On the no-gap second-order optimality conditions for a non-smooth semilinear elliptic optimal control

Abstract

This work is concerned with second-order necessary and sufficient optimality conditions for optimal control of a non-smooth semilinear elliptic partial differential equation, where the nonlinearity is the non-smooth max-function and thus the associated control-to-state operator is, in general, not Gâteaux-differentiable. In addition to standing assumptions, two main hypotheses are imposed. The first one is the Gâteaux-differentiability at the considered control of the objective functional and it is precisely characterized by the vanishing of an adjoint state on the set of all zeros of the corresponding state. The second one is a structural assumption on the sets of all points at which the values of the interested state are ‘close’ to the non-differentiability point of the max-function. We then derive a ‘no-gap’ theory of second-order optimality conditions in terms of a second-order generalized derivative of the cost functional, i.e. for which the only change between necessary and sufficient second-order optimality conditions are between a strict and non-strict inequality.

Keywords:

• Non-smooth semilinear elliptic equation
• optimal control
• second-order necessary and sufficient optimality condition
• no-gap theory
• control constraints

AMS SUBJECT Classifications:

• 49K20
• 49J20
• 35J25
• 35J61
• 35J91
• 90C48

Acknowledgments

The author would like to thank the anonymous reviewers for their constructive comments, which led to notable improvements to the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research was partially supported by International Centre for Research and Postgraduate Training in Mathematics, Institute of Mathematics, VAST, under grant number ICRTM01_2021.01.