\(L^\infty\)-stability of a parametric optimal control problem governed by semilinear elliptic equations
The paper studies the local \(L^\infty\)-stability of a parametric optimal control problem governed by semilinear elliptic equations with mixed pointwise constraints. The present work expands on the results presented in [\textit{B. T. Kien} and \textit{J.-C. Yao}, ``Local stability of solutions to parametric semilinear elliptic optimal control problems'', Appl. Anal. Optim. 1, No. 3, 361--379 (2017)]. Under the assumption that that the zero-gap second-order optimality condition is satisfied, it is shown that the solution map is locally upper Hölder continuous at the reference parameter in the \(L^\infty\)-norm (and not only in the \(L^2\)-norm) of the control variable. First, the paper presents the key assumptions and the main theorem (local Hölder continuity in the \(L^\infty\)-norm; Theorem 2.1), followed by some auxiliary results. Subsequently, the authors present their main body of work -- the proof of Theorem 2.1.
second-order sufficient optimality condition
scientific article; zbMATH DE number 7381023