Lyapunov stabilizability of controlled diffusions via a superoptimality principle for viscosity solutions ∗
نویسنده
چکیده
We prove optimality principles for semicontinuous bounded viscosity solutions of HamiltonJacobi-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.
منابع مشابه
M ay 2 00 4 Lyapunov stabilizability of controlled diffusions via a superoptimality principle for viscosity solutions ∗
We prove optimality principles for continuous bounded nonnegative 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 representation formula is applied to extend the Lyapunov direct method for stability to controlled Ito stochastic differential equa...
متن کاملAlmost Sure Stabilizability of Controlled Degenerate Diffusions
We develop a direct Lyapunov method for the almost sure open-loop stabilizability and asymptotic stabilizability of controlled degenerate diffusion processes. The infinitesimal decrease condition for a Lyapunov function is a new form of Hamilton-Jacobi-Bellman partial differential inequality of 2nd order. We give local and global versions of the First and Second Lyapunov Theorems assuming the e...
متن کاملA converse Lyapunov theorem for almost sure stabilizability
We prove a converse Lyapunov theorem for almost sure stabilizability of controlled diffusions: given a stochastic system a.s. open loop stabilizable at the origin, we construct a lower semicontinuous positive definite function whose level sets form a local basis of viable neighborhoods of the equilibrium. This result provides, with the direct Lyapunov theorem proved in a companion paper, a comp...
متن کاملAlmost sure properties of controlled diffusions and worst case properties of deterministic systems
We compare a general controlled diffusion process with a deterministic system where a second controller drives the disturbance against the first controller. We show that the two models are equivalent with respect to two properties: the viability (or controlled invariance, or weak invariance) of closed smooth sets, and the existence of a smooth control Lyapunov function ensuring the stabilizabil...
متن کاملSub- and superoptimality principles and construction of almost optimal strategies for differential games in Hilbert spaces
We prove suband superoptimality principles of dynamic programming and show how to use the theory of viscosity solutions to construct almost optimal strategies for two-player, zero-sum differential games driven by abstract evolution equations in Hilbert spaces.
متن کامل