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A Region-Dependent Gain Condition for Asymptotic Stability
A sufficient condition for the stability of a system resulting from the
interconnection of dynamical systems is given by the small gain theorem.
Roughly speaking, to apply this theorem, it is required that the gains
composition is continuous, increasing and upper bounded by the identity
function. In this work, an alternative sufficient condition is presented for
the case in which this criterion fails due to either lack of continuity or the
bound of the composed gain is larger than the identity function. More
precisely, the local (resp. non-local) asymptotic stability of the origin
(resp. global attractivity of a compact set) is ensured by a region-dependent
small gain condition. Under an additional condition that implies convergence of
solutions for almost all initial conditions in a suitable domain, the almost
global asymptotic stability of the origin is ensured. Two examples illustrate
and motivate this approach
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