Volume 27, 2021
|Number of page(s)||35|
|Published online||21 June 2021|
No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation
Faculty of Mathematics, University of Duisburg-Essen,
2 Current address: Faculty of Fundamental Sciences, PHENIKAA University, Yen Nghia, Ha Dong, Hanoi 12116, Vietnam.
* Corresponding author: firstname.lastname@example.org
Accepted: 22 December 2020
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.
Mathematics Subject Classification: 49K20 / 49B22 / 35J62
Key words: Optimal control / non-smooth optimization / second-order necessary optimality condition / second-order sufficient optimality condition / quasilinear elliptic equation / piecewise differentiable function
© EDP Sciences, SMAI 2021
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