) with v(t) = 0, i.e. the disturbance is absent in this equation. Assuming that, along with equation (1) we have the equation for the opponent

Remark 5. Let the equation for the partner have the form
and the control u satisfies the condition: u ∈ P . At the moments τ i ∈ ∆, the phase states x(τ i ) and x (1) (τ i ) are measured with errors. The results of the measurements ξ h i ∈ (H 1 (Ω)) * and ψ h i ∈ (H 1 (Ω)) * , satisfy the inequalities It is necessary to construct a law (a strategy) for forming the control (1) with the properties: whatever a value ε > 0, one can specify (explicitly) h 1 > 0 and δ 1 > 0 such that the inequality is fulfilled uniformly with respect to all measurements ξ h i and ψ h i with the properties listed above if h ≤ h 1 and δ = δ(∆) ≤ δ 1 . If Condition 2 is valid, then the algorithm described above is applied to solve the problem in question. In this case, it is necessary to use ψ h 1 and ψ h i instead w(τ 1 ) and w(τ i ) in formulas (14), (16) and (17) respectively. 4. Algorithm for solving Problem 2. We specify an algorithm for solving Problem 2. Assume that, as everywhere above, Condition 1 is fulfilled.
5. Algorithm for solving Problem 3. Let us specify an algorithm for solving Problem 3. We assume that there is the function F (t, u, v) = F 1 (t, u) + F 2 (t, v) instead of the difference u − v + f in the right-hand part of equation (1); i.e., the equation takes the form Here, F j : T ×H 1 (Ω) → H 1 (Ω), j = 1, 2, are given functions satisfying the Lipschitz conditions in the metric of the space H. Consider the ordinary differential equatioṅ g(t) = φ(t, x(t), u(t), v(t)), g(0) = 0, t ∈ T, g ∈ R. (44) Introducing this new variable g, we reduce the robust control problem of Bolza type to a control problem with a terminal quality criterion of the form I 1 (x(·), g(·)) = σ(x(ϑ)) + g(ϑ). In this case, the controlled system consists of equation (43) in the space H and ordinary differential equation (44) in the space R.
By virtue of [4], the following lemma takes place.

VYACHESLAV MAKSIMOV
Here, |a| stands for the module of the number a, c = 1 + 2ϑ 1/2 (d (1) ) 1/2 . If there does not exist an approximate numberg for the given quadruple, then we take an arbitrary number asg(τ i ).
Then, the algorithms for solving problems 1 and 3 described in Section 3 and 5 can be applied to solving corresponding problems in the case of additional constraints specified above.