ASYMPTOTIC SOLUTION OF THE PROBLEM OF OPTIMAL CONTROL OF LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS WITH DEGENERATIONS

Abstract

Singularly perturbed optimal control systems with variable parameters are integrated by asymptotic methods. The asymptotic solution of this system depends on the spectrum of the main matrix of the system. For systems of linear algebraic-differential equations, the asymptotic solutions depend on the spectrum of the main matrix pencil. Optimization problems of controlling systems of singularly perturbed algebraic-differential equations have been studied in the present century. The theory of asymptotic integration of systems with degeneracies was developed in the works of A.M. Samoilenko, M.I. Shkil, G.S. Zhukova, and V.P. Yakovets at the end of the last century. The developed methods made it possible to construct asymptotic solutions of systems with degeneracies for the case of a stable spectrum of the boundary value of matrices. The solutions of these control systems were constructed in the works of V.P. Yakovets and O.V. Tarasenko. Systems of singularly perturbed equations with turning points are important in practical applications. For systems of ordinary differential equations with turning points, asymptotic solutions were constructed in the works of M. Iwano, Y. Shibuya, and W. Wasow. Asymptotic solutions of systems with turning points are multiscale. Two-scale asymptotic solutions of systems of algebraic-differential equations were constructed by A.M. Samoilenko and P.F. Samusenko. Optimal control systems for ordinary differential equations with unstable spectrum were studied by V.M. Leifura. In this article, the results obtained by the above-mentioned authors are applied to the problem of optimal control of a system of singularly perturbed algebraic-differential equations with a simple turning point. An asymptotic representation of the matrix of impulse transition functions of the system of equations with a simple turning point is constructed, and asymptotic estimates of the constructed approximations are given. In the general post, the optimal control problem is considered without obtaining specific estimates. The error estimate is affected by both the multiplicity and the type of pivot point. The system formed by applying the Pontryagins maximum principle will also have an unstable spectrum, but the type of turning point may change. Therefore, specific estimates need to be considered. This will be the task of future research.