Optimal sampled-data controls with running inequality state constraints: Pontryagin maximum principle and bouncing trajectory phenomenon

In the present paper we derive a Pontryagin maximum principle for general nonlinear optimal sampled-data control problems in the presence of running inequality state constraints. We obtain, in particular, a nonpositive averaged Hamiltonian gradient condition associated with an adjoint vector being a...

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Vydané v:	Mathematical programming Ročník 191; číslo 2; s. 907 - 951
Hlavní autori:	Bourdin, Loïc, Dhar, Gaurav
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2022 Springer Springer Nature B.V
Predmet:	Bouncing Calculus of Variations and Optimal Control; Optimization Combinatorics Equality Full Length Paper Inequality Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Maximum principle Nonlinear control Numerical Analysis Numerical methods Sampling Theoretical Trajectory optimization Sampled-data control 93C57 Indirect numerical method 93C10 Optimal control State constraints Pontryagin maximum principle Ekeland variational principle 49M05 34H05 Shooting method
ISSN:	0025-5610, 1436-4646
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Abstract	In the present paper we derive a Pontryagin maximum principle for general nonlinear optimal sampled-data control problems in the presence of running inequality state constraints. We obtain, in particular, a nonpositive averaged Hamiltonian gradient condition associated with an adjoint vector being a function of bounded variation. As a well known challenge, theoretical and numerical difficulties may arise due to the possible pathological behavior of the adjoint vector (jumps and singular part lying on parts of the optimal trajectory in contact with the boundary of the restricted state space). However, in our case with sampled-data controls, we prove that, under certain general hypotheses, the optimal trajectory activates the running inequality state constraints at most at the sampling times. Due to this so-called bouncing trajectory phenomenon, the adjoint vector experiences jumps at most at the sampling times (and thus in a finite number and at precise instants) and its singular part vanishes. Taking advantage of these informations, we are able to implement an indirect numerical method which we use to solve three simple examples.
AbstractList	In the present paper we derive a Pontryagin maximum principle for general nonlinear optimal sampled-data control problems in the presence of running inequality state constraints. We obtain, in particular, a nonpositive averaged Hamiltonian gradient condition associated with an adjoint vector being a function of bounded variation. As a well known challenge, theoretical and numerical difficulties may arise due to the possible pathological behavior of the adjoint vector (jumps and singular part lying on parts of the optimal trajectory in contact with the boundary of the restricted state space). However, in our case with sampled-data controls, we prove that, under certain general hypotheses, the optimal trajectory activates the running inequality state constraints at most at the sampling times. Due to this so-called bouncing trajectory phenomenon, the adjoint vector experiences jumps at most at the sampling times (and thus in a finite number and at precise instants) and its singular part vanishes. Taking advantage of these informations, we are able to implement an indirect numerical method which we use to solve three simple examples.
Audience	Academic
Author	Bourdin, Loïc Dhar, Gaurav
Author_xml	– sequence: 1 givenname: Loïc surname: Bourdin fullname: Bourdin, Loïc organization: Institut de recherche XLIM, UMR CNRS 7252, Université de Limoges – sequence: 2 givenname: Gaurav orcidid: 0000-0002-5849-1127 surname: Dhar fullname: Dhar, Gaurav email: gaurav.dhar@unilim.fr organization: Institut de recherche XLIM, UMR CNRS 7252, Université de Limoges
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Keywords	Sampled-data control 93C57 Indirect numerical method 93C10 Optimal control State constraints Pontryagin maximum principle Ekeland variational principle 49M05 34H05 Shooting method
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Snippet	In the present paper we derive a Pontryagin maximum principle for general nonlinear optimal sampled-data control problems in the presence of running inequality...
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SubjectTerms	Bouncing Calculus of Variations and Optimal Control; Optimization Combinatorics Equality Full Length Paper Inequality Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Maximum principle Nonlinear control Numerical Analysis Numerical methods Sampling Theoretical Trajectory optimization
Title	Optimal sampled-data controls with running inequality state constraints: Pontryagin maximum principle and bouncing trajectory phenomenon
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