Nonlinear Optimal Control of Thermal Processes in a Nonlinear Inverse Problem

Abstract

The questions of weakly generalized solvability of a nonlinear inverse problem in nonlinear optimal control of thermal processes for a parabolic differential equation are studied. The parabolic equation is considered under initial and boundary conditions. To determine the recovery function, a nonlocal integral condition is specified. Moreover, the recovery function nonlinearly enters into the differential equation. Is applied the method of variable separation based on the search for a solution to the mixed inverse problem in the form of a Fourier series. It is assumed that the recovery function and nonlinear term of the given differential equation are also expressed as a Fourier series. For fixed values of the control function, the unique solvability of the inverse problem is proved by the method of compressive mappings. The quality functional has a nonlinear form. The necessary optimality conditions for nonlinear control are formulated. The determination of the optimal control function is reduced to a complicated functional-integral equation, the process of solving which consists of solving separately taken two nonlinear functional and nonlinear 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. Is proved the absolutely and uniformly convergence of the obtained Fourier series.