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

Second-Order Analysis for Control Constrained Optimal Control Problems of Semilinear Elliptic... 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 extend the theory of second-order optimality conditions 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 \$(\Omega)\$ and L s \$(\Omega)\$ , with s>n/2 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

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

, Volume 38 (3) – Aug 1, 2090
23 pages

/lp/springer_journal/second-order-analysis-for-control-constrained-optimal-control-problems-LSYaU4BbRX
Publisher
Springer-Verlag
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s002459900093
Publisher site
See Article on Publisher Site

### 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 extend the theory of second-order optimality conditions 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 \$(\Omega)\$ and L s \$(\Omega)\$ , with s>n/2 .

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Aug 1, 2090

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