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.Key words. long run average optimal control, singularly perturbed control systems, occupational measures, averaging method, linear programming AMS subject classifications. 34E15, 34C29, 34A60, 93C70 DOI. 10.1137/0406168021. 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 finitedimensional 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] 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 systemẏ

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.

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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

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