Monotonicity and differential properties of the value functions in optimal control (Q1777870)

Using monotonicity properties satisfied by the value function along trajectories of a control system, it is possible to characterize the value function of an optimal control problem. This can be viewed as a ``generalized dynamic programming principle''. The Hamilton-Jacobi-Bellman equation can be considered as two inequalities which are the infinitesimal characterization of the monotonicity properties of the value function. It is well known that this infinitesimal characterization could be expressed using (viscosity) sub- and supersolution, contingent solution, generalized Subotin's solution etc. In the present paper, the author suggests to use contingent and peritangent derivatives to express infinitesimal versions of the monotonicity properties. Doing this, he obtains a new characterization of the value function in term of -- suitably generalized -- solution of the Hamilton-Jacobi equation.