A nonlinear problem of optimal control for a system with parabolic equation if there are several dot mobile sources

In studying many problems of nonlinear optimal control for heat conduction often we have to take into account the auxiliary elements, without which it is impossible to control the studying process. These elements have usually lumped parameters. The behavior of such systems is generally described by a set of nonlinear ordinary differential equations and parabolic equations with initial value and boundary value conditions. It is studied the questions of analytical and approximation solving the nonlinear dot mobile point problem of nonlinear optimal control for a system with parabolic and ordinary differential equations in the case of presence of several dot mobile sources. At that time, the parabolic equation is considered with initial-nonlocal conditions, while ordinary differential equation is considered with initial value condition. A distinctive feature of this work is that nonlocal boundary conditions with respect to the second variable in the parabolic equation is simplified the application of the Fourier method of separation of variables. Functional of quality has nonlinear type and it additionally depends from the square of solution of the given ordinary differential equation. First, it is proved that the function of the state belongs to the class of Sobolev functions. On the base of maximum principle it is formulated the necessary conditions for nonlinear optimal control. Determination of the optimal control function is reduced to the complex functional-integral equation, the solving process of which is composed of solutions of two different equations: nonlinear functional equations and nonlinear integral equations. In the proof of the one-valued solvability of integral equations the method of successive approximations in combination it with the method of compressing mapping is applied. As iterations the Picard iterative process is taken. The formula for approximation calculating the dot mobile nonlinear optimal control and the estimate for the permissible error with respect to optimal control are obtained. The formulas for approximation calculating the nonlinear optimal process and the minimum value of the functional of quality are given. The results obtained in this work can find further application in the development of the mathematical theory of nonlinear optimal control of distributed parameter systems in the presence of mobile sources.