Quasi-optimal control with a general quadratic criterion in a special norm for systems described by parabolic-hyperbolic equations with non-local boundary conditions

In this work, we consider a dynamical system generated by a parabolic-hyperbolic equation with non-local boundary conditions. The optimal control problem for this system is studied using a notion of quasi-optimal solution. Existence and uniqueness of quasi-optimal control are proved.


1.
Introduction. At present, there is a need for generalization of the classical problems of mathematical physics, as well as the formulating qualitatively new problems, arising in the study of different nature objects. The results of research in this direction can be found in such areas as economics, physics, etc [8] [9]. An example of such a problem is mentioned as follows.
Let some medium be filled with gas, and at some instant of time an ionizing radiation, for example it could be X-rays, has an effect on this gas. As the result of sufficient ionization, we obtain a medium with a higher conductivity. Thus, the determination of the electric field strength at the time of changing is related to the solution of the boundary value problem for two equations: parabolic and hyperbolic types [11]. This paper is a continuation of the work on the study of parabolic-hyperbolic equations with non-local boundary conditions and is devoted to the construction of optimality conditions for quasi-optimal distributed control with a general quadratic criterion in a special norm. The elliptic and parabolic case were considered in [5], [6].
The paper is organized as follows. In Section 2, we give a statement of the problem. Section 3 contains the main results, where we give the conditions for finding control and prove the theorems of existence of the solution. 2. Statement of the problem. Let the controlled process y(x, t) ∈ C 1 (D) ∩ C 2 (D − ) ∩ C 2,1 (D + ) in D satisfy the equation with initial condition y(x, −α) = ϕ(x) (2) and boundary conditions where D = {(x, t) : 0 < x < 1, −α < t ≤ T, α, T > 0}, This boundary value problem was solved in [2]. The differential operator Ly and the non-local conditions (3) generate a biorthogonal in L 2 (0, 1) Riesz basis: Thus, the functions of the problem are expanded as follows: y i (t) = (y(., t), It is required to find the piecewise continuous by t controlû * (x, t), for which the following functional takes a minimum value: where ψ(x) is fixed function,α,β i ≥ 0,γ i > 0, i = 1, 2;α +β 1 +β 2 > 0.

QUASI-OPTIMAL CONTROL WITH A GENERAL QUADRATIC CRITERION 1245
The expressionγ 2 (u 2 i (0) + T 0u 2 i (t)dt) is used for the functional (4), since the classical formγ 2 ( T 0 u 2 i (t)dt) leads to the impossibility of finding a continuous solution y(x, t) of the problem.
The origin problem was solved in [3]. Based on the type of functional (4) and solution of (1) -(3) for the problem (1) -(3), (4) we can construct a quasi-optimal control. Definition 2.1. Quasi-optimal control is defined as the control for which the solution for the odd membersû 2k−1 (t) of decomposition in the Riesz basis is found firstly, and then for the even onesû 2k (t).
In this case the problem is formally equivalent to a sequence of finite-dimensional problems [1]: on the basis of the boundary problem y 0 (0−) = y 0 (0+),ẏ 0 (0−) =ẏ 0 (0+) = u 0 (0); on the basis of the boundary problem on the basis of the boundary problem Hereu i is denoted as ξ i , λ k = 2πk.
3. Optimality conditions. Optimality conditions for problem 1 . The functional (5) is strictly convex by the controls. Therefore the problem (5) - (6) has no more than one minimum point in C[−α, 0) × R 1 × L 2 (0, T ), which is characterized by optimality conditionŝ where We establish the unique solvability of the system (11). Let us define an operator operator A 0 is determined by the remainder terms of the left-hand sides of the equation system (11).
It is linear and continuous. Let's prove the following theorem.
We select the quadratic by controls v 0 (t), t ∈ [−α, 0); u 0 (0), ξ 0 (t) ∈ [0, T ] part and subtract from it the value that is, we consider the functional It is clear thatĨ 0 ≥ 0. Now the values of the operator A 0 θ 0 (.) scalar multiply by θ 0 (t), that is, consider a quadratic form Substituting into the quadratic form Π 0 the explicit form of kernel K (j) 0,i , i, j = 1, 3, we obtain equality Π 0 = 2Ĩ 0 . This implies the positive definiteness of the operator A 0 and the unique solvability of the system (11) From the first equation of (11) we find the estimation It follows that v 0 (t) is absolutely continuous.
Let's consider the following case.
Proof. Coincides with the proof of the theorem 3.1, if we replace the system (11) by (12).