Abstract
An optimal control problem for a semilinear elliptic equation is discussed, where the control appears nonlinearly in the state equation but is not included in the objective functional. The existence of optimal controls is proved by a measurable selection technique. First-order necessary optimality conditions are derived and two types of second-order sufficient optimality conditions are established. A first theorem invokes a well-known assumption on the set of zeros of the switching function. A second relies on coercivity of the second derivative of the reduced objective functional. The results are applied to the convergence of optimal state functions for a finite element discretizion of the control problem.
Highlights
In this paper, we consider the following optimal control problem:(P) min J(u), u\in \scrU ad where\scrU ad = \{ u \in L\infty (\Omega ) : \alpha \leq u(x) \leq \beta for a.a. x \in \Omega \}with - \infty < \alpha < \beta < +\infty, and\int J(u) = L(x, yu(x)) dx. \OmegaAbove L : \Omega \times \BbbR - \rightar \BbbR is a given function, and yu is the solution of the following elliptic equation: (1.1)\bigl\{ Ay = f (x, y, u) in \Omega, \partialnA y = on \Gama
By using a structural assumption on the optimal adjoint state, we show that the control satisfying the first-order optimality conditions is locally optimal in the sense of L\infty (\Omega )
We show the existence of optimal controls by a measurable selection theorem
Summary
We consider the following optimal control problem:. \scrU ad = \{ u \in L\infty (\Omega ) : \alpha \leq u(x) \leq \beta for a.a. x \in \Omega \}. By using a structural assumption on the optimal adjoint state, we show that the control satisfying the first-order optimality conditions is locally optimal in the sense of L\infty (\Omega ). Due to the monotonicity of f with respect to y, the existence of a unique solution of (1.1) in H1(\Omega ) \cap L\infty (\Omega ) is a classical result; see, for instance, [3] or [25, Theorem 4.7]. \leq Cf,M \| uk - u\| Lp\=(\Omega )\| yuk - yu\| Lp\=\prime (\Omega ) \leq Cf,M Cp\=,\Omega \| uk - u\| Lp\=(\Omega )\| yuk - yu\| H1(\Omega ), where we have used that p\= >
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