A variant of nonsmooth maximum principle for state constrained problems

This work has developed a new robust and reliable O(N) algorithm for solving general inequality/equality constrained minimum-time problems. To our knowledge, no one has ever applied an O(N) algorithm for solving such minimum time problems. Moreover, the algorithm developed here is new and unique and does not suffer the inevitable ill-conditioning problems that pre-existing O(N) methods for inequality-constrained problems do. Herein we demonstrate the new algorithm by solving several cases of a tip path constrained three-link redundant robotic arm problem with torque bounds and joint angle bounds. Results are consistent with Pontryagin’s Maximum Principle. We include a speed/robustness/complexity comparison with a sequential quadratic programming (SQP) code. Here, the O(N) complexity and the significant speed, robustness, and complexity improvements over an SQP code are demonstrated. These numerical results are complemented with a rigorous theoretical convergence proof of the new O(N) algorithm.

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Maximum principle for state and mixed constrained problems with equality and inequality mixed constraints: Convex case

10.1063/1.4825572 ◽

2013 ◽

Author(s):

M. H. A. Biswas ◽

MdR de Pinho

Keyword(s):

Maximum Principle ◽

Mixed Constraints ◽

Constrained Problems ◽

Convex Case

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When Is the Maximum Principle for State Constrained Problems Nondegenerate?

Journal of Mathematical Analysis and Applications ◽

10.1006/jmaa.1994.1366 ◽

1994 ◽

Vol 187 (2) ◽

pp. 438-467 ◽

Cited By ~ 51

Author(s):

M.M.A. Ferreira ◽

R.B. Vinter

Keyword(s):

Maximum Principle ◽

The Maximum Principle ◽

Constrained Problems

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Relationship between maximum principle and dynamic programming in presence of intermediate and final state constraints

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2021084 ◽

2021 ◽

Author(s):

Olivier Bokanowski ◽

Anya Desilles ◽

Hasnaa Zidani

Keyword(s):

Maximum Principle ◽

Control Problem ◽

State Constraints ◽

Value Function ◽

Regularity Property ◽

Initial Time ◽

Final State ◽

Lipschitz Regularity ◽

Constrained Problems ◽

The Value Function

In this paper, we consider a class of optimal control problems governed by a differential system. We analyse the sensitivity relations satisfied by the co-state arc of the Pontryagin maximum principle and the value function that associates the optimal value of the control problem to the initial time and state. Such a relationship has been already investigated for state-constrained problems under some controllability assumptions to guarantee Lipschitz regularity property of the value function. Here, we consider the case with intermediate and final state constraints, without any controllability assumption on the system, and without Lipschitz regularity of the value function. Because of this lack of regularity, the sensitivity relations cannot be expressed with the sub-differentials of the value function. This work shows that the constrained problem can be reformulated with an auxiliary value function which is more regular and suitable to express the sensitivity of the adjoint arc of the original state-constrained control problem along an optimal trajectory. Furthermore, our analysis covers the case of normal optimal solutions, and abnormal solutions as well.

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