This paper addresses the problem of establishing stability of nonlinear interconnected systems. This paper introduces a mathematical formulation of the state-dependent scaling problems whose solutions directly provide Lyapunov functions proving stability properties of interconnected dissipative systems in a unified manner. Stability criteria are interpreted as sufficient conditions for the existence of solutions to the state-dependent scaling problems. Computing solutions to the problems is straightforward for systems covered by classical stability criteria. It, however, could be too difficult for systems with strong nonlinearity. The main purpose of this paper is to demonstrate the effectiveness beyond formal applicability by focusing on interconnected integral input-to-state stable (iISS) systems and input-to-state stable (ISS) systems. This paper derives small-gain-type theorems for interconnected systems involving iISS systems from the state-dependent scaling formulation. This paper provides solutions and Lyapunov functions explicitly. The new framework seamlessly generalizes the ISS smallgain theorem and classical stability criteria such as the smallgain theorem, the passivity theorems, the circle, and Popov criteria. State-dependence of the scaling is crucial for effective treatment of essential nonlinearities, while constants are sufficient for classical nonlinearities.Index Terms-Dissipation, input-to-state stability, integral input-to-state stability, Lyapunov function, nonlinear interconnected system, small-gain condition.

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International audienceThis 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

Abstract. This paper is devoted to two issues. One is to provide Lyapunov-based tools to establish integral input-to-state stability (iISS) and input-to-state stability (ISS) for some classes of nonlinear parabolic equations. The other is to provide a stability criterion for interconnections of iISS parabolic systems. The results addressing the former problem allow us to overcome obstacles arising in tackling the latter one. The results for the latter problem are a small-gain condition and a formula of Lyapunov functions which can be constructed for interconnections whenever the smallgain condition holds. It is demonstrated that for interconnections of partial differential equations, the choice of a right state and input spaces is crucial, in particular for iISS subsystems which are not ISS. As illustrative examples, stability of two highly nonlinear reaction-diffusion systems is established by the the proposed small-gain criterion.Key words. nonlinear control systems, infinite-dimensional systems, integral input-to-state stability, Lyapunov methods AMS subject classifications. 93C20, 93C25, 37C75, 93D30, 93C10.

For monotone systems evolving on the positive orthant of R n + two types of Lyapunov functions are considered: Sum-and max-separable Lyapunov functions. One can be written as a sum, the other as a maximum of functions of scalar arguments. Several constructive existence results for both types are given. Notably, one construction provides a max-separable Lyapunov function that is defined at least on an arbitrarily large compact set, based on little more than the knowledge about one trajectory. Another construction for a class of planar systems yields a global sum-separable Lyapunov function, provided the right hand side satisfies a small-gain type condition. A number of examples demonstrate these methods and shed light on the relation between the shape of sublevel sets and the right hand side of the system equation. Negative examples show that there are indeed globally asymptotically stable systems that do not admit either type of Lyapunov function.