Infinite Horizon Sparse Optimal Control

Abstract

A class of infinite horizon optimal control problems involving L p -type cost functionals with 0 < p ≤ 1 is discussed. The existence of optimal controls is studied for both the convex case with p = 1 and the nonconvex case with 0 < p < 1, and the sparsity structure of the optimal controls promoted by the L p -type penalties is analyzed. A dynamic programming approach is proposed to numerically approximate the corresponding sparse optimal controllers.

Sparse and switching infinite horizon optimal controls with mixed-norm penalizations

“…Proof. The case q = p has been dealt with in [21]. Therefore, we focus on the case q > p. Let w n = (w n 1 , .…”

“…They are either active, or zero and thus they join the set of sparse control coordinates. Comparing to the case p = q which was treated in [21,Proposition 5.2], the case (iii) is such that the control is necessarily active. Thus p < q enhances additional sparsity compared to p = q.…”

“…Let us mention previous related work on sparse and switching control. Closed-loop infinite horizon sparse optimal control problems with L p (0 < p ≤ 1) functionals were analyzed in [21]. Open-loop, finite horizon L 1 sparse optimal control for dynamical systems have been studied in e.g.…”

A Hybrid Finite-Dimensional RHC for Stabilization of Time-Varying Parabolic Equations

“…On the other hand, there is very little research dealing with infinite horizon nonsmmoth problems, see e.g., [18,35]. In [35] infinite horizon sparse optimal control problems governed by ordinary differential equations are investigated. In this work, the corresponding sparse optimal controller is approximated by a dynamic programming approach.…”

Strong Local Optimality for Generalized L1 Optimal Control Problems

“…The occurrence of sparse controls in this kind of minimization problems is well known also in the framework of infinite-dimensional optimal control, see for instance [7,8,9], where usually the absolute value of the control is added to the integral cost, in order to induce sparse solutions.…”