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.