# Characterization of local quadratic growth for strong minima in the optimal control of semi-linear elliptic equations

2 Commands - Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées
Abstract : In this article we consider an optimal control problem of a semi-linear elliptic equation, with bound constraints on the control. Our aim is to characterize local quadratic growth for the cost function $J$ in the sense of strong solutions. This means that the function $J$ growths quadratically over all feasible controls whose associated state is close enough to the nominal one, in the uniform topology. The study of strong solutions, classical in the Calculus of Variations, seems to be new in the context of PDE optimization. Our analysis, based on a decomposition result for the variation of the cost, combines Pontryagin's principle and second order conditions. While these two ingredients are known, we use them in such a way that we do not need to assume that the Hessian of Lagrangian of the problem is a Legendre form, or that it is uniformly positive on an extended set of critical directions.
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Journal articles

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Térence Bayen, J. Frederic Bonnans, Francisco J. Silva. Characterization of local quadratic growth for strong minima in the optimal control of semi-linear elliptic equations. Transactions American Mathematical Society, American Mathematical Society, 2014, 366 (4), pp.2063--2087. ⟨10.1090/S0002-9947-2013-05961-2⟩. ⟨inria-00632308v2⟩

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