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Register a new userSufficient Stability Conditions for Time-varying Networks of Telegraphers Equations or Difference Delay Equations
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We give a sufficient condition for exponential stability of a network of lossless telegraphers equations, coupled by linear time-varying boundary conditions. The sufficient conditions is in terms of dissipativity of the couplings, which is natural for instance in the context of microwave circuits. Exponential stability is with respect to any $L^p$-norm, $1leq pleqinfty$. This also yields a sufficient condition for exponential stability to a special class of linear time-varying difference delay systems which is quite explicit and tractable. One ingredient of the proof is that $L^p$ exponential stability for such difference delay systems is independent of $p$, thereby reproving in a simpler way some results from [3].
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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.
This paper proposes a unified approach for studying global exponential stability of a general class of switched systems described by time-varying nonlinear functional differential equations. Some new delay-independent criteria of global exponential stability are established for this class of systems under arbitrary switching which satisfies some assumptions on the average dwell time. The obtained criteria are shown to cover and improve many previously known results, including, in particular, sufficient conditions for absolute exponential stability of switched time-delay systems with sector nonlinearities. Some simple examples are given to illustrate the proposed method.
We consider ergodic backward stochastic differential equations in a discrete time setting, where noise is generated by a finite state Markov chain. We show existence and uniqueness of solutions, along with a comparison theorem. To obtain this result, we use a Nummelin splitting argument to obtain ergodicity estimates for a discrete time Markov chain which hold uniformly under suitable perturbations of its transition matrix. We conclude with an application of this theory to a treatment of an ergodic control problem.
We study stability of so-called synchronous slowly oscillating periodic solutions (SOPSs) for a system of identical delay differential equations (DDEs) with linear decay and nonlinear delayed negative feedback that are coupled through their nonlinear term. Under a row sum condition on the coupling matrix, existence of a unique SOPS for the corresponding scalar DDE implies existence of a unique synchronous SOPS for the coupled DDEs. However, stability of the SOPS for the scalar DDE does not generally imply stability of the synchronous SOPS for the coupled DDEs. We obtain an explicit formula, depending only on the spectrum of the coupling matrix, the strength of the linear decay and the values of the nonlinear negative feedback function near plus/minus infinity, that determines the stability of the synchronous SOPS in the asymptotic regime where the nonlinear term is heavily weighted. We also treat the special cases of so-called weakly coupled systems, near uniformly coupled systems, and doubly nonnegative coupled systems, in the aforementioned asymptotic regime. Our approach is to estimate the characteristic (Floquet) multipliers for the synchronous SOPS. We first reduce the analysis of the multidimensional variational equation to the analysis of a family of scalar variational-type equations, and then establish limits for an associated family of monodromy-type operators. We illustrate our results with examples of systems of DDEs with mean-field coupling and systems of DDEs arranged in a ring.
We prove the theorem of linearized asymptotic stability for fractional differential equations. More precisely, we show that an equilibrium of a nonlinear Caputo fractional differential equation is asymptotically stable if its linearization at the equilibrium is asymptotically stable. As a consequence we extend Lyapunovs first method to fractional differential equations by proving that if the spectrum of the linearization is contained in the sector ${lambda in C : |arg lambda| > frac{alpha pi}{2}}$ where $alpha > 0$ denotes the order of the fractional differential equation, then the equilibrium of the nonlinear fractional differential equation is asymptotically stable.
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