Abstract
The paper deals with the problem of optimal control with a convex integral quality index for a linear steady-state control system in the class of piecewise continuous controls with a smooth control constraints. In a general case, for solving such a problem, the Pontryagin maximum principle is applied as the necessary and sufficient optimum condition. In this work, we deduce an equation to which an initial vector of the conjugate system satisfies. Then, this equation is extended to the optimal control problem with the convex integral quality index for a linear system with a fast and slow variables. It is shown that the solution of the corresponding equation as \(\varepsilon\to 0\) tends to the solution of an equation corresponding to the limit problem. The results received are applied to study of the problem which describes the motion of a material point in \(\mathbb{R}^n\) for a fixed period of time. The asymptotics of the initial vector of the conjugate system that defines the type of optimal control is built. It is shown that the asymptotics is a power series of expansion.
Highlights
The paper is devoted to studying the asymptotics of the initial vector of a conjugated state and an optimal value of the quality index in the optimal control problem [1]–[3] for a linear system with a fast and slow variables, convex integral quality index [3, Chapter 3], and smooth geometrical constraints for control.Singularly perturbed problems of optimal control have been considered in different settings in [5]–[7].The method of boundary function that was developed in [4, 10] allows effectively constructing an asymptotics of solutions for problems with an open control area and smooth controlling actions.The solving of problems with a closed and bounded control area meets certain difficulties
Let us consider a problem that belongs to the class of piecewise continuous controls – optimal control problem for a linear stationary system with a convex integral quality index:
Formulas (2.5) and (1.5) show that if one manages to gain the complete asymptotic expansion of vector lε, which determines the optimal control in problem (2.1), this vector can be used for the asymptotic expansions of the above values as well
Summary
The paper is devoted to studying the asymptotics of the initial vector of a conjugated state and an optimal value of the quality index in the optimal control problem [1]–[3] for a linear system with a fast and slow variables (see review [4]), convex integral quality index [3, Chapter 3], and smooth geometrical constraints for control. Problems of fast operation and terminal control with constraints for control in the form of a polygon are dealt with in [5, 7]. The structure of such optimal control is a relay function with values in the apexes of the polygon. Asymptotic expansion of a solution for one singularly perturbed optimal control problem
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