Abstract
In this paper, we consider optimal control problems associated with semilinear elliptic equation equations, where the states are subject to pointwise constraints but there are no explicit constraints on the controls. A term is included in the cost functional promoting the sparsity of the optimal control. We prove existence of optimal controls and derive first and second order optimality conditions. In addition, we establish some regularity results for the optimal controls and the associated adjoint states and Lagrange multipliers.
Highlights
In this paper, we analyze the following optimal control problem (P) min J(u) u∈Uad with J(u) = F (u) + κj(u), κ > 0, (yu(x) Ω − yd(x))2 dx + ν 2u2(x) dx and j(u) =where yd ∈ L2(Ω) and ν > 0 are given
With some γ > 0, where yu is the solution of the semilinear elliptic partial differential equation
An additional difficulty to get the optimality conditions is the presence of the non-differentiable term j(u) in the cost functional that promotes the sparsity of the optimal control
Summary
We analyze the following optimal control problem (P) min J(u) u∈Uad with J(u) = F (u) + κj(u), κ > 0,. |u(x)| dx, with some γ > 0, where yu is the solution of the semilinear elliptic partial differential equation. Semilinear elliptic equations, first and second order optimality conditions, sparse controls.
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