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The paper deals with a problem of control of a system characterized by the fact that the influence of controls on the dynamics of certain functions of state variables (called observables) is relatively weak, and the rates of change of these observables are much slower than the rates of change of the state variables themselves. The contributions of the paper are twofold. First, the averaged system whose solutions approximate the trajectories of the slow observables is introduced, and it is shown that the optimal value of the problem of optimal control with time discounting criterion considered on the solutions of the system with slow observables (this problem is referred to as perturbed) converges to the optimal value of the corresponding problem of optimal control of the averaged system. Second, a method for constructing an asymptotically optimal control of the perturbed problem on the basis of an optimal solution of the averaged problem is indicated, sufficient and necessary optimality conditions for the averaged problem are stated, and a way for constructing numerically a near optimal solution of the latter is outlined (the construction being illustrated with an example). [ABSTRACT FROM AUTHOR]

The paper aims at the development of an apparatus for analysis and construction of near optimal solutions of singularly perturbed (SP) optimal controls problems (that is, problems of optimal control of SP systems) considered on the infinite time horizon. We mostly focus on problems with time discounting criteria but a possibility of the extension of results to periodic optimization problems is discussed as well. Our consideration is based on earlier results on averaging of SP control systems and on linear programming formulations of optimal control problems. The idea that we exploit is to first asymptotically approximate a given problem of optimal control of the SP system by a certain averaged optimal control problem, then reformulate this averaged problem as an infinite-dimensional linear programming (LP) problem, and then approximate the latter by semi-infinite LP problems. We show that the optimal solution of these semi-infinite LP problems and their duals (that can be found with the help of a modification of an available LP software) allow one to construct near optimal controls of the SP system. We demonstrate the construction with two numerical examples. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to study singularly perturbed control systems. Firstly, we provide linearized formulation version for the calculus of the value function associated with the averaged dynamics. Secondly, we obtain necessary and sufficient conditions in order to identify the optimal trajectory of the averaged system. [Copyright &y& Elsevier]

A representation of the limit occupational measures set of a control system in terms of the vector function defining the system's dynamics is established. Applications in averaging of singularly perturbed control systems are demonstrated. [ABSTRACT FROM AUTHOR]