New problems in nonlinear control

This research will pursue some new directions in the area of control and differential games. Three main topics will be investigated: (i) Nash equilibrium solutions in feedback form, for non-cooperative differential games. These will be studied by looking at the system of Hamilton-Jacobi equations for the value functions, and using a homotopy approach to connect solutions of the differential game to the solution of a corresponding optimal control problem. (ii) The control of mechanical systems by means of active constraints. In particular, it is proposed to study the issues of global controllability, optimal control, and the asymptotic stabilization to an equilibrium position. (iii) A new mathematical model describing the spreading of a wild fire in terms of a differential inclusion. It is here assumed that fire propagation can be stopped by constructing barriers, in real time, along rectifiable curves in the plane. In this connection the existence and the properties of optimal strategies, which minimize the total area of the burned region, will be investigated.

The research on differential games will specifically focus on price-inventory games, providing a better understanding of what should the ?rational strategies? be in a competitive economic situation, where different groups of producers and consumers try to manipulate the market to their own advantage. The study of mechanical systems is primarily motivated by applications to swim-like motion in fluids. Maneuverability and optimal strokes for a swimmer that achieves locomotion by periodically changing its body shape will be investigated. Finally, the analysis of fire propagation models will provide indications on how to optimally manage fire-fighting resources in real time, in the presence of a forest fire advancing along a large front.