517 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
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
Notions of Input to Output Stability
This paper deals with several related notions of output stability with
respect to inputs. The inputs may be thought of as disturbances; when there are
no inputs, one obtains generalizations of the classical concepts of partial
stability. The main notion studied is called input to output stability (IOS),
and it reduces to input to state stability (ISS) when the output equals the
complete state. Several variants, which formalize in different manners the
transient behavior, are introduced. The main results provide a comparison among
these notions. A companion paper establishes necessary and sufficient
Lyapunov-theoretic characterizations.Comment: 16 pages See http://www.math.rutgers.edu/~sontag/ for many related
paper
3 sampled-data control of nonlinear systems
This chapter provides some of the main ideas resulting from recent developments in sampled-data control of nonlinear systems. We have tried to bring the basic parts of the new developments within the comfortable grasp of graduate students. Instead of presenting the more general results that are available in the literature, we opted to present their less general versions that are easier to understand and whose proofs are easier to follow. We note that some of the proofs we present have not appeared in the literature in this simplified form. Hence, we believe that this chapter will serve as an important reference for students and researchers that are willing to learn about this area of research
Mather sets for sequences of matrices and applications to the study of joint spectral radii
The joint spectral radius of a compact set of d-times-d matrices is defined
?to be the maximum possible exponential growth rate of products of matrices
drawn from that set. In this article we investigate the ergodic-theoretic
structure of those sequences of matrices drawn from a given set whose products
grow at the maximum possible rate. This leads to a notion of Mather set for
matrix sequences which is analogous to the Mather set in Lagrangian dynamics.
We prove a structure theorem establishing the general properties of these
Mather sets and describing the extent to which they characterise matrix
sequences of maximum growth. We give applications of this theorem to the study
of joint spectral radii and to the stability theory of discrete linear
inclusions.
These results rest on some general theorems on the structure of orbits of
maximum growth for subadditive observations of dynamical systems, including an
extension of the semi-uniform subadditive ergodic theorem of Schreiber, Sturman
and Stark, and an extension of a noted lemma of Y. Peres. These theorems are
presented in the appendix
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