On the problem of control synthesis: the Pontryagin alternating integral and the Hamilton-Jacobi equation

This paper deals with the problem of control synthesis under unknown, but bounded disturbances for a system with linear structure and hard (geometric) bounds on the control and the disturbance inputs. It emphasizes the role of set-valued methods and, in particular, of the Pontryagin multivalued alternating integral in the corresponding solution schemes. Close ties with the Hamilton-Jacobi techniques are discussed.This paper also discusses an approach producing effective numerical solutions on the basis of appropriate ellipsoidal techniques. It presents a framework for going over from the abstract theory to numerically realizable ellipsoidal representations.

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