On an optimal control of inverse thermal processes with an integral condition of redefinition

The questions of weakly generalized solvability of a nonlinear inverse problem in nonlinear optimal control of thermal processes for one type of parabolic differential equation are studied. The parabolic differential equation with respect to the state function is linear, with respect to the recovery function is nonlinear and with respect to the control function is implicit. The parabolic equation is considered under initial boundary conditions. To determine the recovery function, a nonlocal integral condition is specified. A system of two countable systems of integral and functional equations is obtained with respect to the state function and the recovery function. For fixed values of the control function, the unique solvability of the inverse problem by the method of compressing mappings is proved. The quality functional is nonlinear. The necessary optimality conditions for nonlinear control are formulated. The determination of the optimal control function is reduced to a complex functional-integral equation, the process of solving which consists of solving separately taken two nonlinear functional and integral equations. Nonlinear functional and integral equations are solved by the method of successive approximations. Formulas are obtained for the approximate calculation of the state function of the controlled process, the recovery function, and the optimal control function.