Abstract. We discuss control systems defined on an infinite horizon, where typically all the associated costs become unbounded as the time grows indefinitely. It is proved, under certain lower semicontinuity and controllability assumptions, that a linear time function can be subtracted from the cost, resulting in a modified cost, which is bounded on the infinite time interval. The cost evaluated over one sampling interval has a simple representation in terms of the initial and final states. Applying this representation we obtain an optimality result for control systems represented by ordinary differential equations whose cost integrand contains a discounting factor.
We discuss conditions for the existence of the limit occupational measures set for a control system. We approximate slow components of the trajectories of a singularly perturbed control system by the solutions of a differential inclusion. The differential inclusion is obtained via averaging the slow subsystem over measures from the limit occupational measures set constructed for the associated system describing the ''fast'' motions with ''frozen'' slow ones. ᮊ
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