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42
Hybrid systems: Generalized solutions and robust stability
 In IFAC Symposium on Nonliear Control Systems
, 2004
"... Abstract: Robust asymptotic stability for hybrid systems is considered. For this purpose, a generalized solution concept is developed. The first step is to characterize a hybrid time domain that permits an efficient description of the convergence of a sequence of solutions. Graph convergence is used ..."
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Cited by 46 (13 self)
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Abstract: Robust asymptotic stability for hybrid systems is considered. For this purpose, a generalized solution concept is developed. The first step is to characterize a hybrid time domain that permits an efficient description of the convergence of a sequence of solutions. Graph convergence is used. Then a generalized solution definition is given that leads to continuity with respect to initial conditions and perturbations of the system data. This property enables new results on necessary conditions for asymptotic stability in hybrid systems.
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.
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.
Flow stability of patchy vector fields and robust feedback stabilization
 SIAM J. Control Optim
, 2002
"... Abstract. The paper is concerned with patchy vector fields, a class of discontinuous, piecewise smooth vector fields that were introduced by the authors to study feedback stabilization problems. We prove the stability of the corresponding solution set w.r.t. a wide class of impulsive perturbations. ..."
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Cited by 18 (3 self)
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Abstract. The paper is concerned with patchy vector fields, a class of discontinuous, piecewise smooth vector fields that were introduced by the authors to study feedback stabilization problems. We prove the stability of the corresponding solution set w.r.t. a wide class of impulsive perturbations. These results yield the robustness of patchy feedback controls in the presence of measurement errors and external disturbances.
Singularities of stabilizing feedbacks
, 1998
"... This paper is concerned with the stabilization problem for a control system of the form (1) ˙x = f (x, u), u ∈ K, assuming that the set of control values K ⊂ m is compact and that the map f: n × m ↦ → n ..."
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Cited by 11 (3 self)
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This paper is concerned with the stabilization problem for a control system of the form (1) ˙x = f (x, u), u ∈ K, assuming that the set of control values K ⊂ m is compact and that the map f: n × m ↦ → n
Global stabilization for systems evolving on manifolds
 Journal of Dynamical and Control Systems
, 2006
"... We show that any globally asymptotically controllable system on any smooth manifold can be globally stabilized by a state feedback. Since we allow discontinuous feedbacks, we interpret the solutions of our systems in the “sample and hold ” sense introduced by ClarkeLedyaevSontagSubbotin (CLSS). O ..."
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Cited by 10 (0 self)
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We show that any globally asymptotically controllable system on any smooth manifold can be globally stabilized by a state feedback. Since we allow discontinuous feedbacks, we interpret the solutions of our systems in the “sample and hold ” sense introduced by ClarkeLedyaevSontagSubbotin (CLSS). Our work generalizes the CLSS Theorem which is the special case of our result for systems on Euclidean space. We apply our result to the inputtostate stabilization of systems on manifolds relative to actuator errors, under small observation noise. Key Words: Asymptotic controllability, control systems on manifolds, inputtostate stabilization 1
Quantum control design by Lyapunov trajectory tracking for dipole and polarizability coupling
 New. J. Phys
"... Abstract. We analyse in this paper the Lyapunov trajectory tracking of the Schrödinger equation for a coupling control operator containing both a linear (dipole) and a quadratic (polarizability) term. We show numerically that the contribution of the quadratic part cannot be exploited by standard tr ..."
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Cited by 10 (1 self)
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Abstract. We analyse in this paper the Lyapunov trajectory tracking of the Schrödinger equation for a coupling control operator containing both a linear (dipole) and a quadratic (polarizability) term. We show numerically that the contribution of the quadratic part cannot be exploited by standard trajectory tracking tools and propose two improvements: discontinuous feedback and periodic (timedependent) feedback. For both cases we present theoretical results and support them by numerical illustrations.
Singularities of viscosity solutions and the stabilization problem in the plane
 Indiana Univ. Math. J
, 2003
"... ABSTRACT. We study the general problem of globally asymptotically controllable affine systems in the plane. As preliminaries we present some results of independent interest. We study the regularity of some sets related to semiconcave viscosity supersolutions of HamiltonJacobiBellman equations. T ..."
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Cited by 8 (3 self)
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ABSTRACT. We study the general problem of globally asymptotically controllable affine systems in the plane. As preliminaries we present some results of independent interest. We study the regularity of some sets related to semiconcave viscosity supersolutions of HamiltonJacobiBellman equations. Then we deduce a construction of stabilizing feedbacks in the plane. 1.
State Constrained Feedback Stabilization
 SIAM J. Control Optim
, 2003
"... Abstract A standard finite dimensional nonlinear control system is considered, along with a state constraint set S and a target set Σ. It is proven that open loop Sconstrained controllability to Σ implies closed loop Sconstrained controllability to the closed δneighborhood of Σ, for any specified ..."
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Cited by 8 (3 self)
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Abstract A standard finite dimensional nonlinear control system is considered, along with a state constraint set S and a target set Σ. It is proven that open loop Sconstrained controllability to Σ implies closed loop Sconstrained controllability to the closed δneighborhood of Σ, for any specified δ > 0. When the target set Σ satisfies a small time Sconstrained controllability condition, conclusions on closed loop Sconstrained stabilizability ensue. The (necessarily discontinuous) feedback laws in question are implemented in the sampleandhold sense and possess a robustness property with respect to state measurement errors. The feedback constructions involve the quadratic infimal convolution of a control Lyapunov function with respect to a certain modification of the original dynamics. The modified dynamics in effect provide for constraint removal, while the convolution operation provides a useful semiconcavity property.