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

Robust and Resilient State Dependent Control of Discrete-Time Nonlinear Systems with General Performance Criteria

A novel state dependent control approach for discrete-time nonlinear systems with general performance criteria is presented. This controller is robust for unstructured model uncertainties, resilient against bounded feedback control gain perturbations in achieving optimality for general performance criteria to secure quadratic optimality with inherent asymptotic stability property together with quadratic dissipative type of disturbance reduction. For the system model, unstructured uncertainty description is assumed, which incorporates commonly used types of uncertainties, such as norm-bounded and positive real uncertainties as special cases. By solving a state dependent linear matrix inequality at each time step, sufficient condition for the control solution can be found which satisfies the general performance criteria. The results of this paper unify existing results on nonlinear quadratic regulator, H∞ and positive real control to provide a novel robust control design. The effectiveness of the proposed technique is demonstrated by simulation of the control of inverted pendulum

Large time behavior for some nonlinear degenerate parabolic equations

We study the asymptotic behavior of Lipschitz continuous solutions of nonlinear degenerate parabolic equations in the periodic setting. Our results apply to a large class of Hamilton-Jacobi-Bellman equations. Defining S as the set where the diffusion vanishes, i.e., where the equation is totally degenerate, we obtain the convergence when the equation is uniformly parabolic outside S and, on S, the Hamiltonian is either strictly convex or satisfies an assumption similar of the one introduced by Barles-Souganidis (2000) for first-order Hamilton-Jacobi equations. This latter assumption allows to deal with equations with nonconvex Hamiltonians. We can also release the uniform parabolic requirement outside S. As a consequence, we prove the convergence of some everywhere degenerate second-order equations

Robust and Resilient State-dependent Control of Continuous-time Nonlinear Systems with General Performance Criteria

A novel state-dependent control approach for continuous-time nonlinear systems with general performance criteria is presented in this paper. This controller is optimally robust for model uncertainties and resilient against control feedback gain perturbations in achieving general performance criteria to secure quadratic optimality with inherent asymptotic stability property together with quadratic dissipative type of disturbance reduction. For the system model, unstructured uncertainty description is assumed, which incorporates commonly used types of uncertainties, such as norm-bounded and positive real uncertainties as special cases. By solving a state-dependent linear matrix inequality at each time, sufficient condition for the control solution can be found which satisfies the general performance criteria. The results of this paper unify existing results on nonlinear quadratic regulator, H∞ and positive real control. The efficacy of the proposed technique is demonstrated by numerical simulations of the nonlinear control of the inverted pendulum on a cart system

On the Large Time Behavior of Solutions of Hamilton-Jacobi Equations Associated with Nonlinear Boundary Conditions

In this article, we study the large time behavior of solutions of first-order Hamilton-Jacobi Equations, set in a bounded domain with nonlinear Neumann boundary conditions, including the case of dynamical boundary conditions. We establish general convergence results for viscosity solutions of these Cauchy-Neumann problems by using two fairly different methods : the first one relies only on partial differential equations methods, which provides results even when the Hamiltonians are not convex, and the second one is an optimal control/dynamical system approach, named the "weak KAM approach" which requires the convexity of Hamiltonians and gives formulas for asymptotic solutions based on Aubry-Mather sets

Uniqueness Results for Second Order Bellman-Isaacs Equations under Quadratic Growth Assumptions and Applications

In this paper, we prove a comparison result between semicontinuous viscosity sub and supersolutions growing at most quadratically of second-order degenerate parabolic Hamilton-Jacobi-Bellman and Isaacs equations. As an application, we characterize the value function of a finite horizon stochastic control problem with unbounded controls as the unique viscosity solution of the corresponding dynamic programming equation

Aubry sets for weakly coupled systems of Hamilton--Jacobi equations

We introduce a notion of Aubry set for weakly coupled systems of Hamilton--Jacobi equations on the torus and characterize it as the region where the obstruction to the existence of globally strict critical subsolutions concentrates. As in the case of a single equation, we prove the existence of critical subsolutions which are strict and smooth outside the Aubry set. This allows us to derive in a simple way a comparison result among critical sub and supersolutions with respect to their boundary data on the Aubry set, showing in particular that the latter is a uniqueness set for the critical system. We also highlight some rigidity phenomena taking place on the Aubry set.Comment: 35 pages v.2 the introduction has been rewritten and shortened. Some proofs simplified. Corrections and references added. Corollary 5.3 added stating antisymmetry of the Ma\~n\'e matrix on points of the Aubry set. Section 6 contains a new example

user's guide to viscosity solutions of second order partial differential equations

The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions.Comment: 67 page