an example of a gas system which can be destabilized by an integrable perturbation a construction of a globally asymptotically stable time invariant system which can be destabilized by some integrable perturbation is given besides its intrinsic interest this serves to provide counterexamples to an open question regarding lyapunov functions

weighted lipschitz estimates for on convex domains of finite type abstract in this paper we introduce the scaled weighted growth spaces and the scaled weighted lipschitz spaces we prove the norm estimates for and the boundedness of the bergman projection in these scaled weighted function spaces on a convex domain of finite type in c n

integral input to state stable systems in cascade the integral input to state stability iiss property is studied in the context of nonlinear time invariant systems in cascade some sufficient conditions for the preservation of the iiss property under a cascade interconnection are presented these are first given as growth restrictions on the supply functions of the storage function associated with each subsystem and are then expressed as solutions based requirements a lyapunov based condition guaranteeing that the cascade composed of an iiss system driven by a globally asymptotically stable gas one remains gas is also provided we also show that some of these results extend to cascades composed of more than two subsystems

combining iiss and iss with respect to small inputs the strong iiss property this paper studies the notion of strong iiss which is defined as the combination of input to state stability iss with respect to small inputs and integral input to state stability iiss this notion char acterizes the robustness property that the state remains bounded as long as the magnitude of exogenous inputs is reasonably small but may diverge for stronger disturbances we provide several lyapunov based sufficient conditions for strong iiss one of them relies on iiss lyapunov functions admitting a radially non vanishing class k dissipation rate although such dissipation inequality appears natural in view of the existing lyapunov characterization of iiss and iss we show through a counter example that it is not a necessary condition for strong iiss less conservative conditions are then provided as well as lyapunov tools to estimate the tolerated input magnitude that preserves solutions boundedness

relaxed conditions for the stability of switched nonlinear triangular systems under arbitrary switching this paper addresses the stability of a class of nonlinear switching systems under arbitrary switching it focuses on switching systems whose modes are made of a cascade of nonlinear scalar systems it relaxes the stability conditions proposed by angeli and liberzon by relying on the strong iiss property which is known to be preserved under cascade interconnection its applicability is illustrated by the study of switched bilinear systems in triangular form

suboptimal decentralized control over noisy communication channels a b s t r a c t in this paper we present a technique for the design of decentralized controllers for mean square stability of a large scale system with cascaded clusters of subsystems each subsystem is linear and time invariant and both system and measurement are subject to gaussian noise for stability analysis of this system we consider the effects of additive white gaussian noise awgn channels used for exchanging information between subsystems

a unifying integral iss framework for stability of nonlinear cascades we analyze nonlinear cascades in which the driven subsystem is integral input to state stable iss and we characterize the admissible integral iss gains for stability this characterization makes use of the convergence speed of the driving subsystem and allows a larger class of gain functions when the convergence is faster we show that our integral iss gain characterization unifies different approaches in the literature which restrict the nonlinear growth of the driven subsystem and the convergence speed of the driving subsystem the result is used to develop a new observer based backstepping design in which the growth of the nonlinear damping terms is reduced

a characterization of integral input to state stability the notion of input to state stability iss is now recognized as a central concept in nonlinear systems analysis it provides a nonlinear generalization of finite gains with respect to supremum norms and also of finite l sup 2 gains it plays a central role in recursive design coprime factorizations controllers for nonminimum phase systems and many other areas in this paper a newer notion that of integral input to state stability iiss is studied the notion of iiss generalizes the concept of finite gain when using an integral norm on inputs but supremum norms of states in that sense generalizing the linear h sup 2 theory it allows one to quantify sensitivity even in the presence of certain forms of nonlinear resonance we obtain several necessary and sufficient characterizations of the iiss property expressed in terms of dissipation inequalities and other alternative and nontrivial characterizations

a lyapunov approach to cascade interconnection of integral input to state stable systems this technical note deals with the stability of cascade interconnection of integral input to state iiss time varying systems a new technique to construct smooth lyapunov functions of cascaded systems is proposed from the construction sufficient conditions for internal stability and stability with respect to external signals are derived one of the derived conditions is a trade off between slower convergence of the driving system and steeper input growth of the driven system the trade off is no more necessary if the speed of convergence of the driven system is not radially vanishing the results are related to trajectory based approaches and small gain techniques for feedback interconnection the difference between the feedback case and the cascade case is viewed from the requirement on convergence speed of autonomous parts

examples of ges systems that can be driven to infinity by arbitrarily small additive decaying exponentials examples are given of nonlinear time invariant systems in continuous time discrete time and of hybrid type that have linear sector growth the origin globally exponentially stable and that can be driven to infinity by arbitrarily small additive decaying exponentials resulting observations about additive cascades and lyapunov functions are discussed these observations extend those derived from an example which recently appeared in the literature of a globally asymptotically stable continuous time system destabilized by a piecewise constant integrable disturbance

comparison of cobordism theories abstract relying on results of hopkins morel we show that for x a quasi projective variety over a field of characteristic zero the canonical map n x mgl 2 n n x is an isomorphism

on global uniform asymptotic stability of nonlinear time varying systems in cascade abstract in this short paper we deal with the stability analysis problem of nonautonomous nonlinear systems in cascade in particular we give sufficient conditions to guarantee that i a globally uniformly stable gus nonlinear time varying nltv system remains gus when it is perturbed by the output of a globally uniformly asymptotically stable guas nltv system under the assumption that the perturbing signal is absolutely integrable ii if in addition the perturbed system is guas it remains guas under the cascaded interconnection iii two guas systems yield a guas cascaded system under some growth restrictions over the lyapunov function our proofs rely on the second method of lyapunov roughly speaking on a e stability analysis

brief growth rate conditions for uniform asymptotic stability of cascaded time varying systems in the last decade several results on cascaded autonomous systems have appeared in the literature the sufficient conditions for the stability and stabilizability of these systems are often related to certain growth rate conditions on the functions which define the dynamics of the system in this paper we analyze three complementary classes of nonlinear time varying systems according to these growth rates for each case we give further results to guarantee uniform global asymptotic stability of the cascade and relate our contributions to input to state stability and other growth rate conditions previously reported the advantage of this type of analysis is that it is often easier to verify such conditions than to find a lyapunov function with a negative definite derivative

strong iiss is preserved under cascade interconnection in a recent paper we have introduced the notion of strong iiss as a compromise between the strength of input to state stability iss and the generality of integral iss iiss in this note we continue the investigations around this property by studying its behavior in an interconnection context in particular we show that the cascade of strongly iiss systems is itself strongly iiss and we recall some useful tools to study strongly iiss systems in feedback interconnection

global stabilization of nonlinear cascade systems abstract we present sufficient conditions for the global stabilizability of two cascade connected nonlinear systems these are based on general results concerning global asymptotic stability of triangular systems which are proved in the last section for polynomial systems in particular the stabilizing feedback is given explicitly

connection between cooperative positive systems and integral input to state stability of large scale systems we consider a class of continuous time cooperative systems evolving on the positive orthant r n we show that if the origin is globally attractive then it is also globally stable and furthermore there exists an unbounded invariant manifold where trajectories strictly decay this leads to a small gain type condition which is sufficient for global asymptotic stability gas of the origin we establish the following connection to large scale interconnections of integral input to state stable iss subsystems if the cooperative system is integral iss and arises as a comparison system associated with a large scale interconnection of i iss systems then the composite nominal system is also i iss we provide a criterion in terms of a lyapunov function for integral input to state stability of the comparison system furthermore we show that if a small gain condition holds then the classes of systems participating in the large scale interconnection are restricted in the sense that certain iiss systems cannot occur moreover this small gain condition is essentially the same as the one obtained previously by dashkovskiy ruffer and wirth 2007 in press for large scale interconnections of iss systems

separation principles for input output and integral input to state stability we present new characterizations of input output to state stability this is a notion of detectability formulated in the iss input to state stability framework equivalent properties are presented in terms of asymptotic estimates of the state trajectories based on the magnitudes of the external input and output signals these results provide a set of separation principles for input output to state stability characterizations of the property as conjunctions of weaker stability notions when applied to the notion of integral iss these characterizations yield analogous results

stability of stochastic nonlinear systems in cascade with not necessarily unbounded decay rates this paper develops tools to verify stability and robustness of cascaded nonlinear stochastic systems based on lyapunov functions constituent systems are formulated in terms of integral input to state stability iiss and input to state stability iss which are popular notions for both stochastic and deterministic systems this paper highlights differences between the stochastic and the deterministic cases in contrast to deterministic systems it is demonstrated that assuming iss systems having unbounded decay rates in dissipation inequalities is restrictive taking this fact into account stability criteria are formulated without assuming unboundedness of decay rates so that iss systems with bounded decay rates and iiss systems which are not iss are covered in a unified manner stability criteria for stochastic cascades involve the growth rate conditions at connecting channels this paper clarifies how noise diffusion fields affect the growth rate conditions and the influence depends on definition of stochastic robustness

nonsmooth stabilization of a class of nonlinear cascaded systems a global nonsmooth stabilization scheme is presented for a class of nonlinear cascaded systems with uncontrollable linearization a stepwise constructive control methodology is proposed for the driving subsystem by using the adding a power integrator technique under suitable conditions and based on homogeneous properties it is proved that the stabilization of the driving subsystem implies the stabilization of the overall cascaded system due to the versatility of the adding a power integrator technique and homogeneous properties the proposed controller not only can be used to stabilize the cascaded system asymptotically but also is able to lead to an interesting result of finite time stabilization under appropriate conditions

relationships between subclasses of integral input to state stability a certain qualitative equivalence has been recently demonstrated between integral input to state stability iiss and a nonlinear mathcal l 2 gain property furthermore it has been observed that the properties of strong iiss and nonlinear mathcal l 2 gain are preserved when multiple systems satisfying the respective property are connected in cascade this technical note clarifies the relationships between various input to state stability and non linear mathcal l p gain properties