No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation

Clason, Christian; Nhu, Vu Huu; Rösch, Arnd

ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 62

This paper deals with second-order optimality conditions for a quasilinear elliptic control problem with a nonlinear coefficient in the principal part that is finitely PC² (continuous and C² apart from finitely many points). We prove that the control-to-state operator is continuously differentiable even though the nonlinear coefficient is non-smooth. This enables us to establish “no-gap” second-order necessary and sufficient optimality conditions in terms of an abstract curvature functional, i.e., for which the sufficient condition only differs from the necessary one in the fact that the inequality is strict. A condition that is equivalent to the second-order sufficient optimality condition and could be useful for error estimates in, e.g., finite element discretizations is also provided.

Classification : 49K20, 49B22, 35J62
Keywords: Optimal control, non-smooth optimization, second-order necessary optimality condition, second-order sufficient optimality condition, quasilinear elliptic equation, piecewise differentiable function

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