New approximation method in the proof of the Maximum Principle for nonsmooth optimal control problems with state constraints

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Abstract

Traditional proofs of the Pontryagin Maximum Principle (PMP) require the continuous differentiability of the dynamics with respect to the state variable on a neighborhood of the minimizing state trajectory, when arbitrary values of control variable are inserted into the dynamic equations. Sussmann has drawn attention to the fact that the PMP remains valid when the dynamics are differentiable with respect to the state variable, merely when the minimizing control is inserted into the dynamic equations. This weakening of earlier hypotheses has been referred to as the Lojasiewicz refinement. Arutyunov and Vinter showed that these extensions of early versions of the PMP can be simply proved by finite-dimensional approximations, application of a Lagrange multiplier rule in finite dimensions and passage to the limit. This paper generalizes the finite-dimensional approximation technique to a problem with state constraints, where the use of needle variations of the optimal control had not been successful. Moreover, the cost function and endpoint constraints are not assumed to be differentiable, but merely locally Lipschitz continuous. The Maximum Principle is expressed in terms of Michel-Penot subdifferential.

Original languageEnglish (US)
Pages (from-to)974-1000
Number of pages27
JournalJournal of Mathematical Analysis and Applications
Volume326
Issue number2
DOIs
StatePublished - Feb 15 2007

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Pontryagin Maximum Principle
Maximum principle
State Constraints
Maximum Principle
Approximation Methods
Optimal Control Problem
Finite-dimensional Approximation
Dynamic Equation
Differentiable
Lagrange multiplier Rule
Subdifferential
Differentiability
Lipschitz
Cost Function
Lagrange multipliers
Optimal Control
Refinement
Cost functions
Needles
Valid

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

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abstract = "Traditional proofs of the Pontryagin Maximum Principle (PMP) require the continuous differentiability of the dynamics with respect to the state variable on a neighborhood of the minimizing state trajectory, when arbitrary values of control variable are inserted into the dynamic equations. Sussmann has drawn attention to the fact that the PMP remains valid when the dynamics are differentiable with respect to the state variable, merely when the minimizing control is inserted into the dynamic equations. This weakening of earlier hypotheses has been referred to as the Lojasiewicz refinement. Arutyunov and Vinter showed that these extensions of early versions of the PMP can be simply proved by finite-dimensional approximations, application of a Lagrange multiplier rule in finite dimensions and passage to the limit. This paper generalizes the finite-dimensional approximation technique to a problem with state constraints, where the use of needle variations of the optimal control had not been successful. Moreover, the cost function and endpoint constraints are not assumed to be differentiable, but merely locally Lipschitz continuous. The Maximum Principle is expressed in terms of Michel-Penot subdifferential.",
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