Applied Mathematics and Optimization

, Volume 38, Issue 3, pp 303–325

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

  • J. F. Bonnans

DOI: 10.1007/s002459900093

Cite this article as:
Bonnans, J. Appl Math Optim (1998) 38: 303. doi:10.1007/s002459900093


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 L2\((\Omega)\) and Ls\((\Omega)\) , with s>n/2 .

Key words. Optimal control, Elliptic systems, Sensitivity analysis, Expansion of solutions, Second-order optimality conditions, Legendre forms, Polyhedricity, Two norms approach. AMS Classification. Primary 49K40, Secondary 49K20, 35B30, 35J60, 90C31. 

Copyright information

© 1998 Springer-Verlag New York Inc.

Authors and Affiliations

  • J. F. Bonnans
    • 1
  1. 1.INRIA, B.P. 105, 78153 Rocquencourt, France FR

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