Integration of Non-Stationary Optimal Control Systems with Delay and an Unstable Spectrum

Purpose. 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. Design / Method / Approach. In this paper, asymptotic methods are used, in particular, the Feshchenko-Shkil’ method for integrating singularly perturbed systems and the Wasow’s method for systems with an unstable spectrum. Findings. In this paper, we construct a transformation that reduces the optimal delay control system to a system that does not contain terms without rejecting the argument. This transformation makes it possible to integrate the system by the method of steps without solving systems of differential equations at each step. Theoretical Implications. 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. Practical Implications. 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. Originality / Value. The delayed control systems in the described formulation are studied for the first time. Research Limitations / Future Research. 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.