On the Optimal-Control Problem for Dynamical Processes With Variable Delays

Abstract

The principles of calculus of variations are used to obtain necessary conditions for optimal control of dynamical systems that involve nonconstant time lags. Consideration is given to systems that can be represented mathematically by a finite set of ordinary nonlinear differential-difference equations with one or more time-dependent argument lags. Application of the general results to classes of linear systems with finite-time quadratic performance criteria is considered in detail. Optimal feedback control laws are given. Discussion of a proposed method for obtaining computational solutions for nonlinear systems with variable delays is included in the paper.

Discussion: “Optimum Control of Linear Systems With Transportation Lag” (Khatri, H. C., 1967, ASME J. Basic Eng., 89, pp. 385–390)

“…ill the paper. In particular, we can consider (see references [19], [20]) systems described by the linear vector equation.…”

“…as given by Athans and Falb [16], is 6 The n X " nonlinear matrix equation may be transformed into 2 it X h linear equations. The details are shown in Appendix C of reference [19].…”

“…In equation (67), substituting for </,-, interchanging the order of integration and carrying out the integration gives where the quantities W,(s), (7;*(s), and Xd*(s) are defined in equations (19), (20), and (21) and…”

Optimal control theory with general constraints

“…We shall denote the problem defined by (1), (2), (3a) and (4) by problem B. For some relevant literature, see Asher and Sebesta [1], Hadley and Kemp [3], Hestenes [4,Chapter 11], Long and Vousden [5], Mangasarian [6], Neustadt [7,Chapters 5 and 6], Schwartzkopf [9], Sebesta and Clark [10], and Teo and Craven [11]. However, it is fair to say that the results available for problem B are either hard to apply, or require very stringent conditions for their validity, or both.…”

Optimal control of systems with state-dependent time delay†

“…Sebesta (61) has obtained an approximate feedback control law for sufficiently small time delay. However, no results are available when the time delay is large.…”