Appl Math Optim 49:1–26 (2004)
2003 Springer-Verlag New York Inc.
Viscosity Solutions of HJB Equations with Unbounded Data
and Characteristic Points
Dipartimento di Matematica Pura e Applicata, Universit`adiPadova,
via Belzoni 7, 35131 Padova, Italy
Communicated by M. Nisio
Abstract. We study a class of inﬁnite horizon and exit-time control problems for
nonlinear systems with unbounded data using the dynamic programming approach.
We prove local optimality principles for viscosity super- and subsolutions of degen-
erate Hamilton–Jacobi equations in a very general setting. We apply these results
to characterize the (possibly multiple) discontinuous solutions of Dirichlet and free
boundary value problems as suitable value functions for the above-mentioned con-
Key Words. Viscosity solutions, Uniqueness and characterization, Unbounded
control problems, Impulsive controls.
AMS Classiﬁcation. 49L25, 49L20, 34A37.
We consider a nonlinear control system
˙y(t) = f (y(t), α(t)), y(0) = x, (1.1)
This research was partially supported by EEC-TMR network project “Viscosity solutions and their
applications” and by MURST-Coﬁn project “Analisi e controllo di equazioni di evoluzione deterministiche e