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Stability and local growth near bounded-strong optimal controls. (English) Zbl 1050.49021

Sachs, E. W. (ed.) et al., System modeling and optimization XX. IFIP TC7, 20th conference on system modeling and optimization, Trier, Germany, July 23–37, 2001. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-7565-0/hbk). IFIP, Int. Fed. Inf. Process. 130, 213-227 (2003).
Summary: Nonlinear constrained optimal control problems as a rule suffer from the so-called two-norm discrepancy, which in particular says that under stable optimality conditions the objective functionals satisfy a quadratic local growth estimate in terms of the \(L_2\) norms but in \(L_\infty\) neighborhoods of the solution only. Furthermore, in the case of weak local optima with continuous control functions, stability w.r.t. parameter changes usually can be expected to hold in \(L_\infty\) sense rather than in \(L_p\).
Whenever we consider problems with discontinuous optimal control behavior, these results are too restrictive to discuss general variations of the solution including changes in the break points or switches in the active sets. In the paper we show how the use of certain integrated optimality criteria obtained via a duality approach allows for estimates also in the case of discontinuous controls. We consider \(L_2\) and \(L_1\) quadratic growth estimates and discuss consequences for the behavior of minimzing sequences.
For the entire collection see [Zbl 1024.00076].

MSC:

49K40 Sensitivity, stability, well-posedness
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