Weak Exponential Stability for Time-Periodic Differential Inclusions via First Approximation Averaging

In this work we propose a method to study a weak exponential stability for time-varying differential inclusions applying an averaging procedure to a first approximation. Namely, we show that a weak exponential stability of the averaged first approximation to the differential inclusion implies the weak exponential stability of the original time-varying inclusion. The result is illustrated by an example.     

Decay characteristics for differential equations without linear terms

The asymptotic stability of isolated critical points of differential systems without linear terms is related to the signum of the type numbers introduced by Coleman. This leads to asymptotic estimates for solutions of a class of differential equations with homogeneous first approximation.     

Stability of stationary solutions of piecewise affine differential equations describing gene regulatory networks

We study stability properties of a class of piecewise affine systems of ordinary differential equations arising in the modeling of gene regulatory networks. Our method goes back to the concept of a Filippov stationary solution (in the narrow sense) to a differential inclusion corresponding to the system in question. The main result of the paper justifies a reduction principle in the stability analysis enabling to omit the variables that are not singular, i.e. that stay away from the discontinuity set of the system. We suggest also “the first approximation method” to study asymptotic stability of stationary solutions based on calculating the principal part of the system, which is 0-homogeneous rather than linear. This leads to an efficient algorithm of how to check asymptotic stability without calculating the eigenvalues of the system?s Jacobian. In Appendix A we discuss and compare two other concepts of stationary solutions to the system in question.     

On the stability of time-varying differential equations *

This paper deals with stability of time-varying differential equations. New asymptotic stability conditions for time-varying continuous systems with more general assumptions are given. The Gronwall's inequality and its discrete-time versions play a crucial role in our investigation.     

Pathwise convergence rates for numerical solutions of Markovian switching stochastic differential equations

This work develops numerical approximation algorithms for solutions of stochastic differential equations with Markovian switching. The existing numerical algorithms all use a discrete-time Markov chain for the approximation of the continuous-time Markov chain. In contrast, we generate the continuous-time Markov chain directly, and then use its skeleton process in the approximation algorithm. Focusing on weak approximation, we take a re-embedding approach, and define the approximation and the solution to the switching stochastic differential equation on the same space. In our approximation, we use a sequence of independent and identically distributed (i.i.d.) random variables in lieu of the common practice of using Brownian increments. By virtue of the strong invariance principle, we ascertain rates of convergence in the pathwise sense for the weak approximation scheme.     

Higher-order spectral analysis and weak asymptotic stability of convex processes

This paper deals with the asymptotic stability analysis of a discrete dynamical inclusion whose right-hand side is a convex process. We provide necessary and sufficient conditions for weak asymptotic stability, and obtain sharp estimates for the asymptotic null-controllability set. These estimates involve not only standard, but also higher-order spectral information on the convex process and its adjoint.     

Contrast structures in the reaction-diffusion-advection equations in the case of balanced advection

For a singularly perturbed parabolic equation termed in applications as the reaction-diffusion-advection equation, stationary solutions with internal transition layers (contrast structures) are studied. An arbitrary-order asymptotic approximation of such solutions is constructed, and an existence theorem is proved. An efficient algorithm for constructing an asymptotic approximation of the transition point is proposed. The constructed asymptotic approximation is justified by applying the asymptotic method of differential inequalities, which is extended to the class of problems under study. This method is also used to establish the Lyapunov stability of such stationary solutions.     

Stochastic recursive inclusions with non-additive iterate-dependent Markov noise

In this paper we study the asymptotic behaviour of stochastic approximation schemes with set-valued drift function and non-additive iterate-dependent Markov noise. We show that a linearly interpolated trajectory of such a recursion is an asymptotic pseudotrajectory for the flow of a limiting differential inclusion obtained by averaging the set-valued drift function of the recursion w.r.t. the stationary distributions of the Markov noise. The limit set theorem by Benaim is then used to characterize the limit sets of the recursion in terms of the dynamics of the limiting differential inclusion. We then state two variants of the Markov noise assumption under which the analysis of the recursion is similar to the one presented in this paper. Scenarios where our recursion naturally appears are presented as applications. These include controlled stochastic approximation, subgradient descent, approximate drift problem and analysis of discontinuous dynamics all in the presence of non-additive iterate-dependent Markov noise.     

Stability analysis based on nonlinear inhomogeneous approximation

The asymptotic stability of zero solutions for essentially nonlinear systems of differential equations in triangular inhomogeneous approximation is studied. Conditions under which perturbations do not affect the asymptotic stability of the zero solution are determined by using the direct Lyapunov method. Stability criteria are stated in the form of inequalities between perturbation orders and the orders of homogeneity of functions involved in the nonlinear approximation system under consideration.     

Diagonally drift-implicit Runge–Kutta methods of weak order one and two for Itô SDEs and stability analysis

The class of stochastic Runge–Kutta methods for stochastic differential equations due to Rößler is considered. Coefficient families of diagonally drift-implicit stochastic Runge–Kutta (DDISRK) methods of weak order one and two are calculated. Their asymptotic stability as well as mean-square stability (MS-stability) properties are studied for a linear stochastic test equation with multiplicative noise. The stability functions for the DDISRK methods are determined and their domains of stability are compared to the corresponding domain of stability of the considered test equation. Stability regions are presented for various coefficients of the families of DDISRK methods in order to determine step size restrictions such that the numerical approximation reproduces the characteristics of the solution process.     

Stability criteria for a class of differential inclusion systems with discrete and distributed time delays

In this paper, the global asymptotic stability for a class of differential inclusion systems with discrete and distributed time delays is investigated. Some delay-dependent criteria are proposed to guarantee the global asymptotic stability of the systems. Finally, a numerical example is provided to illustrate the use of the main results.     

Stochastic differential inclusions and diffusion processes

Connections between weak solutions of stochastic differential inclusions and solutions of partial differential inclusions, generated by given set-valued mappings are considered. The main results are based on some continuous approximation selection theorem and weak compactness of the set of all weak solutions to a given stochastic differential inclusion.     

On stability of discrete-time switched systems

This article deals with stability of discrete-time switched systems. Given a family of nonlinear systems and the admissible switches among the systems in the family, we first propose a class of switching signals under which the resulting switched system is globally asymptotically stable. We allow unstable systems in the family and our stability condition depends solely on asymptotic behaviour of the switching signals. We then discuss algorithmic construction of the above class of switching signals, and show that in the presence of exogenous inputs and outputs, a switching signal so constructed also ensures input/output-to-state stability for discrete-time switched nonlinear systems. We finally show that our class of switching signals that ensures global asymptotic stability also extends to the continuous-time setting with minor modifications under standard assumptions. We employ multiple Lyapunov-like functions and graph theoretic tools as the main apparatuses for our analysis.     

Discounted cost optimality problem: stability with respect to weak metrics

We find inequalities to estimate the stability (robustness) of a discounted cost optimization problem for discrete-time Markov control processes on a Borel state space. The one stage cost is allowed to be unbounded. Unlike the known results in this area we consider a perturbation of transition probabilities measured by the Kantorovich metric, closely related to the weak convergence. The results obtained make possible to estimate the vanishing rate of the stability index when approximation is made through empirical measures.     

Singularly perturbed differential inclusions: An averaging approach

We consider nonlinear, singularly perturbed differential inclusions and apply the averaging method in order to construct a limit differential inclusion for slow motion. The main approximation result states that the existence and regularity of the limit differential inclusion suffice to describe the limit behavior of the slow motion. We give explicit approximation rates for the uniform convergence on compact time intervals. The approach works under controllability or stability properties of fast motion.     

Contrast structures in singularly perturbed quasilinear reaction-diffusion-advection equations

We consider stationary solutions with internal transition layers (contrast structures) for a singularly perturbed elliptic equation that is referred to in applications as the stationary reaction-diffusion-advection equation. We construct an asymptotic approximation of arbitrary-order accuracy to such solutions and prove the existence theorem. We suggest an efficient algorithm for constructing an asymptotic approximation to the localization curve of the transition layer. To justify the constructed asymptotics, we use and develop, to this class of problems, an asymptotic method of differential inequalities, which also permits one to prove the Lyapunov stability of such stationary solutions.     

Global stability of discrete-time competitive population models

We develop practical tests for the global asymptotic stability of interior fixed points for discrete-time competitive population models. Our method constitutes the extension to maps of the Split Lyapunov method developed for differential equations. We give ecologically-motivated sufficient conditions for global stability of an interior fixed point of a map of Kolmogorov form. We introduce the concept of a principal reproductive mode, which is linked to a normal at the interior fixed point of a hypersurface of vanishing weighted-average growth. Our method is applied to establish new global stability results for 3-species competitive systems of May-Leonard type, where previously only parameter values for local stability was known.     

Stability of equilibria of discrete-time systems in terms of invariant sets

We suggest a new approach to the verification of the stability (or asymptotic stability) of the equilibria of time-invariant discrete-time systems based on stability and asymptotic stability criteria stated in terms of invariant sets. A set-theoretic method for the verification of the conditions in these criteria is presented.     

Second Order Asymptotics of Aggregated Log-Elliptical Risk

In this paper we establish the error rate of first order asymptotic approximation for the tail probability of sums of log-elliptical risks. Our approach is motivated by extreme value theory which allows us to impose only some weak asymptotic conditions satisfied in particular by log-normal risks. Given the wide range of applications of the log-normal model in finance and insurance our result is of interest for both rare-event simulations and numerical calculations. We present numerical examples which illustrate that the second order approximation derived in this paper significantly improves over the first order approximation.     

Asymptotics of a Class of Singularly Perturbed Weak Nonlinear Boundary Value Problem with a Multiple Root of the Degenerate Equation

A singularly perturbed boundary value problem with weak nonlinearity in the case when the degenerate equation has a multiple root is studied. The asymptotic approximation of the solution is constructed by the modified boundary layer function method. Based on the comparison principle, there exist multizonal boundary layers in the neighborhood of the endpoints. The existence of a solution is proved by using the method of asymptotic differential inequalities.