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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Bibliographic Details
Published in:Mathematical programming Vol. 191; no. 2; pp. 907 - 951
Main Authors: Bourdin, Loïc, Dhar, Gaurav
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2022
Springer
Springer Nature B.V
Subjects:
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
Online Access:Get full text
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Summary: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.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-020-01574-2