Problems of Hard Control for a Class of Degenerate Fractional Order Evolution Equations

Abstract

We find conditions of unique strong solution existence for the generalized Showalter—Sidorov problem to semilinear evolution equations with a degenerate operator at the highest fractional Gerasimov—Caputo derivative and with some constraint on the image of the nonlinear operator. Then we consider a class of optimal control problems for systems, whose dynamics is described by such equations endowed with the respective initial value conditions. Target functional is assumed not to take into account control costs. In such situation we used the additional condition of the admissible controls set boundedness. The obtained result of the initial problem unique solvability and properties of some functions spaces are applied to the proof of optimal control existence for such class of problems. Abstract results are applied to study of a control problem for a system, which is described by an initial-boundary value problem to a nonlinear partial differential equation, not solvable with respect to the highest time fractional derivative.