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arXiv · 2009.12124

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

Vu Huu Nhu
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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 G\^{a}teaux-differentiable. In addition to standing assumptions, two main hypotheses are imposed. The first one is the G\^{a}teaux-differentiability at the considered control of the objective functional and it is precisely characterized by the vanishing of an adjoint state on the active set. The second one is a structural assumption on the 'almost' active sets, i.e., 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 an abstract curvature functional, i.e., for which the only change between necessary or sufficient second-order optimality conditions is between a strict and non strict inequality.

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Vu Huu Nhu. 2020-09-25. On the no-gap second-order optimality conditions for a non-smooth semilinear elliptic optimal control. https://doi.org/10.1080/02331934.2021.1939698

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