Linear Programming Approach to Deterministic Long Run Average Problems of Optimal Control

Abstract

We establish that deterministic long run average problems of optimal control are "asymptotically equivalent" to infinite-dimensional linear programming problems (LPPs) and we establish that these LPPs can be approximated by finite-dimensional LPPs, the solutions of which can be used for construction of the optimal controls. General results are illustrated with numerical examples. 1. Introduction and description of the problems. In this paper we show that, under some conditions, deterministic long run average problems of optimal control are "asymptotically equivalent" to infinite-dimensional linear programming problems (LPPs) and we establish that these LPPs can be approximated by finite-dimensional LPPs, the solutions of which can be used for numerical construction of the optimal controls. Infinite horizon problems of optimal control have been studied intensively in both deterministic and stochastic settings (see Anderson and Kokotovic [3], Arisawa, Ishii, and Lions [5], Bardi and Capuzzo-Dolcetta [10], Bensoussan [12], Carlson, Haurie, and Leizarowitz [14], Colonius and Kliemann [17], Fleming and Soner [21], Grüne [29], Kushner [34], Kushner and Dupuis [35], Vigodner [46], and references therein). In the stochastic setting, the linear programming formulation is a common tool for treating the problems (see, e.g., Basak, Borkar, and Ghosh [11], Borkar [13], Hernandez-Lerma and Lasserre [31], Stockbridge [44], Yin and Zhang [48]). Finite-dimensional approximations of LPPs arising in stochastic optimal control problems were considered by Helmes and Stockbridge [30] and by Mendiondo and Stockbridge [38]. A linear programming approach to long run average optimal control problems in the deterministic setting appears to be new and, to the best of our knowledge, there are no publications devoted to this topic (under different assumptions and for a different problem, a linear programming formulation was discussed in Evans and Gomes [20]). A linear programming approach to deterministic optimal control problems on a finite time interval has been studied in Rubio [42]. Let us introduce the problems that we will be dealing with. Consider the control system

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

“…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.…”

“…Namely, the optimal values of these two problems are related by the inequality 9) and also, under certain conditions (see [35] and [47]),…”

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]). …”

Linearization Techniques

“…As consequence, using (2.10), we have 12) for all γ ∈ co (Γ (0, t, x 0 )). Using the equality (2.11) and the inequality (2.12), one gets…”

“…A wide literature is available on the subject both in the deterministic setting ( [11,12,17,22]) and in the stochastic framework ( [6][7][8][9]14]). Recently, the authors of [8,11,17] have provided linearized versions of the standard continuous infinite horizon discounted control problems.…”