On a Representation of the Limit Occupational Measures Set of a Control System with Applications to Singularly Perturbed Control Systems

Optimality Issues for a Class of Controlled Singularly Perturbed Stochastic Systems

“…Linear programming techniques have proved to be very useful in dealing with deterministic and stochastic control problems. A wide literature is available on the subject both in the deterministic and in the stochastic setting ( [1,2,3,4,5,6,7,8]). …”

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

“…Introduction and preliminaries. Problems of optimal control of singularly perturbed (SP) systems have been studied intensively in both deterministic and stochastic settings (see [2], [8], [17], [18], [23], [26], [28], [29], [33], [41], [52], [55], [56], [58], [61], [63], [68], [67], [65], [74], [75], [77] for a sample of the literature). Originally, the most common approaches to SP control systems, especially in the deterministic case, were related to an approximation of the slow dynamics by the solutions of the systems obtained via equating of the singular perturbations parameter to zero, with further application of the boundary layer method (see [64], [73]) for an asymptotical description of the fast dynamics.…”

“…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.…”

“…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.…”

“…Also, it has been established that, under some additional conditions (see Theorem 2.1(ii),(iii) and Proposition 4.1 in [41])), 16) with the convergence being uniform with respect to y ∈ Y .…”