STUDY OF NON-STATIONARY AUTOMATIC CONTROL SYSTEMS WITH DELAY

We considered a non-stationary optimal control system with delay. Non-stationary optimal control systems are described by systems of differential or differential-algebraic equations. Variable coefficients do not allow, in general, to construct a solution to such systems in quadratures. Numerical or asymptotic methods are used to solve such systems. In this paper, asymptotic methods are used, in particular, the Feshchenko-Shkil' method for integrating of singularly perturbed systems and the Wasow’s method for systems with an unstable spectrum. In this paper, we construct a transformation that reduces the optimal delay control system to a system that does not contain terms without delay. This transformation makes it possible to integrate the system by the method of steps without solving systems of differential equations at each step. The system of equations obtained as a result of the transformation of the original system is somewhat easier to study in terms of building a solution. However, the problem of optimizing the control of both systems requires both a separate mathematical study and clarification of the practical reality of spectrum instability in such systems. If the instability of the spectrum is caused by the degeneracy of the main matrix, this leads to the unboundedness of the system solution as the small parameter approaches zero. The aforementioned growth of the solution can create emergency situations in real systems. Future research concerns solving the problem of optimal control of systems with an unstable spectrum and studying the question of the reality and physical meaning of turning points in specific systems. In future studies, it is also planned to construct an asymptotic representation of the solution of a control system with delays and degeneracies. Keywords: optimal control system, systems of differential equations with delay, asymptotic solution, turning point.