Appl Math Optim 53:1–29 (2006)
2005 Springer Science+Business Media, Inc.
Lyapunov Stabilizability of Controlled Diffusions via a
Superoptimality Principle for Viscosity Solutions
Dipartimento di Matematica P. e A., Universit`adiPadova,
via Belzoni 7, 35131 Padova, Italy
Abstract. We prove optimality principles for semicontinuous bounded viscosity
solutions of Hamilton–Jacobi–Bellman equations. In particular, we provide a repre-
sentation formula for viscosity supersolutions as value functions of suitable obstacle
control problems. This result is applied to extend the Lyapunov direct method for
stability to controlled Ito stochastic differential equations. We deﬁne the appropri-
ate concept of the Lyapunov function to study stochastic open loop stabilizability in
probability and local and global asymptotic stabilizability (or asymptotic controlla-
bility). Finally, we illustrate the theory with some examples.
Key Words. Controlled degenerate diffusion, Hamilton–Jacobi–Bellman inequal-
ities, Viscosity solutions, Dynamic programming, Superoptimality principles, Ob-
stacle problem, Stochastic control, Stability in probability, Asymptotic stability.
AMS Classiﬁcation. 49L25, 93E15, 93D05, 93D20.
We consider an N -dimensional stochastic differential equation
= f (X
) dt + σ(X
is a standard M-dimensional Brownian motion. Since the sixties, a stochas-
tic Lyapunov method for the analysis of the qualitative properties of the solutions of
stochastic differential equations, in analogy to the deterministic Lyapunov method, was
This research was partially supported by the Young-Researcher-Project CPDG034579 ﬁnanced by the
University of Padova.