781 research outputs found
Stability of Planar Nonlinear Switched Systems
We consider the time-dependent nonlinear system , where , and are two
% smooth vector fields, globally asymptotically stable at the origin
and is an arbitrary measurable function. Analysing the
topology of the set where and are parallel, we give some sufficient and
some necessary conditions for global asymptotic stability, uniform with respect
to . Such conditions can be verified without any integration or
construction of a Lyapunov function, and they are robust under small
perturbations of the vector fields
Stability Criteria for SIS Epidemiological Models under Switching Policies
We study the spread of disease in an SIS model. The model considered is a
time-varying, switched model, in which the parameters of the SIS model are
subject to abrupt change. We show that the joint spectral radius can be used as
a threshold parameter for this model in the spirit of the basic reproduction
number for time-invariant models. We also present conditions for persistence
and the existence of periodic orbits for the switched model and results for a
stochastic switched model
Extremal norms for positive linear inclusions
For finite-dimensional linear semigroups which leave a proper cone invariant
it is shown that irreducibility with respect to the cone implies the existence
of an extremal norm. In case the cone is simplicial a similar statement applies
to absolute norms. The semigroups under consideration may be generated by
discrete-time systems, continuous-time systems or continuous-time systems with
jumps. The existence of extremal norms is used to extend results on the
Lipschitz continuity of the joint spectral radius beyond the known case of
semigroups that are irreducible in the representation theory interpretation of
the word
Stability of uniformly bounded switched systems and Observability
This paper mainly deals with switched linear systems defined by a pair of
Hurwitz matrices that share a common but not strict quadratic Lyapunov
function. Its aim is to give sufficient conditions for such a system to be
GUAS.We show that this property of being GUAS is equivalent to the uniform
observability on of a bilinear system defined on a subspace whose
dimension is in most cases much smaller than the dimension of the switched
system.Some sufficient conditions of uniform asymptotic stability are then
deduced from the equivalence theorem, and illustrated by examples.The results
are partially extended to nonlinear analytic systems
Review on computational methods for Lyapunov functions
Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function
Uniform global stability of switched nonlinear systems in the Koopman operator framework
In this paper, we provide a novel solution to an open problem on the global
uniform stability of switched nonlinear systems. Our results are based on the
Koopman operator approach and, to our knowledge, this is the first theoretical
contribution to an open problem within that framework. By focusing on the
adjoint of the Koopman generator in the Hardy space on the polydisk (or on the
real hypercube), we define equivalent linear (but infinite-dimensional)
switched systems and we construct a common Lyapunov functional for those
systems, under a solvability condition of the Lie algebra generated by the
linearized vector fields. A common Lyapunov function for the original switched
nonlinear systems is derived from the Lyapunov functional by exploiting the
reproducing kernel property of the Hardy space. The Lyapunov function is shown
to converge in a bounded region of the state space, which proves global uniform
stability of specific switched nonlinear systems on bounded invariant sets.Comment: 29 pages, 3 figure
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