TY - CHAP

T1 - Strong maximum principle for systems of differential inclusions

AU - Sussmann, Hector J.

PY - 1996

Y1 - 1996

N2 - We present a version of the finite-dimensional Maximum Principle for systems of differential inclusions, that contains as particular cases previous versions for control systems as well as for a single differential inclusion. The result incorporates high-order point variations and is valid under minimal technical assumptions, weaker than those of most classical and non-smooth versions. The proof follows the classical idea of using needle variations and applying an appropriate open mapping theorem to a multiparameter variation, whose effect is computed in terms of those of the needle variations by means of the chain rule. However, this has to be carried out in a new setting, namely, the class of `semidifferentiable maps,' that contains all maps arising in the optimal control problem and has a concept of generalized differential with all the right properties. We depart from the most common approaches to the problem by reducing the differential inclusions case to the vector field system case, in spite of the fact that set-valued functions - even if they are continuous or Lipschitz or smooth - in general fail to have continuous selections. The key technical tool enabling us to carry out the reduction anyhow is a generalization of a selection theorem due to A. Bressan, which yields the existence, for almost lower semicontinuous set-valued maps, of selections that, while not necessarily continuous, are time-varying vector fields with good properties, such as local existence of trajectories and upper semi-continuity of the (set-valued) flow.

AB - We present a version of the finite-dimensional Maximum Principle for systems of differential inclusions, that contains as particular cases previous versions for control systems as well as for a single differential inclusion. The result incorporates high-order point variations and is valid under minimal technical assumptions, weaker than those of most classical and non-smooth versions. The proof follows the classical idea of using needle variations and applying an appropriate open mapping theorem to a multiparameter variation, whose effect is computed in terms of those of the needle variations by means of the chain rule. However, this has to be carried out in a new setting, namely, the class of `semidifferentiable maps,' that contains all maps arising in the optimal control problem and has a concept of generalized differential with all the right properties. We depart from the most common approaches to the problem by reducing the differential inclusions case to the vector field system case, in spite of the fact that set-valued functions - even if they are continuous or Lipschitz or smooth - in general fail to have continuous selections. The key technical tool enabling us to carry out the reduction anyhow is a generalization of a selection theorem due to A. Bressan, which yields the existence, for almost lower semicontinuous set-valued maps, of selections that, while not necessarily continuous, are time-varying vector fields with good properties, such as local existence of trajectories and upper semi-continuity of the (set-valued) flow.

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M3 - Chapter

AN - SCOPUS:0030389003

T3 - Proceedings of the IEEE Conference on Decision and Control

SP - 1809

EP - 1814

BT - Proceedings of the IEEE Conference on Decision and Control

A2 - Anon, null

T2 - Proceedings of the 35th IEEE Conference on Decision and Control. Part 4 (of 4)

Y2 - 11 December 1996 through 13 December 1996

ER -