International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 65, Pages 3513-3540
doi:10.1155/S0161171204310306
Monotonicity and differential properties of the value functions in
optimal control
Faculty of Mathematics and computer science, University of Bucharest, Academiei 14, Bucharest 70109, Romania
Received 28 October 2003
Copyright © 2004 Ştefan Mirică. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Using the “basic monotonicity property” along locally
admissible trajectories, we extend to very general problems
certain existing results concerning the differential inequalities
verified by the value function of an optimal control problem;
these differential inequalities are expressed in terms of
its contingent, quasitangent, and peritangent
(Clarke's) directional derivatives and in terms of
certain sets of “generalized tangent directions” to the
“locally admissible trajectories.” Under additional, rather
restrictive hypotheses on the data, which allow suitable estimates
(and even exact characterizations) of the sets of generalized
tangent directions to the trajectories, the differential
inequalities are shown to imply previous results according to
which the value function is a “generalized solution” (in the
“contingent,” “viscosity,” or “Clarke” sense) of the
associated Hamilton-Jacobi-Bellman (HJB) equation.