Yuri S. Ledyaev

A Lyapunov characterization of robust stabilization

One of the fundamental facts in control theory (Artstein’s theorem) is the equivalence, for systems affine in controls, between continuous feedback stabilizability to an equilibrium and the existence of smooth control Lyapunov functions. This equivalence breaks down for general nonlinear systems, not affine in controls. One of the main results in this paper establishes that the existence of smooth Lyapunov functions implies the existence of, in general discontinuous, feedback stabilizers which are insensitive (or robust) to small errors in state measurements. Conversely, it is shown that the existence of such stabilizers in turn implies the existence of smooth control Lyapunov functions. Moreover, it is established that, for general nonlinear control systems under persistently acting disturbances, the existence of smooth Lyapunov functions is equivalent to the existence of (in general, discontinuous) feedback stabilizers which are robust with respect to small measurement errors and small additive external disturbances. ∗Supported in part by Russian Fund for Fundamental Research Grant 96-01-00219 and by the Rutgers Center for Systems and Control (SYCON). Work done while visiting Rutgers University, Mathematics Department. On leave from Steklov Institute of Mathematics, Moscow 117966, Russia †Supported in part by US Air Force Grant AFOSR-94-0293

Feedback Stabilization and Lyapunov Functions

Given a locally defined, nondifferentiable but Lipschitz Lyapunov function, we employ it in order to construct a (discontinuous) feedback law which stabilizes the underlying system to any given tolerance. A converse result shows that suitable Lyapunov functions of this type exist under mild assumptions. We also establish that the feedback in question possesses a robustness property relative to measurement error, despite the fact that it may not be continuous.

Implicit Multifunction Theorems

We prove a general implicit function theorem for multifunctions with a metric estimate on the implicit multifunction and a characterization of its coderivative. Traditional open covering theorems, stability results, and sufficient conditions for a multifunction to be metrically regular or pseudo-Lipschitzian can be deduced from this implicit function theorem. We prove this implicit multifunction theorem by reducing it to an implicit function/solvability theorem for functions. This approach can also be used to prove the Robinson–Ursescu open mapping theorem. As a tool for this alternative proof of the Robinson–Ursescu theorem, we also establish a refined version of the multidirectional mean value inequality which is of independent interest.

On Global Optimality Conditions for Nonlinear Optimal Control Problems

AbstractLet a trajectory and control pair

Stabilization under measurement noise: Lyapunov characterization

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Robustness of discontinuous feedback in control under disturbance

maximize globally the functional g(x(T)) in the basic optimal control problem. Then (evidently) any pair (x,u) from the level set of the functional g corresponding to the value g(

Asymptotic controllability implies feedback stabilization

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