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An almost global notion of input-to-state stability



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Proceedings of the 41st IEEE Conference on Decision and Control Las Vegas Nevada USA December 2002 FrM09 1 An almost global notion of input to state stability David Angeli Dip di Sistemi e Informatica UniversitS di Firenze Via di S Marta 3 50139 FIRENZE Italy angeliCdsi unifi it A new definition of almost global Input tostate stability for systems on differentiable manifolds is discussed Some of its properties are investigated and specific examples are shown which can be t r e a t e d by means of dual Lyapunov techniques with continuous right hand side and we are willing to investigate stability of some attractor It is well known in fact t h a t the only possible global attractors 1 in euclidean space are equilibria Other times as in the case of the inverted pendulum although the system might be controllable obstructions arise to global stabilization by means of continuous state feedback Under such circumstances it is reasonable to relax the requirement from global to almost global asymptotic stability viz to allow in 2 for a zero measure set of initial conditions which are not a t t r a c t e d to A Abstract 1 Introduction and motivations One of the central topics in control theory has been since its very beginning the study of stability A fundamental chapter in the study of such notions is the quest for checkable conditions which may serve in order to estabilish stability of a system without knowing explicitly its solutions A key contribution in this respect dates back to the works of Lyapunov 3 Its second m e t h o d for stability analysis allows to check local and global stability properties on the basis of a test on the positivity of the derivative along systems trajectories of an energy like function Much later the Passivity theory and the Input to State stability paradigm see 2 9 extended to systems with inputs the massive body of results which followed those first major intuitions Dissipation inequalities and Lyapunov second m e t h o d have played a central



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