Suboptimization of Singularly Perturbed Control Systems

Abstract

Abstract. An averaging technique for nonlinear multiscale singularly perturbed control systems is developed. Issues concerning the existence and structure of limit occupational measures sets generated by such systems are discussed. General results are illustrated with special cases.

On Average Control Generating Families for Singularly Perturbed Optimal Control Problems with Long Run Average Optimality Criteria

“…Also, as can be readily verified, Assumption 6.2(i) implies that the solutions y(τ, u(·), y 1 ) and y(τ, u(·), y 2 ) of the associated system (2. From the validity of (6.4) and (6.5) it follows that the associated system (2.11) and the averaged system (2.18) have unique forward invariant sets which also are global attractors for the solutions of these systems (see Theorem 3.1(ii) in [39]). For simplicity, we will assume that Y and Z are these sets.…”

“…As in [48], our consideration is based on earlier results on averaging of SP control systems obtained in [39], [41], [42], [44], [47] (see also [6], [7], [8], [10], [51], [52], [67]) and on results obtained in [35], [45], [46], [47] that establish the equivalence of optimal control problems to certain infinite dimensional (ID) linear programming (LP) problems (related results on IDLP formulations of optimal control problems in both deterministic and stochastic settings can be found in [5], [15], [16], [21], [31], [37], [50], [53], [57], [60], [69], [71], [72] and [76]). In contrast to [48], where mostly optimal control problems with time discounting criteria were dealt with, this paper is devoted to the consideration of the problems with long run average optimality criteria.…”

“…Various averaging type approaches allowing a consideration of more general classes of SP problems, in which the optimal and near optimal controls take the form of rapidly oscillating functions and in which equating of the small parameter to zero does not lead to a right approximation, were studied in [1], [2], [6], [7], [8], [9], [10], [18], [19], [20], [27], [28], [33], [34], [38], [39], [41], [42], [44], [51], [52], [68], [67], [70], [75] (see also references therein). This research lead to a good understanding of what the "true limit" problems, optimal solutions of which approximate optimal solutions of the SP problems, are.…”

Optimality Issues for a Class of Controlled Singularly Perturbed Stochastic Systems

“…We emphasize that, in general, the averaged system is set-valued. We refer the reader to [14,20] for averaging methods. It is important to notice that only the behavior of the "slow" variable X x;y;u ;" ( ) is concerned by this approach.…”

“…Indeed, these criteria involve Pontryagin's maximum principle which is di¢cult to study if one does not fully understand the averaged dynamics. We recall [12,13,14] and references therein dealing with this kind of problems.…”