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Linearization and Boundary Trajectories of Nonsmooth Control Systems

Published online by Cambridge University Press:  20 November 2018

H. Frankowska  and
B. Kaśkosz
H. Frankowska
Affiliation:
Université de Paris-Dauphine, Paris, France
B. Kaśkosz
Affiliation:
University of Rhode Island, Kingston, Rhode Island
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This paper deals with boundary trajectories of non-smooth control systems and differential inclusions.

Consider a control system

(1.1)

and denote by R(t) its reachable set at time t. Let (z, u*) be a trajectory-control pair. If for every t from the time interval [0, 1], z(t) lies on the boundary of R(t) then z is called a boundary trajectory. It is known that for systems with Lipschitzian in x right-hand side, z is a boundary trajectory if and only if z(1) belongs to the boundary of the set R(1). If z is not a boundary trajectory, that is, z(1) ∊ Int R(1) then the system is said to be locally controllable around z at time 1.

A first-order necessary condition for boundary trajectories of smooth systems comes from the Pontriagin maximum principle, (see e.g. [12]).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 3 , 01 June 1988 , pp. 589 - 609
Copyright
Copyright © Canadian Mathematical Society 1988

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