7 research outputs found
Almost Global Stochastic Stability
We develop a method to prove almost global stability of stochastic
differential equations in the sense that almost every initial point (with
respect to the Lebesgue measure) is asymptotically attracted to the origin with
unit probability. The method can be viewed as a dual to Lyapunov's second
method for stochastic differential equations and extends the deterministic
result in [A. Rantzer, Syst. Contr. Lett., 42 (2001), pp. 161--168]. The result
can also be used in certain cases to find stabilizing controllers for
stochastic nonlinear systems using convex optimization. The main technical tool
is the theory of stochastic flows of diffeomorphisms.Comment: Submitte
Feedback control of quantum state reduction
Feedback control of quantum mechanical systems must take into account the probabilistic nature of quantum measurement. We formulate quantum feedback control as a problem of stochastic nonlinear control by considering separately a quantum filtering problem and a state feedback control problem for the filter. We explore the use of stochastic Lyapunov techniques for the design of feedback controllers for quantum spin systems and demonstrate the possibility of stabilizing one outcome of a quantum measurement with unit probability
Lyapunov stabilizability of controlled diffusions via a superoptimality principle for viscosity solutions
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.Comment: 22 page
p
The problem of pth mean exponential stability and stabilizability of a class of stochastic nonlinear and bilinear hybrid systems with unstable and stable subsystems is considered. Sufficient conditions for the pth mean exponential stability and stabilizability under a feedback control and stabilizing switching rules are derived. A method for the construction of stabilizing switching rules based on the Lyapunov technique and the knowledge of regions of decreasing the Lyapunov functions for subsystems is given. Two cases, including single Lyapunov function and a a single Lyapunov-like function, are discussed. Obtained results are illustrated by examples