Infinite Horizon Autonomous Systems with Unbounded Cost

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.