Abstract
We consider infinite heterogeneous networks, consisting of input-to-state stable subsystems of possibly infinite dimension. We show that the network is input-to-state stable, provided that the gain operator satisfies a certain small-gain condition. We show that for finite networks of nonlinear systems this condition is equivalent to the so-called strong small-gain condition of the gain operator (and thus our results extend available results for finite networks), and for infinite networks with a linear gain operator they correspond to the condition that the spectral radius of the gain operator is less than one. We provide efficient criteria for input-to-state stability of infinite networks with linear gains, governed by linear and homogeneous gain operators, respectively.
Highlights
We live in a hyperconnected world, where the size of networks and the number of connections between their components are rapidly growing
We show that the network is input-to-state stable, provided that the gain operator satisfies a certain small-gain condition
We show that for finite networks of nonlinear systems this condition is equivalent to the so-called strong small-gain condition of the gain operator, and for infinite networks with a linear gain operator they correspond to the condition that the spectral radius of the gain operator is less than one
Summary
We live in a hyperconnected world, where the size of networks and the number of connections between their components are rapidly growing. Theorems 6.1 and 6.4 are our small-gain results for uniform global stability of infinite networks They use the monotone bounded invertibility (MBI) property of the gain operator, which is equivalent for finite networks (see Proposition 7.12) to the strong small-gain condition, employed in the small-gain analysis of finite networks in [18, Thm. 8] and [37]. In “Appendix A”, we derive a characterization of exponential ISS (eISS) for discrete-time systems with a generating and normal cone, induced by homogeneous of degree one and subadditive operators (Proposition A.1) We apply this and recent results in [19], to show in Proposition 7.16 that for linear infinite-dimensional systems with a generating and normal cone the MBI, MLIM and the uniform small-gain condition all are equivalent to the spectral small-gain condition (saying that the spectral radius of the gain operator is less than one). Propositions 7.16, 7.17 and A.1 are useful, in particular, to obtain efficient smallgain theorems for infinite networks with linear gains, see Corollaries 6.3, 6.6
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