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Asymptotic Controllability Implies Feedback Stabilization
 IEEE Trans. Autom. Control
, 1999
"... It is shown that every asymptotically controllable system can be globally stabilized by means of some (discontinuous) feedback law. The stabilizing strategy is based on pointwise optimization of a smoothed version of a controlLyapunov function, iteratively sending trajectories into smaller and smal ..."
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Cited by 118 (12 self)
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It is shown that every asymptotically controllable system can be globally stabilized by means of some (discontinuous) feedback law. The stabilizing strategy is based on pointwise optimization of a smoothed version of a controlLyapunov function, iteratively sending trajectories into smaller and smaller neighborgoods of a desired equilibrium. A major technical problem, and one of contributions of the present paper, concerns the precise meaning of "solution" when using a discontinuous controller. I. Introduction A longstanding open question in nonlinear control theory concerns the relationship between asymptotic controllability to the origin in R n of a nonlinear system x = f(x; u) (1) by an "open loop" control u : [0; +1) ! U and the existence of a feedback control k : R n ! U which stabilizes trajectories of the system x = f(x; k(x)) (2) with respect to the origin. For the special case of linear control systems x = Ax + Bu, this relationship is well understood: asymptotic cont...
Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions
 ESAIM: CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS
, 1999
"... We study stability and stabilizability properties of systems with discontinuous right handside (with solutions intended in Filippov’s sense) by means of locally Lipschitz continuous and regular Lyapunov functions. The stability result is obtained in the more general context of differential inclusio ..."
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Cited by 68 (9 self)
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We study stability and stabilizability properties of systems with discontinuous right handside (with solutions intended in Filippov’s sense) by means of locally Lipschitz continuous and regular Lyapunov functions. The stability result is obtained in the more general context of differential inclusions. Concerning stabilizability, we focus on systems affine with respect to the input: we give some sufficient conditions for a system to be stabilized by means of a feedback law of the JurdjevicQuinn type.
Stability and stabilization: discontinuities and the effect of disturbances
, 1998
"... This expository paper deals with several questions related to stability and stabilization of nonlinear finitedimensional continuoustime systems. The topics covered include a review of stability and asymptotic controllability, an introduction to the problem of stabilization and obstructions to cont ..."
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Cited by 67 (13 self)
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This expository paper deals with several questions related to stability and stabilization of nonlinear finitedimensional continuoustime systems. The topics covered include a review of stability and asymptotic controllability, an introduction to the problem of stabilization and obstructions to continuous stabilization, the notion of controlLyapunov functions, and a discussion of discontinuous feedback and methods of nonsmooth analysis.An emphasis is placed upon relatively new areas of research which concern stability with respect to noise, including the notion of insensitivity to small measurement and actuator errors as well as the more global notion of inputtostate stability.
A Lyapunov Characterization of Robust Stabilization
 Nonlinear Analysis
, 1997
"... 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 system ..."
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Cited by 37 (4 self)
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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 sma...
A Globalization Procedure for Locally Stabilizing Controllers
, 2000
"... For a nonlinear system with a singular point that is locally asymptotically nullcontrollable we present a class of feedbacks that globally asymptotically stabilizes the system on the domain of asymptotic nullcontrollability. The design procedure is twofold. In a neighborhood of the singular point ..."
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Cited by 36 (0 self)
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For a nonlinear system with a singular point that is locally asymptotically nullcontrollable we present a class of feedbacks that globally asymptotically stabilizes the system on the domain of asymptotic nullcontrollability. The design procedure is twofold. In a neighborhood of the singular point we use linearization arguments to construct a sampled (or discrete) feedback that yields a feedback invariant neighborhood of the singular point and locally exponentially stabilizes without the need for vanishing sampling rate as the trajectory approaches the equilibrium. On the remainder of the domain of controllability we construct a piecewise constant patchy feedback that guarantees that all Carathéodory solutions of the closed loop system reach the previously constructed neighborhood.
Existence of Lipschitz and semiconcave controlLyapunov functions
"... Given a locally Lipschitz control system which is globally asymptotically controllable to the origin, we construct a controlLyapunov function for the system which is Lipschitz on bounded sets and we deduce the existence of another one which is semiconcave (and so locally Lipschitz) outside the o ..."
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Cited by 29 (7 self)
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Given a locally Lipschitz control system which is globally asymptotically controllable to the origin, we construct a controlLyapunov function for the system which is Lipschitz on bounded sets and we deduce the existence of another one which is semiconcave (and so locally Lipschitz) outside the origin. The proof relies on value functions and nonsmooth calculus.
Clocks and Insensitivity to Small Measurement Errors
, 1999
"... This paper deals with the problem of stabilizing a system in the presence of small measurement errors. It is known that, for general stabilizable systems, there may be no possible memoryless state feedback which is robust with respect to such errors. In contrast, a precise result is given here, sho ..."
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Cited by 28 (1 self)
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This paper deals with the problem of stabilizing a system in the presence of small measurement errors. It is known that, for general stabilizable systems, there may be no possible memoryless state feedback which is robust with respect to such errors. In contrast, a precise result is given here, showing that, if a (continuoustime, finitedimensional) system is stabilizable in any way whatsoever (even by means of a dynamic, time varying, discontinuous, feedback) then it can also be semiglobally and practically stabilized in a way which is insensitive to small measurement errors, by means of a hybrid strategy based on the idea of sampling at a "slow enough" rate.
SEMICONCAVE CONTROLLYAPUNOV FUNCTIONS AND STABILIZING FEEDBACKS
, 2002
"... We study the general problem of stabilization of globally asymptotically controllable systems. We construct discontinuous feedback laws, and particularly we make it possible to choose these continuous outside a small set (closed with measure zero) of discontinuity in the case of control systems wh ..."
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Cited by 26 (7 self)
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We study the general problem of stabilization of globally asymptotically controllable systems. We construct discontinuous feedback laws, and particularly we make it possible to choose these continuous outside a small set (closed with measure zero) of discontinuity in the case of control systems which are affine in the control; moreover this set of singularities is shown to be repulsive for the Carathéodory solutions of the closedloop system under an additional assumption.
Feedback stabilization and Lyapunov functions
"... Given a locally defined, nondifferentiable but Lipschitz Lyapunov function, we construct a (discontinuous) feedback law which stabilizes the underlying system to any given tolerance. A further result shows that suitable Lyapunov functions of this type exist under mild assumptions. We also establish ..."
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Cited by 25 (7 self)
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Given a locally defined, nondifferentiable but Lipschitz Lyapunov function, we construct a (discontinuous) feedback law which stabilizes the underlying system to any given tolerance. A further result shows that suitable Lyapunov functions of this type exist under mild assumptions. We also establish a robustness property of the feedback relative to measurement error commensurate with the sampling rate of the control implementation scheme. 0Key words:Asymptotic stabilizability, discontinuous feedback law, system sampling, locally Lipschitz Lyapunov function, nonsmooth analysis, robustness
Feedback stabilization and Lyapunov functions
 SIAM J. CONTROL OPTIM
"... 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 assump ..."
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Cited by 20 (2 self)
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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.