Optimal almost-periodic control problem: a trigonometric approximation approach

The problem of finding an almost-periodic control that is optimal with respect to a certain time-averaged criterion for the dynamic system operated over a long period of time is considered. The existence of the optimal solution, spectral properties of which satisfy certain regularity conditions, is hypothesized. The problem is approximated by a sequence of finite-dimensional optimization problems containing trigonometric sums for the approximation of the state and control variables, and using a Fejér-Riesz type representation for a positive trigonometric sum to handle the instantaneous constraints for these variables. Sufficient conditions for the sequence of approximate optimal solutions of the discretized problems to be an approximately minimizing sequence for the basic problem are given. The constructive character of the proposed approach and its potential applications are pointed out both for dynamic systems affected by irregularly pulsating disturbances and for stationary systems, the non-linear dynamics of which can be exploited by a non-stationary control to improve the averaged system performance.