14,745 research outputs found
Relaxed ISS Small-Gain Theorems for Discrete-Time Systems
In this paper ISS small-gain theorems for discrete-time systems are stated,
which do not require input-to-state stability (ISS) of each subsystem. This
approach weakens conservatism in ISS small-gain theory, and for the class of
exponentially ISS systems we are able to prove that the proposed relaxed
small-gain theorems are non-conservative in a sense to be made precise. The
proofs of the small-gain theorems rely on the construction of a dissipative
finite-step ISS Lyapunov function which is introduced in this work.
Furthermore, dissipative finite-step ISS Lyapunov functions, as relaxations of
ISS Lyapunov functions, are shown to be sufficient and necessary to conclude
ISS of the overall system.Comment: input-to-state stability, Lyapunov methods, small-gain conditions,
discrete-time non-linear systems, large-scale interconnection
Converse Lyapunov Theorems for Switched Systems in Banach and Hilbert Spaces
We consider switched systems on Banach and Hilbert spaces governed by
strongly continuous one-parameter semigroups of linear evolution operators. We
provide necessary and sufficient conditions for their global exponential
stability, uniform with respect to the switching signal, in terms of the
existence of a Lyapunov function common to all modes
Lower Bounds on Complexity of Lyapunov Functions for Switched Linear Systems
We show that for any positive integer , there are families of switched
linear systems---in fixed dimension and defined by two matrices only---that are
stable under arbitrary switching but do not admit (i) a polynomial Lyapunov
function of degree , or (ii) a polytopic Lyapunov function with facets, or (iii) a piecewise quadratic Lyapunov function with
pieces. This implies that there cannot be an upper bound on the size of the
linear and semidefinite programs that search for such stability certificates.
Several constructive and non-constructive arguments are presented which connect
our problem to known (and rather classical) results in the literature regarding
the finiteness conjecture, undecidability, and non-algebraicity of the joint
spectral radius. In particular, we show that existence of an extremal piecewise
algebraic Lyapunov function implies the finiteness property of the optimal
product, generalizing a result of Lagarias and Wang. As a corollary, we prove
that the finiteness property holds for sets of matrices with an extremal
Lyapunov function belonging to some of the most popular function classes in
controls
Rate of Converrgence for ergodic continuous Markov processes : Lyapunov versus Poincare
We study the relationship between two classical approaches for quantitative
ergodic properties : the first one based on Lyapunov type controls and
popularized by Meyn and Tweedie, the second one based on functional
inequalities (of Poincar\'e type). We show that they can be linked through new
inequalities (Lyapunov-Poincar\'e inequalities). Explicit examples for
diffusion processes are studied, improving some results in the literature. The
example of the kinetic Fokker-Planck equation recently studied by H\'erau-Nier,
Helffer-Nier and Villani is in particular discussed in the final section
Exponential dichotomies of evolution operators in Banach spaces
This paper considers three dichotomy concepts (exponential dichotomy, uniform
exponential dichotomy and strong exponential dichotomy) in the general context
of non-invertible evolution operators in Banach spaces. Connections between
these concepts are illustrated. Using the notion of Green function, we give
necessary conditions and sufficient ones for strong exponential dichotomy. Some
illustrative examples are presented to prove that the converse of some
implication type theorems are not valid
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