Lyapunov Stabilizability of Controlled Diffusions via a Superoptimality Principle for Viscosity Solutions

Abstract

We prove optimality principles for semicontinuous bounded viscosity solutions of Hamilton-Jacobi-Bellman equations. In particular we provide a representation 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 appropriate concept of Lyapunov function to study the stochastic open loop stabilizability in probability and the local and global asymptotic stabilizability (or asymptotic controllability). Finally we illustrate the theory with some examples.

Nonexistence of nonconstant solutions of some degenerate Bellman equations and applications to stochastic control

“…If the boundary is smooth, (19) is equivalent to (11) whereas (20) is slightly stronger than the first condition in (10), so, combined, they imply condition (10) (see Remark 2). The normal cone N (z) and the set of projections…”

“…Proposition 3.4. Let (8), (9), (19), (20) hold. Then, for any M ≥ 0, there exists δ > 0 such that − log(d(x)) satisfies in viscosity sense…”

“…Note that (8), (9) assure that the operator F [u] is continuous. For a fixed x ∈ ∂Ω take α ∈ A and the neighborhood U such that (19) and (20) hold. For x ∈ Ω ∩ U consider (p, Y ) ∈ J 2,+ d(x).…”

Zubov’s method for controlled diffusions with state constraints

“…In this case, however, compactness assumption on the dynamics (considering weak solutions of the SDE) are necessary in order to guarantee the last passage to the limit (see [15]). The version of the super-optimality principle we state below avoids this kind of assumption by taking into account only continuous super-solutions.…”

Un théorème de représentation des solutions de viscosité d'une équation d'Hamilton–Jacobi–Bellman ergodique dégénérée sur le tore

“…,x k }, telles que inf V = inf W = 0 et que W soit solution de viscosité de − 1 2 tr σ x, u(x) σ T x, u(x) D 2 W (x) + g x, u(x) , DW (x) + V (x) 0. (7)Il suffit en effet d'appliquer le Théorème 18 de[9] et la remarque qui le précède. On peut prendre ici W := W (i) , u := u (i) et V (x) = L(x, u (i) (x)) si L(x, u (i) (x)) > 0pour tout x n'appartenant pas à {x 1 , .…”