5,435 research outputs found
On the stability of some linear nonautonomous systems
Stability of linear nonautonomous systems described by differential equations with time varying coefficient
Stability radius and internal versus external stability in Banach spaces: an evolution semigroup approach
In this paper the theory of evolution semigroups is developed and used to
provide a framework to study the stability of general linear control systems.
These include time-varying systems modeled with unbounded state-space operators
acting on Banach spaces. This approach allows one to apply the classical theory
of strongly continuous semigroups to time-varying systems. In particular, the
complex stability radius may be expressed explicitly in terms of the generator
of a (evolution) semigroup. Examples are given to show that classical formulas
for the stability radius of an autonomous Hilbert-space system fail in more
general settings. Upper and lower bounds on the stability radius are provided
for these general systems. In addition, it is shown that the theory of
evolution semigroups allows for a straightforward operator-theoretic analysis
of internal stability as determined by classical frequency-domain and
input-output operators, even for nonautonomous Banach-space systemsComment: Also at http://www.math.missouri.edu/~stephen/preprint
On the stability of nonautonomous systems
International audienceIn (Kalitine, 1982), the use of semi definite Lyapunov functions for exploring the local stability of autonomous dynamical systems has been introduced. In this paper we give an extension of the results of (Kalitine, 1982) that allows to study the local stability of nonautonomous differential systems. We give an application to the Algebraic Riccati Equation
Characterizations of safety in hybrid inclusions via barrier functions
This paper investigates characterizations of safety in terms of barrier functions for hybrid systems modeled by hybrid inclusions. After introducing an adequate definition of safety for hybrid inclusions, sufficient conditions using continuously differentiable as well as lower semicontinuous barrier functions are proposed. Furthermore, the lack of existence of autonomous and continuous barrier functions certifying safety, guides us to propose, inspired by converse Lyapunov theorems for only stability, nonautonomous barrier functions and conditions that are shown to be both necessary as well as sufficient, provided that mild regularity conditions on the system's dynamics holds
Finite-time stability results for fractional damped dynamical systems with time delays
This paper is explored with the stability procedure for linear nonautonomous multiterm fractional damped systems involving time delay. Finite-time stability (FTS) criteria have been developed based on the extended form of Gronwall inequality. Also, the result is deduced to a linear autonomous case. Two examples of applications of stability analysis in numerical formulation are described showing the expertise of theoretical prediction
The Bohl spectrum for nonautonomous differential equations
We develop the Bohl spectrum for nonautonomous linear differential equation
on a half line, which is a spectral concept that lies between the Lyapunov and
the Sacker--Sell spectrum. We prove that the Bohl spectrum is given by the
union of finitely many intervals, and we show by means of an explicit example
that the Bohl spectrum does not coincide with the Sacker--Sell spectrum in
general. We demonstrate for this example that any higher-order nonlinear
perturbation is exponentially stable, although this not evident from the
Sacker--Sell spectrum. We also analyze in detail situations in which the Bohl
spectrum is identical to the Sacker-Sell spectrum
Generalized chronotaxic systems: time-dependent oscillatory dynamics stable under continuous perturbation
Chronotaxic systems represent deterministic nonautonomous oscillatory systems
which are capable of resisting continuous external perturbations while having a
complex time-dependent dynamics. Until their recent introduction in \emph{Phys.
Rev. Lett.} \textbf{111}, 024101 (2013) chronotaxic systems had often been
treated as stochastic, inappropriately, and the deterministic component had
been ignored. While the previous work addressed the case of the decoupled
amplitude and phase dynamics, in this paper we develop a generalized theory of
chronotaxic systems where such decoupling is not required. The theory presented
is based on the concept of a time-dependent point attractor or a driven steady
state and on the contraction theory of dynamical systems. This simplifies the
analysis of chronotaxic systems and makes possible the identification of
chronotaxic systems with time-varying parameters. All types of chronotaxic
dynamics are classified and their properties are discussed using the
nonautonomous Poincar\'e oscillator as an example. We demonstrate that these
types differ in their transient dynamics towards a driven steady state and
according to their response to external perturbations. Various possible
realizations of chronotaxic systems are discussed, including systems with
temporal chronotaxicity and interacting chronotaxic systems.Comment: 9 pages, 8 figure
Transit times and mean ages for nonautonomous and autonomous compartmental systems
We develop a theory for transit times and mean ages for nonautonomous
compartmental systems. Using the McKendrick-von F\"orster equation, we show
that the mean ages of mass in a compartmental system satisfy a linear
nonautonomous ordinary differential equation that is exponentially stable. We
then define a nonautonomous version of transit time as the mean age of mass
leaving the compartmental system at a particular time and show that our
nonautonomous theory generalises the autonomous case. We apply these results to
study a nine-dimensional nonautonomous compartmental system modeling the
terrestrial carbon cycle, which is a modification of the Carnegie-Ames-Stanford
approach (CASA) model, and we demonstrate that the nonautonomous versions of
transit time and mean age differ significantly from the autonomous quantities
when calculated for that model
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