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.