Abstract
Stabilization problems for parabolic equations with polynomial nonlinearities are investigated in the context of an optimal control formulation with a sparsity enhancing cost functional. This formulation allows that the optimal control completely shuts down once the trajectory is sufficiently close to a stable steady state. Such a property is not present for commonly chosen control mechanisms. To establish these results it is necessary to develop a function space framework for a class of optimal control problems posed on infinite time horizons, which is otherwise not available.
Highlights
In recent years there has been significant interest in the topic of sparse optimal controls
Up to now optimal control problems with sparsity constraints have typically been investigated for tracking problems on finite time horizons
The focus in previous work was set on controlling the sparsity structure in spatial directions
Summary
In recent years there has been significant interest in the topic of sparse optimal controls. In the present paper we focus on optimal controls which exhibit temporal sparsity This can be achieved by choosing a cost functional for the control variable which is nonsmooth in time. If ye is a stable equilibrium of a dynamical system, optimal control strategy typically provides controls which asymptotically steer the system to ye with the control not shutting down to zero even if the controlled trajectory is already in the close vicinity of ye. In the present paper we develop the necessary concepts for a class of semilinear parabolic equations This will include, in particular, proposing a function space framework for open loop infinite time horizon nonlinear optimal control problems. This topic has received very little attention in the literature even in cases of smooth cost functionals. U(x, t) if (x, t) ∈ ω × (0, ∞), 0 otherwise
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