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# Second Order Analysis for Control Constrained Optimal Control Problems of Semilinear Elliptic Systems

1 PROMATH - Mathematical Programming
Inria Paris-Rocquencourt
Abstract : This paper presents a second-order analysis for a simple model optimal control problem of a partial differential equation, namely a well-posed semilinear elliptic system with constraints on the control variable only. The cost to be minimized is a standard quadratic functional. Assuming the feasible set to be polyhedric, we state necessary and sufficient second order optimality conditions, including a characterization of the quadratic growth condition. Assuming that the second order sufficient condition holds, we give a formula for the second order expansion of the value of the problem as well as the directional derivative of the optimal control, when the cost function is perturbed. Then we partially extend these results to the case of vector valued controls when the feasible set is defined by local and smooth convex constraints. When the space dimension $n$ is greater than 3, the results are based on a two norms approach, involving spaces $L^2(Øm)$ and $L^s(Øm)$, with $s>n/2$.
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https://hal.inria.fr/inria-00073680
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Submitted on : Wednesday, May 24, 2006 - 1:29:51 PM
Last modification on : Friday, February 4, 2022 - 3:08:43 AM
Long-term archiving on: : Sunday, April 4, 2010 - 11:54:35 PM

### Identifiers

• HAL Id : inria-00073680, version 1

### Citation

J. Frederic Bonnans. Second Order Analysis for Control Constrained Optimal Control Problems of Semilinear Elliptic Systems. [Research Report] RR-3014, INRIA. 1996. ⟨inria-00073680⟩

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