68,391 research outputs found
Stability and Boundedness of Impulsive Systems with Time Delay
The stability and boundedness theories are developed for impulsive differential equations with time delay. Definitions, notations and
fundamental theory are presented for delay differential systems with both fixed and state-dependent impulses. It is usually more
difficult to investigate the qualitative properties of systems with state-dependent impulses since different solutions have
different moments of impulses. In this thesis, the stability problems of nontrivial solutions of systems with state-dependent impulses are ``transferred" to those of the trivial solution of systems with fixed impulses by constructing the so-called ``reduced system". Therefore, it is enough to investigate the
stability problems of systems with fixed impulses. The exponential stability problem is then discussed for the system with fixed
impulses. A variety of stability criteria are obtained and`numerical examples are worked out to illustrate the results, which shows that impulses do contribute to the stabilization of some delay differential equations. To unify various stability concepts and to offer a general framework for the investigation of
stability theory, the concept of stability in terms of two measures is introduced and then several stability criteria are developed for impulsive delay differential equations by both the single and multiple Lyapunov functions method. Furthermore, boundedness and periodicity results are discussed for impulsive differential systems with time delay. The Lyapunov-Razumikhin technique, the Lyapunov functional method, differential
inequalities, the method of variation of parameters, and the partitioned matrix method are the main tools to obtain these results. Finally, the application of the stability theory to neural networks is presented. In applications, the impulses are considered as either means of impulsive control or perturbations.Sufficient conditions for stability and stabilization of neural
networks are obtained
Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications
In this work we study certain invariant measures that can be associated to
the time averaged observation of a broad class of dissipative semigroups via
the notion of a generalized Banach limit. Consider an arbitrary complete
separable metric space which is acted on by any continuous semigroup
. Suppose that possesses a global
attractor . We show that, for any generalized Banach limit
and any distribution of initial
conditions , that there exists an invariant probability measure
, whose support is contained in , such that for all
observables living in a suitable function space of continuous mappings
on .
This work is based on a functional analytic framework simplifying and
generalizing previous works in this direction. In particular our results rely
on the novel use of a general but elementary topological observation, valid in
any metric space, which concerns the growth of continuous functions in the
neighborhood of compact sets. In the case when does not
possess a compact absorbing set, this lemma allows us to sidestep the use of
weak compactness arguments which require the imposition of cumbersome weak
continuity conditions and limits the phase space to the case of a reflexive
Banach space. Two examples of concrete dynamical systems where the semigroup is
known to be non-compact are examined in detail.Comment: To appear in Communications in Mathematical Physic
Statistical solutions of hyperbolic conservation laws I: Foundations
We seek to define statistical solutions of hyperbolic systems of conservation
laws as time-parametrized probability measures on -integrable functions. To
do so, we prove the equivalence between probability measures on spaces
and infinite families of \textit{correlation measures}. Each member of this
family, termed a \textit{correlation marginal}, is a Young measure on a
finite-dimensional tensor product domain and provides information about
multi-point correlations of the underlying integrable functions. We also prove
that any probability measure on a space is uniquely determined by certain
moments (correlation functions) of the equivalent correlation measure.
We utilize this equivalence to define statistical solutions of
multi-dimensional conservation laws in terms of an infinite set of equations,
each evolving a moment of the correlation marginal. These evolution equations
can be interpreted as augmenting entropy measure-valued solutions, with
additional information about the evolution of all possible multi-point
correlation functions. Our concept of statistical solutions can accommodate
uncertain initial data as well as possibly non-atomic solutions even for atomic
initial data.
For multi-dimensional scalar conservation laws we impose additional entropy
conditions and prove that the resulting \textit{entropy statistical solutions}
exist, are unique and are stable with respect to the -Wasserstein metric on
probability measures on
Optimal linear stability condition for scalar differential equations with distributed delay
Linear scalar differential equations with distributed delays appear in the
study of the local stability of nonlinear differential equations with feedback,
which are common in biology and physics. Negative feedback loops tend to
promote oscillations around steady states, and their stability depends on the
particular shape of the delay distribution. Since in applications the mean
delay is often the only reliable information available about the distribution,
it is desirable to find conditions for stability that are independent from the
shape of the distribution. We show here that for a given mean delay, the linear
equation with distributed delay is asymptotically stable if the associated
differential equation with a discrete delay is asymptotically stable. We
illustrate this criterion on a compartment model of hematopoietic cell dynamics
to obtain sufficient conditions for stability
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