Abstract
A class of infinite horizon optimal control problems involving mixed quasi-norms of Lp-type cost functionals for the controls is discussed. These functionals enhance sparsity and switching properties of the optimal controls. The existence of optimal controls and their structural properties are analyzed on the basis of first order optimality conditions. A dynamic programming approach is used for numerical realization.
Highlights
In this work we continue our investigations of infinite horizon optimal control problems with nonconvex cost functionals which we started in [23]
To investigate the optimality conditions satisfied by the optimal controllers, we introduce firstly the adjoint equation associated to (y, u) satisfying (5.20):
In this paper we have studied infinite horizon optimal control problems with a control cost of the form u qp, where 0 < p ≤ q ≤ 1, leading to a non-convex, non-smooth optimization problem
Summary
In this work we continue our investigations of infinite horizon optimal control problems with nonconvex cost functionals which we started in [23]. Decreasing q we expect that the subdomain over which the optimal control vanishes (in all coordinates) increases These properties will be illustrated by numerical experiments. In the context of partial differential equations optimal control of systems switching among different modes were analysed in [18, 19], problems with convex switching enhancing functionals were investigated in [13, 11], and problems with nonconvex switching penalization in [12]. The sparsity and switching structure of the optimal controls is analyzed on the basis of the optimality conditions for the time-continuous as well as the time discrete problems in sections 4 and 5, respectively, and section 6 contains numerical results
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