International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 65, Pages 3513-3540

Monotonicity and differential properties of the value functions in optimal control

Ştefan Mirică

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.


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.