Appl Math Optim 38:109–120 (1998)
1998 Springer-Verlag New York Inc.
Adaptive Stabilization of Nonlinear Stochastic Systems
URA CNRS No. 399,
D´epartement de Math´ematiques, UFR MIM, Universit´e de Metz,
Ile du Saulcy, F 57045 Metz Cedex, France
Abstract. The purpose of this paper is to study the problem of asymptotic sta-
bilization in probability of nonlinear stochastic differential systems with unknown
parameters. With this aim, we introduce the concept of an adaptivecontrol Lyapunov
function for stochastic systems and we use the stochastic version of Artstein’s the-
orem to design an adaptive stabilizer. In this framework the problem of adaptive
stabilization of a nonlinear stochastic system is reduced to the problem of asymp-
totic stabilization in probability of a modiﬁed system. The design of an adaptive
control Lyapunov function is illustrated by the example of adaptively quadratically
stabilizable in probability stochastic differential systems.
Key Words. Stochastic differential equation, Asymptotic stability in probability,
Adaptive stabilization, Adaptive control Lyapunov function.
AMS Classiﬁcation. 60H10, 93C10, 93D05, 93D15, 93E15.
The aim of this paper is to study the problem of asymptotic feedback stabilization
in probability of nonlinear stochastic differential systems with an unknown constant
parameter in the drift. Since in general this problem is not solvable by means of static
feedback laws we introduce the concept of an adaptive control Lyapunovfunction and we
use the stochastic version of Artstein’s theorem established in  to design an adaptive
stabilizer. In our framework the problem of adaptive stabilization of nonlinear stochastic
differential systems is reduced to the problem of dynamic feedback stabilization, for all
values of the unknown parameter, of a modiﬁed system.
The concept of a control Lyapunov function for stochastic differential systems has
been introduced in  in order to prove a stochastic version of Artstein’s theorem