Abstract
A problem of sparse optimal control for the heat equation is considered, where pointwise bounds on the control and mixed pointwise control-state constraints are given. A standard quadratic tracking type functional is to be minimized that includes a Tikhonov regularization term and the \begin{document}$ L^1 $\end{document} -norm of the control accounting for the sparsity. Special emphasis is laid on existence and regularity of Lagrange multipliers for the mixed control-state constraints. To this aim, a duality theorem for linear programming problems in Hilbert spaces is proved and applied to the given optimal control problem.
Highlights
In a bounded domain Ω ⊂ RN with Lipschitz boundary Γ, we investigate the following problem of optimal sparse control: min J(y, u) := T 0 Ω 1 |y − yQ|2 + ν |u|2 2 κ |u|dxdt subject to the parabolic initial-boundary value problem (1.1)
We resolve two main difficulties: First, we show the existence of a Lagrange multiplier for inequality (1.4) that belongs to L∞(Q)
In Theorem 3.4, we introduced Lagrange multipliers only for the two upper constraints, while the lower bound u ≥ ua was not “eliminated” by a multiplier
Summary
(i) Let Assumption 3.2 be satisfied, ube optimal for the control problem (1.1)-(1.4), and ybe the associated state. If uis an optimal control and (μ1, μ2) ∈ (L2(Q))2 is an associated pair of Lagrange multipliers that exists according to Theorem 3.4, for almost all (x, t) ∈ Q the solution u of the problem min (φ + ν u + κ λ − S∗μ2)(x, t) u subject to ua ≤ u ≤ min{ub, ud + y(x, t)}
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