About the solvability of the problem of nonlinear optimization of oscillatory processes with the appearance of special controls

Abstract

The article deals with the cases when special controls appear in nonlinear optimization of oscillatory processes, when the function of external influence depends non-linearly on the control parameter. Quality-managed process is to minimize the integral of the functional, that is, in the final moment of time square deviation of a controlled process from a given desired state was minimal. The study was conducted using a generalized solution of the boundary value problem, which more or less adequately describes the actual process. According to the well-known method of optimal control theory, the increment of the functional is calculated and the Pontryagin function is written out, which is studied for the maximum in the range of acceptable values of the control function. The optimal control conditions are written out in the form of equality and differential inequality, which must be fulfilled simultaneously. The case when the maximum principle degenerates is studied. It is established that the desired control satisfies an infinite-dimensional system of linear integral Fredholm equations of the first kind, the solvability of which is studied by operator methods. It is proved that the operator equation has infinitely many solutions and an algorithm for their construction is developed. Then the functional is minimized on the set of solutions of the operator equation. This problem has at least one solution that is the desired special optimal control.