### OPTIMAL CONTROL OF SOLUTIONS TO THE SHOWALTER--SIDOROV PROBLEM IN A MODEL OF LINEAR WAVES IN PLASMA

#### Abstract

In this article an optimal control problem for a high order Sobolev type equation is investigated under the assumption that the operator pencil is relatively polynomially bounded. The results are applied to the study of the optimal control of the solutions to the Showalter--Sidorov problem for a model of linear waves in a plasma in an external magnetic field. Showalter--Sidorov conditions are a generalization of Cauchy conditions. As it is well known, the Cauchy problem for Sobolev type equations is fundamentally insoluble for arbitrary initial values. We use the phase space method developed by G. А. Sviridyuk, a theory of relatively polynomially bounded operator pencils developed by A. A. Zamyshlyaeva. The mathematical model considered in the article describes ion-acoustic waves in plasma in an external magnetic field and was first obtained by Yu. D. Pletner.

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