G.S. Ladde

Multiplex control systems: Stochastic stability and dynamic reliability

The problem of dynamic reliability of multiplex control systems under Markovian structural perturbations is reformulated, analyzed, and resolved in the context of stochastic stability and Liapunovs direct method. The failures of controllers are identified as states of a Markov process, and stability conditions are provided for a reliable performance of the multiplex control system alternating between operating (stable) and failure (unstable) structural configurations.

Cellular systems—II. stability of compartmental systems

Abstract In this work, by employing the concept of vector Lyapunov functions, and the theory of differential inequalities, the stability analysis of compartmental systems is initiated in a systematic and unified way. Furthermore, an attempt is made to formulate and partially resolve the “complexity vs. stability” problem in open compartmental systems. The recent stability results obtained for open chemical systems are applied to open compartmental systems in a natural way. As a byproduct of our analysis, we obtain an estimate for washout functions. Finally, it has been demonstrated that the stability analysis of compartmental systems provides a common conceptual framework for innumerable, apparently unrelated systems in biological, medical, and physical sciences.

Cellular systems—I. stability of chemical systems

Abstract In this work, we attempt to formulate and partially resolve the “complexity vs. stability” problem in open chemical systems, in the framework of Lyapunovs second method. Sufficient conditions are given for stability of the steady state of a chemical system under structural perturbations caused by interactions among the chemical species in the system. As a byproduct of this analysis, we will show important properties of the self-inhibitory chemical systems, and establish the tolerance of the stability to a broad class of perturbations. Finally, biological open systems are exhibited in order to show the scope of our stability analysis.

Differential inequalities and stochastic functional differential equations

Consider the system of stochastic functional differential equations dx=f(t,xt)dt+σ(t,xt)dz(t),xt0=φ0, where σ is a n×m matrix, column vectors of σ, f are continuous, and z(t) is a normalized m‐vector Wiener process with E[(z(t)−z(s))·(z(t)−z(s))T]=I|t−−s|. By developing a comparison principle, sufficient conditions are given for stability and boundedness in the mean of solutions of (S). The main technique here is the theory of functional differential inequalities and Lyapunov‐like functions.

Stability of model ecosystems with time-delay.

Abstract In this work, we attempt to formulize and partially resolve the “time-delay versus stability” as well as “complexity versus stability” problems in model ecosystems, in the framework of Lyapunovs stability theory. Sufficient conditions are given for stability of models with time-lag under structural perturbations caused by nonlinear interactions among species in the community. As a byproduct of this analysis, we will show important structural properties of the density-dependent models with time-lag, and establish tolerance of community stability to a broad class of nonlinear interactions and to a class of time-delays.

Diagonalization and stability of multi-time-scale singularly perturbed linear systems

In this paper, an alternate approach to the method of asymptotic expansions for the study of a singularly perturbed, linear system with multiparameters and multi-time scales, is developed. The method consists of developing a linear, nonsingular transformation that enables to decouple the original system completely. This process of diagonalization thus provides a very simple and suitable tool to investigate (i) the stability analysis and (ii) approximations of solutions of original system in terms of the overall reduced system and the corresponding boundary layer systems.

Averaging principle and systems of singularly perturbed stochastic differential equations

By developing certain auxiliary results, a modified version of the stochastic averaging principle is developed to investigate dynamical systems consisting of fast and slow phenomena. Moreover, an attempt is made to establish a relationship between the averaging assumption and certain ergodic‐type properties of the random process determined by an auxiliary system of stochastic differential equations. Finally, an example is given to illustrate the scope of the results.

Convergence and stability of distributed stochastic iterative processes

A mathematical framework is proposed for convergence analysis of stochastic iterative processes arising in applications of pseudogradient optimization algorithms. The framework is based upon the concepts of stochastic difference inequalities and vector Lyapunov functions, which are ideally suited for reducing the dimensionality problem arising in testing convergence of distributed parallel schemes. By applying the M-matrix conditions to a test matrix having a dimension equal to the number of the processor in the scheme, one can use the framework to select suitable scaling factors for each individual processor, producing a satisfactory convergence rate of the overall iterative process. The results provide new convergence tests for distributed iterative processes arising in decentralized extremal regulation, adaptation, and parameter estimation schemes. >

An Introduction to Differential Equations: Stochastic Modeling, Methods and Analysis(Volume 2)

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Existence of coupled quasi-solutions of systems of nonlinear reaction-diffusion equations☆

Systems of nonlinear parabolic initial boundary value problems arise in many applications, such as epidemies, ecology, biochemistry, biology, chemical and nuclear engineering. Constructive methods of proving existence results for such problems, which can also provide numerical procedures for the computation of solutions, are of greater value than theoretical existence results. The method of upper and lower solutions coupled with monotone iterative technique has been employed successfully to prove existence of multiple solutions of nonlinear reaction-diffusion equations, in special cases, by various authors [335, 10, 11, 15, 181. Recently, in [6, 171 weakly coupled systems of reaction-diffusion equations, when the nonlinear terms are independent of gradient terms, are discussed and some special type of results are obtained. We, in this paper, investigate general systems of nonlinear reaction-diffusion problems when the nonlinear terms possess a mixed quasi-monotone property. We discuss a very general situation and obtain coupled extremal quasi-solutions, which in special cases reduce to minimal and maximal solutions. We shall also indicate how one-step cyclic monotone iterative schemes can be generated which yield accelerated rate of convergence of iterates. This work is in the spirit of our recent paper [12] for elliptic systems.