12,202 research outputs found
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
Estimation of Solutions of Differential Systems with Delayed Argument of Neutral Type
Tato disertační práce pojednává o řešení diferenciálních rovnic a systémů diferenciálních rovnic. Hlavní pozornost je věnována asymptotickým vlastnostem rovnic se zpožděním a systémů rovnic se zpožděním. V první kapitole jsou uvedeny fyzikální a technické příklady popsané pomocí diferenciálních rovnic se zpožděním a jejich systémů. Je uvedena klasifikace rovnic se zpožděním a jsou zformulovány základní pojmy stability s důrazem na druhou metodu Ljapunova. Ve druhé kapitole jsou studovány odhady řešení rovnic neutrálního typu. Třetí kapitola se zabývá systémy diferenciálních rovnic neutrálního typu. Jsou odvozeny asymptotické odhady pro řešení i pro derivace řešení. V závěru kapitoly jsou uvedeny příklady a srovnání výsledků s pracemi jiných autorů. Výpočty byly prováděny pomocí programu MATLAB. Poslední, čtvrtá kapitola, se zabývá asymptotickými vlastnostmi systémů se speciálním typem nelinearity, tzv. sektorové nelinearity. Jsou odvozeny vlastnosti řešení a derivace řešení. Základní metodou pro důkazy je v celé práci druhá Ljapunovova metoda a použití funkcionálů Ljapunova-Krasovského.This dissertation discusses the solutions to the differential equation and to systems of differential equations. The main attention is paid to study of asymptotical properties of equations with delay and systems of equations with delay. In the first chapter are given physical and technical examples described by differential equations with delay and their systems. The classification of equations with delay is given and basic notions of theory of stability are formulated (mainly with the emphasis on the Lyapunov second method). In the second chapter estimates of solutions of equations of neutral type are studied. The third chapter deals with systems of differential equations of neutral type. Asymptotic estimates for solutions and their derivatives are proved. At the end of the chapter examples and comparisons of our results and of other authors are given. The calculation were performed with the MATLAB software. Last, the fourth chapter deals with asymptotical properties of systems having a special type of nonlinearities, so called ``sector nonlinearities''. Properties and estimations of solutions and derivatives are derived. The basic tools used in the dissertation are the Lyapunov second method and functionals of Lyapunov-Krasovskii type.
Linearly Solvable Stochastic Control Lyapunov Functions
This paper presents a new method for synthesizing stochastic control Lyapunov
functions for a class of nonlinear stochastic control systems. The technique
relies on a transformation of the classical nonlinear Hamilton-Jacobi-Bellman
partial differential equation to a linear partial differential equation for a
class of problems with a particular constraint on the stochastic forcing. This
linear partial differential equation can then be relaxed to a linear
differential inclusion, allowing for relaxed solutions to be generated using
sum of squares programming. The resulting relaxed solutions are in fact
viscosity super/subsolutions, and by the maximum principle are pointwise upper
and lower bounds to the underlying value function, even for coarse polynomial
approximations. Furthermore, the pointwise upper bound is shown to be a
stochastic control Lyapunov function, yielding a method for generating
nonlinear controllers with pointwise bounded distance from the optimal cost
when using the optimal controller. These approximate solutions may be computed
with non-increasing error via a hierarchy of semidefinite optimization
problems. Finally, this paper develops a-priori bounds on trajectory
suboptimality when using these approximate value functions, as well as
demonstrates that these methods, and bounds, can be applied to a more general
class of nonlinear systems not obeying the constraint on stochastic forcing.
Simulated examples illustrate the methodology.Comment: Published in SIAM Journal of Control and Optimizatio
Non-Smooth Stochastic Lyapunov Functions With Weak Extension of Viscosity Solutions
This paper proposes a notion of viscosity weak supersolutions to build a
bridge between stochastic Lyapunov stability theory and viscosity solution
theory. Different from ordinary differential equations, stochastic differential
equations can have the origins being stable despite having no smooth stochastic
Lyapunov functions (SLFs). The feature naturally requires that the related
Lyapunov equations are illustrated via viscosity solution theory, which deals
with non-smooth solutions to partial differential equations. This paper claims
that stochastic Lyapunov stability theory needs a weak extension of viscosity
supersolutions, and the proposed viscosity weak supersolutions describe
non-smooth SLFs ensuring a large class of the origins being noisily
(asymptotically) stable and (asymptotically) stable in probability. The
contribution of the non-smooth SLFs are confirmed by a few examples;
especially, they ensure that all the linear-quadratic-Gaussian (LQG) controlled
systems have the origins being noisily asymptotically stable for any additive
noises
Local stability of Kolmogorov forward equations for finite state nonlinear Markov processes
The focus of this work is on local stability of a class of nonlinear ordinary
differential equations (ODE) that describe limits of empirical measures
associated with finite-state weakly interacting N-particle systems. Local
Lyapunov functions are identified for several classes of such ODE, including
those associated with systems with slow adaptation and Gibbs systems. Using
results from [5] and large deviations heuristics, a partial differential
equation (PDE) associated with the nonlinear ODE is introduced and it is shown
that positive definite subsolutions of this PDE serve as local Lyapunov
functions for the ODE. This PDE characterization is used to construct explicit
Lyapunov functions for a broad class of models called locally Gibbs systems.
This class of models is significantly larger than the family of Gibbs systems
and several examples of such systems are presented, including models with
nearest neighbor jumps and models with simultaneous jumps that arise in
applications.Comment: Updated to include Acknowledgement
- …