Title:
Differential Inclusions in Optimal Control of Deterministic and Stochastic Systems
Abstract:
Questions of existence of optimal solutions in optimal control theory and of solutions to ODEs with discontinuous right-hand side have naturally led to reformulations of control systems in the form of differential inclusions. For instance, it is well known that uniform limits of sequences of controlled trajectories are solutions to a relaxed differential inclusion. To make a link with the original system, the notion of relaxed controls was then introduced. Further, relaxation theorems in the deterministic finite and infinite dimensional settings have allowed replacing the original optimal control problem by the relaxed one, while preserving the same cost-to-go function.
Differential inclusions involve in their right hand side set-valued maps that, in turn, can be linearized by using set-valued derivatives. Once relaxed, the resulting variational inclusions have a larger family of solutions than the classical linearizations of control systems. The same approach extends to second order variational inclusions, useful in second order analysis of control systems. Variational inclusions are crucial for deriving necessary optimality conditions.
In this talk, for a deterministic optimal control problem, I will illustrate how second order optimality conditions can be deduced from properties of second order set-valued variations and how the stochastic maximum principle can be obtained by using set-valued derivatives.