Abstract
Optimal control problems in measure spaces governed by semilinear elliptic equations are considered. First order optimality conditions are derived and structural properties of their solutions, in particular sparsity, are discussed. Necessary and sufficient second order optimality conditions are obtained as well. On the basis of the sufficient conditions, stability of the solutions is analyzed. Highly nonlinear terms can be incorporated by utilizing an $L^\infty (\Omega)$ regularity result for solutions of the first order necessary optimality conditions.
Highlights
This paper is dedicated to the study of the optimal control problem min J(u) =
M(ω), where y is the unique solution to the Dirichlet problem
To prepare for the second order necessary conditions we introduce the critical cone as follows: (3.9)
Summary
In the case that ω = Ω extra regularity of controls and states which satisfy the first order necessary condition can be obtained We will establish the existence and uniqueness of the solution of the state equation (1.2) as well as the continuity and differentiability properties of the control-to-state mapping. For the sake of completeness, let us give an independent proof of the existence of a solution that illustrates the difficulty of passing from L1 functions to measures and the role played by the growth condition (2.1). The monotonicity of a with respect to the second variable implies g(x, s)s ≥ 0 for all s ∈ R With these properties the existence of a solution w ∈ W01,1(Ω) of (2.6) follows; see [3, Theorem 2].
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