OPTIMAL CONTROL OF SOLUTIONS TO THE INITIAL-FINAL PROBLEM FOR THE MODEL OF LINEAR WAVES IN A PLASMA

The optimal control problem for a Sobolev type equation of higher order with a relatively polynomially bounded operator pencil is investigated in the paper. The results are applied to the study of the optimal control of solutions to the initial-final problem for the model of linear waves in plasma. The first results on the investigation of equation that describes the linear ion-acoustic waves in an unmagnetized plasma and on the study of some properties of these waves were obtained by Yu.D. Pletner. The initial-final conditions posed for the fourth-order Sobolev type equation are the generalization of the conditions in the Cauchy problem that is unsolvable at the arbitrary initial values. The research is based on the phase space method developed by G.A. Sviridiuk and the theory of relatively polynomially bounded operator pencil developed by A.A. Zamyshlyaeva. The article considers an equation that describes ion-acoustic waves in a plasma in an external magnetic field.


Introduction
Let The article investigates the optimal control of solutions to the following problem: (2) Model (1), (2) describes ion-acoustic waves in plasma in an external magnetic field [1]. The parameters in equation (1) relate such physical quantities as the ionic hydro frequency, the Langmuir frequency, and the Debye radius. The function ( , ) x s t represents the generalized potential of an electric field, the function ( , ) u s t represents an external effect. Problem (1), (2) is investigated in the framework of the theory of relatively polynomially bounded pencils of operators [2]. Consider a high-order abstract Sobolev type equation are some projectors in space X . Thus, the optimal control problem is to find a pair ˆ( , ), xu where x is a solution of (1), (2), and ˆa d uU  is the control for which the relation Many non-classical models of mathematical physics [4][5][6][7][8][9] are based on Sobolev type equations. For example, they occur in problems of hydromechanics, plasma physics, atmospheric physics, filtration theory, theory of electrical circuits, and others. In the work, to find the optimal control of solutions to linear Sobolev type equations of high-order, the ideas and methods obtained by G.A. Sviridyuk [10] and his students [11][12][13] in the study of first-order Sobolev type equations are used. Here the initial-final problem [14] is investigated. A distinctive feature of this problem is that one projection of the solution is specified at the initial moment of time, and the other at the final point. The initial-final problem for the first-order Sobolev type equations was considered by G.A. Sviridyuk and S.A. Zagrebina.

Polynomially A -bounded operator pencils and projectors. Strong solutions
By B denote the pencil formed by operators 10 , , .
where  is a contour that bounds the domain containing the relative spectrum of the pencil B . Then the operators 11  and there is a circuit , bounding the domain such, that  Mechanics. Physics, 2019, vol. 11, no. 4, pp. 26-31  28

Optimal control for the model of linear waves in plazma
Consider problem (4) for equation (3), where the functions x , y , u lie in X , Y and , U respectively.
Introduce the control space Here , > 0, lN  Denote by 3  is an inner product in 2 () L  . Lemma 1.
[15] Let one of the following conditions be satisfied: Construct the projector  there exists a unique solution to the optimal control problem for equation (1) with conditions (2), (12) that minimizes functional (10).