Open AccessExit time problems for nonlinear unbounded control systems
Author(s) -
Monica Motta,
Caterina Sartori
Publication year - 1999
Publication title -
discrete and continuous dynamical systems
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.289
H-Index - 70
eISSN - 1553-5231
pISSN - 1078-0947
DOI - 10.3934/dcds.1999.5.137
Subject(s) - controllability , nonlinear system , affine transformation , extension (predicate logic) , mathematics , bellman equation , boundary value problem , path (computing) , function (biology) , mathematical analysis , computer science , mathematical optimization , pure mathematics , physics , quantum mechanics , evolutionary biology , biology , programming language
Given a control system (formulated as a nonconvex and unbounded differential inclusion) we study the problem of reaching a closed target with trajectories of the system. A controllability condition around the target allows us to construct a path that steers each point nearby into it in finite time and using a finite amount of energy. In applications to minimization; problems, limits of such trajectories could be discontinuous. We extend the inclusion so that all the trajectories of the extension can be approached by (graphs of) solutions of the original system. In the extended setting the value function of an exit time problem with Lagrangian affine in the unbounded control can be shown to coincide with the value function of the original problem, to be continuous and to be the unique (viscosity) solution of a Hamilton-Jacobi equation with suitable boundary conditions