Stability Analysis of Optimal Control Problems with a Second-Order State Constraint

This paper gives stability results for nonlinear optimal control problems subject to a regular state constraint of second-order. The strengthened Legendre-Clebsch condition is assumed to hold, and no assumption on the structure of the contact set is made. Under a weak secondorder sufficient conditio...

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Vydané v:SIAM journal on optimization Ročník 20; číslo 1; s. 104 - 129
Hlavný autor: Hermant, Audrey
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Philadelphia Society for Industrial and Applied Mathematics 01.01.2009
Predmet:
Boundary value problems
Contact
Mathematics
Multipliers
Norms
Optimal control
Optimization
Optimization and Control
Ordinary differential equations
Perturbation methods
Regularity
Stability
Studies
alternative formulation
uniform quadratic growth
Optimal control
stability analysis
second-order state constraint
strong regularity
sufficient second-order optimality condition
ISSN:1052-6234, 1095-7189
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Abstract This paper gives stability results for nonlinear optimal control problems subject to a regular state constraint of second-order. The strengthened Legendre-Clebsch condition is assumed to hold, and no assumption on the structure of the contact set is made. Under a weak secondorder sufficient condition (taking into account the active constraints), we show that the solutions are Lipschitz continuous w.r.t. the perturbation parameter in the L^sup 2^ norm, and Holder continuous in the L∞ norm. We use a generalized implicit function theorem in metric spaces by Dontchev and Hager [SIAM J. Control Optim., 36 (1998), pp. 698-718]. The difficulty is that multipliers associated with second-order state constraints have a low regularity (they are only bounded measures). We obtain Lipschitz stability of a "primitive" of the state constraint multiplier. [PUBLICATION ABSTRACT]
AbstractList This paper gives stability results for nonlinear optimal control problems subject to a regular state constraint of second-order. The strengthened Legendre-Clebsch condition is assumed to hold, and no assumption on the structure of the contact set is made. Under a weak secondorder sufficient condition (taking into account the active constraints), we show that the solutions are Lipschitz continuous w.r.t. the perturbation parameter in the L super( 2) norm, and Holder continuous in the L infinity norm. We use a generalized implicit function theorem in metric spaces by Dontchev and Hager [SIAM J. Control Optim., 36 (1998), pp. 698-718]. The difficulty is that multipliers associated with second-order state constraints have a low regularity (they are only bounded measures). We obtain Lipschitz stability of a "primitive" of the state constraint multiplier.
This paper gives stability results for nonlinear optimal control problems subject to a regular state constraint of second-order. The strengthened Legendre-Clebsch condition is assumed to hold, and no assumption on the structure of the contact set is made. Under a weak second-order sufficient condition (taking into account the active constraints), we show that the solutions are Lipschitz continuous w.r.t. the perturbation parameter in the $L^2$ norm, and Hölder continuous in the $L^\infty$ norm. We use a generalized implicit function theorem in metric spaces by Dontchev and Hager [SIAM J. Control Optim., 1998]. The difficulty is that multipliers associated with second-order state constraints have a low regularity (they are only bounded measures). We obtain Lipschitz stability of a ``primitive'' of the state constraint multiplier.
This paper gives stability results for nonlinear optimal control problems subject to a regular state constraint of second-order. The strengthened Legendre-Clebsch condition is assumed to hold, and no assumption on the structure of the contact set is made. Under a weak secondorder sufficient condition (taking into account the active constraints), we show that the solutions are Lipschitz continuous w.r.t. the perturbation parameter in the L^sup 2^ norm, and Holder continuous in the L∞ norm. We use a generalized implicit function theorem in metric spaces by Dontchev and Hager [SIAM J. Control Optim., 36 (1998), pp. 698-718]. The difficulty is that multipliers associated with second-order state constraints have a low regularity (they are only bounded measures). We obtain Lipschitz stability of a "primitive" of the state constraint multiplier. [PUBLICATION ABSTRACT]
Author Hermant, Audrey
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Cites_doi 10.1137/S0363012996299314
10.1137/0130053
10.1007/s10107-007-0167-8
10.1007/BFb0120927
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10.1090/S0025-5718-00-01184-4
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Keywords alternative formulation
uniform quadratic growth
Optimal control
stability analysis
second-order state constraint
strong regularity
sufficient second-order optimality condition
Language English
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SubjectTerms Boundary value problems
Contact
Mathematics
Multipliers
Norms
Optimal control
Optimization
Optimization and Control
Ordinary differential equations
Perturbation methods
Regularity
Stability
Studies
Title Stability Analysis of Optimal Control Problems with a Second-Order State Constraint
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