Convergence in Nonlinear Optimal Sampled-Data Control Problems

Consider, on the one part, a general nonlinear finite-dimensional optimal control problem and assume that it has a unique solution x^*. On the other part, consider the sampled-data control version of it. Under appropriate assumptions, we prove that the optimal state of the sampled-data problem conve...

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Published in:IEEE transactions on automatic control 2024-10, Vol.69 (10), p.7144-7151
Main Authors: Bourdin, Loic, Trelat, Emmanuel
Format: Article
Language:English
Subjects:
Convergence
Costs
Filippov approach
Mathematics
Nonlinear control
Normality
Optimal control
Optimization and Control
Pontryagin maximum principle (PMP)
Pontryagin principle
sampled-data control
Trajectory
Uniqueness
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description Consider, on the one part, a general nonlinear finite-dimensional optimal control problem and assume that it has a unique solution x^*. On the other part, consider the sampled-data control version of it. Under appropriate assumptions, we prove that the optimal state of the sampled-data problem converges uniformly to x^* as the norm of the partition tends to zero. Moreover, applying the Pontryagin maximum principle (PMP) to both problems, we prove that, if x^* has a unique weak extremal lift with a costate p that is normal, then the costate of the sampled-data problem converges uniformly to p. In other words, under a normality assumption, control sampling commutes, at the limit of small partitions, with the application of the PMP.
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On the other part, consider the sampled-data control version of it. Under appropriate assumptions, we prove that the optimal state of the sampled-data problem converges uniformly to <inline-formula><tex-math notation="LaTeX">x^*</tex-math></inline-formula> as the norm of the partition tends to zero. Moreover, applying the Pontryagin maximum principle (PMP) to both problems, we prove that, if <inline-formula><tex-math notation="LaTeX">x^*</tex-math></inline-formula> has a unique weak extremal lift with a costate <inline-formula><tex-math notation="LaTeX">p</tex-math></inline-formula> that is normal, then the costate of the sampled-data problem converges uniformly to <inline-formula><tex-math notation="LaTeX">p</tex-math></inline-formula>. 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subjects Convergence
Costs
Filippov approach
Mathematics
Nonlinear control
Normality
Optimal control
Optimization and Control
Pontryagin maximum principle (PMP)
Pontryagin principle
sampled-data control
Trajectory
Uniqueness
title Convergence in Nonlinear Optimal Sampled-Data Control Problems
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