Equivalent substitution in the control theory

TL;DR

This paper investigates optimal control solutions by replacing the original system with convex envelope-based systems, providing sufficient optimality conditions and methods for evaluating attainability sets using positive definite functions.

Contribution

It introduces a novel approach of using convex envelopes for system substitution in control problems, enhancing the analysis of attainability and optimality conditions.

Findings

Convex envelope substitution simplifies control system analysis.

Sufficient optimality conditions are established for the substituted systems.

Methods for evaluating attainability sets using positive definite functions are provided.

Abstract

In this paper we study a problem of looking for an optimal solution of a system of the differential equations with a control and an optimized function. The system of differential equations is changed for two systems with the upper and lower convex envelopes of a function on the right side of the initial differential system and the lower envelope of the optimized function in a region of attainability. The necessary conditions of optimality are sufficient for the substituted system. The rules for evaluation of the attainability set with the help of positively definite funtions are given.