Abstract
This work concentrates on a class of optimal control problems for semilinear parabolic equations subject to control constraint of the form Vert u(t)Vert _{L^1(varOmega )} le gamma for t in (0,T). This limits the total control that can be applied to the system at any instant of time. The L^1-norm of the constraint leads to sparsity of the control in space, for the time instants when the constraint is active. Due to the non-smoothness of the constraint, the analysis of the control problem requires new techniques. Existence of a solution, first and second order optimality conditions, and regularity of the optimal control are proved. Further, stability of the optimal controls with respect to gamma is investigated on the basis of different second order conditions.
Highlights
We study the optimal control problem (P)
With 0 < γ < +∞, and yu is the solution of the semilinear parabolic equation
Let us denote by ProjBγ : L2(Ω) −→ Bγ ∩ L2(Ω) the L2(Ω) projection, where Bγ = {v ∈ L1(Ω) : v L1(Ω) ≤ γ }
Summary
To deal with the non-linearity of the state equation in the proof of a solution to (P) in L∞(Q), one approach consists in introducing artificial bound constraints on the control and prove that they are inactive as the artificial constraint parameter is large enough; see, for instance [7]. In our case, this would lead to two control constraints with two Lagrange multipliers in the dual of L∞. The L1-norm in space leads to a spatially sparsifying effect for the solutions It is different from the type of sparsification which results when considering such terms in the cost. As a consequence of the second order condition, Hölder and Lipschitz stability of local solutions with respect to the control bound γ is investigated
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