Study of a one-dimensional model,accounting for cell division,of regulation of the renewing zone size in a biological tissue

We describe the modeling of the dynamics of renewing zone structure in biological tissues in the formalism of parameterized L-systems on an example of the shoot apical meristem. Under study is the influence of the ratio of the characteristic times of the cell cycle and diffusion of morphogens on the stability of some spatially distributed molecular-genetic control system. We show that cell division is a perturbing factor for the system regulating the renewing zone structure. Some conditions are found for the loss of stability of the regulation.     

Stabilization of a One-Dimensional Dam-River System: Nondissipative and Noncollocated Case

In this paper, we consider a one-dimensional dam-river system, described by a diffusive-wave equation and often used in hydraulic engineering to model the dynamic behavior of the unsteady flow in a river for shallow water when the flow variations are not important. We propose an integral boundary control which leads to a nondissipative closed-loop system with noncollocated actuators and sensors; hence, two main difficulties arise: first, how to show the C ₀-semigroup generation and second, how to achieve the stability of the system. To overcome this situation, the Riesz basis methodology is adopted to show that the closed-loop system generates an analytic semigroup. Concerning the stability, the shooting method is applied to assign the spectrum of the system in the open left-half plane and ensure its exponential stability as well as the output regulation. Numerical simulations are presented for a family of system parameters. The authors express their sincere thanks to Boumenir Amin for valuable comments and suggestions. The first author acknowledges the support of Sultan Qaboos University. The second author was supported by the National Natural Science Foundation of China.     

Asymptotics of nonlinear processes of a controlled approach

Stability of a terminal system of some class that appears in the projection of a control approach is studied, a mathematical model of which is described by a nonlinear system of ordinary differential equations for which no conditions of the existence theorem for solutions constructed by the Lapunov function are satisfied. We obtain criteria of stability, effectiveness of which is shown in concrete examples. nht]mis|Translated from Dinamicheskie Sistemy, No. 7, pp. 62–71, 1988.     

The first-approximation stability of a nonstationary system with delay

We study the exponential stability of a nonlinear system of differential equations with constant delay such that the right-hand side of one of its subsystems contains the multiplier e ^t . We obtain a sufficient condition for the first-approximation stability of this system.     

Uniform Decay properties of a model in structural acoustics

We investigate decay properties for a system of coupled partial differential equations which model the interaction between acoustic waves in a cavity and the walls of the cavity. In this system a wave equation is coupled to a structurally damped plate or beam equation. The underlying semigroup for this system is not uniformly stable, but when the system is appropriately restricted we obtain some uniform stability. We present two results of this type. For the first result, we assume that the initial wave data is zero, and the initial plate or beam data is in the natural energy space; then the corresponding solution to system decays uniformly to zero. For the second result, we assume that the initial condition is in the natural energy space and the control function is L²(0,∞) (in time) into the control space; then the beam displacement and velocity are both L²(0,∞) into a space with two spatial derivatives.     

On a synthesis of stability of an invariant control system with an astatic gauge

For a synthesis of stability of an invariant control system with a perturbation that acts at the object, we establish that its mathematical model is represented in the form of differential equations with small parameter at the derivative. With the help of the method of functions of flexible structure of N. K. Kulikov for the coordinate of the object we obtain sufficient conditions for invariancy up to , which agree with requirements of absolute stability of the system.Translated from Dinamicheskie Sistemy, No. 8, pp. 54–58, 1989.     

Performance analysis approximation in a queueing system of type M/G/1

In this work, we apply the strong stability method to obtain an estimate for the proximity of the performance measures in the M/G/1 queueing system to the same performance measures in the M/M/1 system under the assumption that the distributions of the service time are close and the arrival flows coincide. In addition to the proof of the stability fact for the perturbed M/M/1 queueing system, we obtain the inequalities of the stability. These results give with precision the error, on the queue size stationary distribution, due to the approximation. For this, we elaborate from the obtained theoretical results, the STR-STAB algorithm which we execute for a determined queueing system: M/Coxian − 2/1. The accuracy of the approach is evaluated by comparison with simulation results.     

On partial stability of a dynamic system that is integrally continuous with respect to a small parameter

For a dynamic system described by Carathéodory ordinary differential equations with a small parameter we introduce the definitions of a type of partial stability, attraction, and asymptotic stability. We state theorems giving sufficient conditions for stability in the new definitions. In particular, in terms of perturbed Lyapunov functions we obtain conditions for partial asymptotic stability that generalize known results. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 29–36.     

Absolute stability criteria for infinite-dimensional discrete Lur'e systems with application to loaded distortionless electric RLCG-transmission line

We study absolute stability of a class of discrete Lur'e control system on an infinite-dimensional Hilbert space. Counterparts of the circle criterion and the Szegö criterion are derived using, respectively, quadratic and non-quadratic Lyapunov functionals. A link with existing finite-dimensional theory of absolute stability is shown. The results are illustrated by an example of the loaded distortionless electric RLCG-transmission line with a nonlinear static feedback. Its stability was previously investigated using the circle criterion for continuous infinite-dimensional systems with unbounded control and observation in the frame of systems in factor form.     

Stability conditions for a pipeline polling scheme in satellite communications

This paper considers the unknown stability conditions of a pipeline polling scheme proposed for satellite communications. This scheme is modelled as a cyclic-service system with limited service and reservation. The walk times and the maximum number of services to be performed during each polling are dependent on the queue lengths of the stations. The main result is the derivation of the necessary and sufficient stability conditions of the system. Our approach is to map the multi-dimensional stability problem into many 1-dimensional stability problems through the concept of the least stable queue. The least stable queue is one that will become unstablefirst when the system load increases in some parameter region. The stability of the least stable queue thus implies stability of the system. The stability region for the whole system is then the union of the queue stability regions of all the least stable queues that are obtained through dominant systems and Loynes' theorem. We also propose a computable sufficient condition that is tighter than the existing result and present some numerical results.     

Analysis of the interaction of inertial forces in a stability problem

Using the plane of characteristic indices we study the stability of an elastic filament with a distributed inertial load moving along it at constant speed. Taking account of the Coriolis inertial forces and assuming the presence of dissipative forces, we carry out a qualitative analysis of the results for different relations among the parameters of the system; this analysis is in agreement with the general stability theorems for elastic structures. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.     

Construction of amplitude-phase-frequency characteristics and definition of stability on a computer

For complex linear control systems we give a method of construction of amplitude- and phase-frequency characteristics and a definition of stability that allow us to choose the set of frequencies according to a conveniency of design. The method is based on the interpolation-dichotomous method and is convenient for realization on a computer, and allows us to exclude jumps of a phase-frequency characteristic and define stability with a high order of precision.Translated from Dinamicheskie Sistemy, No. 8, pp. 65–68, 1989.     

Bifurcation behaviour and limit cycles of a compliant seal-rotor system

This contribution discusses the non-linear dynamic behaviour of a rotor system equipped with a compliant seal. The investigated model consists of a Laval-Rotor and a stiff seal ring which is visco-elastically supported. The fluid forces stemming from the turbulent incompressible lubricant film are accounted for by the non-linear Muszynska model. The added compliance leads to an improved stability behaviour of the steady state. Within the post-critical regime the additional compliance gives rise to bifurcations of the stationary vibration: Hopf bifurcations lead to limit cycles which can loose their stability via Neimark-Sacker bifurcations. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Design of an optimal tuned mass damper for a system with parametric uncertainty

Structural control is becoming an attractive alternative for enhanced performance of civil engineering structures subject to seismic and wind loads. However, in order to guarantee stability and performance of structures when implemented with a passive or active control technique, there is a need to include information of uncertainty in the structural models due to the fact that civil engineering structures are time variant and nonlinear. These variations in the structure are often due to parameters such as variable live loads and inelastic behavior and, in cases, may be modeled as parametric uncertainty. The design of an optimal tuned mass damper (TMD) for a one degree-of-freedom (SDOF) system with parametric uncertainty is presented in this paper. The optimization of the connection between the absorber and the primary structure is cast as a constant feedback problem which is solved using structured singular value, μ, synthesis with D-K iteration and decentralized H _∞ design. Results are presented of the TMD that minimize the harmonic response of the primary structure represented by a set of systems within an uncertainty set.     

Pattern-forming systems for control of large arrays of actuators

Summary. We consider an approach for coordinating the activity of a large array of microactuators via diffusive (i.e., nearest-neighbor) coupling combined with reactive growth and decay, implemented via interconnection templates which have been artificially engineered into the system (for example, in collocated microelectronic circuitry, or through the formulation of active material layers). Such coupled systems can support interesting spatiotemporal patterns, which in turn determine the actuation patterns. Generating such spatiotemporal patterns typically involves stressing the interconnections by raising or lowering a parameter resulting in the crossing of stability thresholds. The possibility of making such parametric adjustments via feedback on a slower timescale offers a solution to the problem of communicating effectively within a large array: The communication is achieved through the interconnection template. The mathematics behind this idea leads us into the rich domain of nonlinear partial differential equations (PDEs) with spatiotemporal pattern solutions. We present a global nonlinear stability analysis that applies to certain model pattern-forming systems. The nonlinear stability analysis could serve as a starting point for control system design for systems containing large microactuator arrays. Received August 1, 2000; accepted August 16, 2001 Online publication November 5, 2001     

Risk sensitive control of pure jump processes on a general state space

We study stochastic control problem for pure jump processes on a general state space with risk sensitive discounted and ergodic cost criteria. For the discounted cost criterion we prove the existence and Hamilton–Jacobi–Bellman characterization of optimal α-discounted control for bounded cost function. For the ergodic cost criterion we assume a Lyapunov type stability assumption and a small cost condition. Under these assumptions we show the existence of the optimal risk-sensitive ergodic control.     

Smooth Lyapunov Functions for Discontinuous Stable Systems

It has been proved that a differential system d x / d t = f(t, x) with a discontinuous right-hand side admits some continuous weak Lyapunov function if and only if it is robustly stable. This paper focuses on the smoothness of such a Lyapunov function. An example of an (asymptotically) stable system for which there does not exist any (even weak) Lyapunov functions of class C ¹ is given. In the more general context of differential inclusions, the existence of a weak Lyapunov function of class C ¹ (or C ) is shown to be equivalent to the robust stability of some perturbed system obtained in introducing measurement error with respect to x and t. This condition is proved to be satisfied by most of the robustly stable systems encountered in the literature. Analogous results are given for the Lagrange stability. As an application to the study of the links between internal and external stability for control systems, an extension of a result by Bacciotti and Beccari is obtained by means of a smooth Lyapunov function associated with a robustly Lagrange stable system.     

Diagnosing departures from stability in a multidimensional multivariate system

This paper deals with a system containing n m-variable elements measured at p points of time. The data matrices for each point of time are represented by the biplot technique. Each biplot provides a map of the system which can be used to determine visually the relations between the elements and variables. The biplots are then compared by superimposing them to obtain an ‘aggregate’ graph. Differences between the biplots indicate departure from stability of the system. An example drawn from the Israeli banking system is provided.     

On the stability of motion of a movable compound structure with a fluid load

We study the stability of uniform rectilinear motion of a system of solid bodies connected by springs and partly filled with an ideal fluid. The frequency equation obtained is studied as a function of the fundamental parameters of the mechanical system. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 111–116.     

A study of a numerical algorithm for a nonlinear transport model

For a nonlinear transport model, we propose a simple and economical two-step algorithm that decreases the dimension of the system of nonlinear equations, as compared with implicit difference schemes. We prove theorems on necessary conditions for stability with respect to the initial data for the nonlinear problem and theorems on sufficient conditions for stability in the case of the linearized model. We also obtain theorems on approximation of the integral conservation law on a grid. The necessary condition obtained is a condition on the coefficients of the differential equation (which singles out an admissible class of equations) but not a condition on the ratio of the grid steps. Bibliography: 3 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 81, 1997, pp. 25–32.