### OPTIMAL CONTROL OF SOLUTIONS TO THE CAUCHY PROBLEM FOR AN INCOMPLETE SEMILINEAR SOBOLEV TYPE EQUATION OF THE SECOND ORDER

#### Abstract

The paper investigates the problem of optimal control of solutions to the Cauchy and Showalter--Sidorov problem for an incomplete semilinear second order Sobolev type equation in Banach spaces. Sobolev type equations are understood as operator-differential equations with an irreversible operator at the highest time derivative. Based on the theorem on the existence and uniqueness of a solution to an inhomogeneous equation, a theorem on the existence of a solution to the optimal control problem is proved. The solution is formally presented as a Galerkin sum and then, based on a priori estimates, the convergence of the Galerkin approximations in the *-weak sense is proved. To illustrate the abstract theory, a study of the optimal control problem in a mathematical model of wave propagation in shallow water under the condition of conservation of mass in the layer and taking into account capillary effects is presented. This mathematical model is based on the IMBq equation and the Dirichlet boundary conditions.

#### Keywords

mathematical model; modified Boussinesq equation; optimal control problem; numerical study; semilinear equation of Sobolev type of the second order.

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