### 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 language | English (US) |
---|---|

Pages (from-to) | 974-1000 |

Number of pages | 27 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 326 |

Issue number | 2 |

DOIs | |

State | Published - Feb 15 2007 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

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**New approximation method in the proof of the Maximum Principle for nonsmooth optimal control problems with state constraints.** / Shvartsman, Ilya.

Research output: Contribution to journal › Article

TY - JOUR

T1 - New approximation method in the proof of the Maximum Principle for nonsmooth optimal control problems with state constraints

AU - Shvartsman, Ilya

PY - 2007/2/15

Y1 - 2007/2/15

N2 - 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.

AB - 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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U2 - 10.1016/j.jmaa.2006.03.056

DO - 10.1016/j.jmaa.2006.03.056

M3 - Article

AN - SCOPUS:33750623445

VL - 326

SP - 974

EP - 1000

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -