Explicit stability conditions for integro-differential equations with periodic coefficients

New explicit stability conditions are derived for a linear integro-differential equation with periodic operator coefficients. The equation under consideration describes oscillations of thin-walled viscoelastic structural members driven by periodic loads. To develop stability conditions two approaches are combined. The first is based on the direct Lyapunov method of constructing stability functionals. It allows stability conditions to be derived for unbounded operator coefficients, but fails to correctly predict the critical loads for high-frequency excitations. The other approach is based on transforming the equation under consideration in such a way that an appropriate ‘differential’ part of the new equation would possess some reserve of stability. Stability conditions for the transformed equation are obtained by using a technique of integral estimates. This method provides acceptable estimates of the critical forces for periodic loads, but can be applied to equations with bounded coefficients only. Combining these two approaches, we derive explicit stability conditions which are close to the Floquet criterion when the integral term vanishes. These conditions are applied to the stability problem for a viscoelastic bar compressed by periodic forces. The effect of material and structural parameters on the critical load is studied numerically. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Stability analysis and stabilization of LPV systems with jumps and (piecewise) differentiable parameters using continuous and sampled-data controllers

Linear Parameter-Varying (LPV) systems with jumps and piecewise differentiable parameters is a class of hybrid LPV systems for which no tailored stability analysis and stabilization conditions have been obtained so far.¹ We fill this gap here by proposing an approach based on a clock- and parameter-dependent Lyapunov function yielding stability conditions under both constant and minimum dwell-times. Interesting adaptations of the latter result consist of a minimum dwell-time stability condition for uncertain LPV systems and LPV switched impulsive systems. The minimum dwell-time stability condition is notably shown to naturally generalize and unify the well-known quadratic and robust stability criteria all together. Those conditions are then adapted to address the stabilization problem via timer-dependent and a timer- and/or parameter-independent (i.e. robust) state-feedback controllers, the latter being obtained from a relaxed minimum dwell-time stability condition involving slack-variables. Finally, the last part addresses the stability of LPV systems with jumps under a range dwell-time condition which is then used to provide stabilization conditions for LPV systems using a sampled-data state-feedback gain-scheduled controller. The obtained stability and stabilization conditions are all formulated as infinite-dimensional semidefinite programming problems which are then solved using sum of squares programming. Examples are given for illustration.     

On delay-dependent stability for a class of nonlinear stochastic systems with multiple state delays

Global asymptotic stability conditions for nonlinear stochastic systems with multiple state delays are obtained based on the convergence theorem for semimartingale inequalities, without assuming the Lipschitz conditions for nonlinear drift functions. The Lyapunov–Krasovskii and degenerate functionals techniques are used. The derived stability conditions are directly expressed in terms of the system coefficients. Nontrivial examples of nonlinear systems satisfying the obtained stability conditions are given.     

Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks

This paper addresses the local and global stability of n-dimensional Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks. Necessary and sufficient conditions for local stability independent of the choice of the delay functions are given, by imposing a weak nondelayed diagonal dominance which cancels the delayed competition effect. The global asymptotic stability of positive equilibria is established under conditions slightly stronger than the ones required for the linear stability. For the case of monotone interactions, however, sharper conditions are presented. This paper generalizes known results for discrete delays to systems with distributed delays. Several applications illustrate the results.     

The stability of the equilibrium position of scleronomous mechanical systems

The problem of the stability of the equilibrium position of a scleronomic mechanical system is considered. The comparison method enables this problem to be reduced to the problem of the stability of scalar differential equations. The stability conditions are found for certain types of scalar comparison equations (Sections 1–4), and the sufficient conditions for the stability of the equilibrium positions of various scleronomous mechanical systems are determined from these (Sections 5–9).     

Stability of regime-switching diffusions

This work is devoted to stability of regime-switching diffusion processes. After presenting the formulation of regime-switching diffusions, the notion of stability is recalled, and necessary conditions for p$p$-stability are obtained. Then main results on stability and instability for systems arising in approximation are presented. Easily verifiable conditions are established. An example is examined as a demonstration. A remark on linear systems is also provided.     

Stability of semilinear stochastic evolution equations

Stability of moments of the mild solution of a semilinear stochastic evolution equation is studied and sufficient conditions are given for the exponential stability of the pth moment in terms of Liapunov function. Sufficient conditions for sample continuity of the solution are also obtained and the exponential stability of sample paths is proved. Three examples are given to illustrate the theory.     

Stability of the Radau IA and Lobatto IIIC methods for delay differential equations

Recently, we have proved that the Radau IA and Lobatto IIIC methods are P-stable, i.e., they have an analogous stability property to A-stability with respect to scalar delay differential equations (DDEs). In this paper, we study stability of those methods applied to multidimensional DDEs. We show that they have a similar property to P-stability with respect to multidimensional equations which satisfy certain conditions for asymptotic stability of the zero solutions. The conditions are closely related to stability criteria for DDEs considered in systems theory. Received October 8, 1996 / Revised version received February 21, 1997     

Stability Conditions of the MMAP [ K ]/ G [ K ]/1/ LCFS Preemptive Repeat Queue

In this paper, we study the stability conditions of the MMAP[K]/G[K]/1/LCFS preemptive repeat queue. We introduce an embedded Markov chain of matrix M/G/1 type with a tree structure and identify conditions for the Markov chain to be ergodic. First, we present three conventional methods for the stability problem of the queueing system of interest. These methods are either computationally demanding or do not provide accurate information for system stability. Then we introduce a novel approach that develops two linear programs whose solutions provide sufficient conditions for stability or instability of the queueing system. The new approach is numerically efficient. The advantages and disadvantages of the methods introduced in this paper are analyzed both theoretically and numerically.     

The Ziegler effect in a non-conservative mechanical system

The destabilization of the stable equilibrium of a non-conservative system under the action of an infinitesimal linear viscous friction force is considered. In the case of low friction, the necessary and sufficient conditions for stability of a system with several degrees of freedom and, as a consequence, the conditions for the existence of the destabilization effect (Ziegler's effect) are obtained. Criteria for the stability of the equilibrium of a system with two degrees of freedom, in which the friction forces take arbitrary values, are constructed. The results of the investigation are applied to the problem of the stability of a two-link mechanism on a plane, and the stability regions and Ziegler's areas are constructed in the parameoter space of the problem.     

Über Die Stabilitätsdefinition Für Differenzengleichungen Die Partielle Differentialgleichungen Approximieren

For difference equations with constant coefficients necessary and sufficient algebraic stability conditions are given for the stability definitions used by G. Forsythe and W. Wasow (A) and P. D. Lax and R. D. Richtmyer (B). The application of these conditions for difference equations with variable coefficients is considered and it is shown that the stability condition of definitionA is not sufficient for stability. The same is true with respect to the definitionB if the difference equations are not parabolic and do not approximate first order systems. Therefore another stability definition is proposed and a number of properties are discussed.     

On the Existence and Stability of Spatially Structured Solutions of the Reaction-Diffusion Equations

The reaction-diffusion equations for the well-known ‘Brusselator’chemical kinetic model are investigated when the model is madeconsistent with the principle of detailed balance. In contrastto the original model, the corrected one does not show solutionswith ‘spatial structure’ i.e. solutions with multipleinternal maxima and multiple internal global minima in bothdependent variables. Sufficient conditions for stability ofthe solutions are given in terms of a Rayleigh quotient forgeneral boundary conditions for nonlinear reaction-diffusionequations in general. For the particular case of the ‘Brusselator’it is shown that conditions for a change of stability are muchmore unlikely to be attained as a result of the restrictionsof detailed balancing. The detailed sufficiency condition forthe stability of any steady-state solution and for no branchingfrom the ‘equilibrium’ branch solution depends onwhether the solution has global extrema inside the region, onits boundary, or both     

Stability of stationary solutions of a semilinear parabolic partial differential equation

We consider a parabolic partial differential equation u_t = u_xx + f(u) on a compact interval of spatial variable x with Dirichlet boundary conditions. The stability of stationary solutions of this system is studied by the use of Liapunov's second method. We obtain necessary and sufficient conditions for the stability, asymptotic stability, neutral stability, instability, and conditional stability. These conditions are closely connected with the conditions for the existence of the stationary solutions.     

On the stability of the σ-scheme with transparent boundary conditions for parabolic equations

Initial-boundary value problems for self-adjoint parabolic equations on a semiaxis and a semibounded strip are considered. For finite-difference σ-schemes, an alternative method for stating approximate transparent boundary conditions is suggested and conditions ensuring unconditional stability in the energy norm with respect to the initial data and free terms for a weight σ ≥ 1/2 are presented. The validity of these stability conditions in the case of discrete transparent boundary conditions is proved (by several methods), and the derivation of the latter conditions is revisited. Published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2007, Vol. 47, No. 4, pp. 671–692. This article was translated by the author.     

Sign-fixedness of third-order forms in a cone

In a study of the stability of solution of differential equations systems, third-order forms can be used. The necessary and sufficient conditions are formulated for the sign-fixedness of third-order forms in a cone; these conditions are used in a study of the monotonic stability of a dynamic system.Translated from Dinamicheskie Sistemy, No. 4, pp. 20–25, 1985.     

On the stability of inhomogeneously ageing viscoelastic plates

A method of investigation is proposed and conditions are set up for the stability of viscoelastic inhomogeneously ageing plates of arbitrary shape with a common creep kernel. The form of the stability conditions is found as a function of the surface forces. The stability problem is examined numerically in a finite time interval. The paper touches on the investigations in /1–3/. (See the bibliography of research on the stability of homogeneous viscoelastic systems in /1–5/, for example.)     

General conditions for global stability in a single species population-toxicant model

We deal with the global stability for a well-known population-toxicant model. We make use of a geometrical approach to the global stability analysis for ordinary differential equation which is based on the use of a higher-order generalization of the Bendixson's criterion. We obtain sufficient conditions for the global stability of the unique nontrivial equilibrium. These conditions are expressed in terms of a generic functional describing the population dynamics. In the special case of a logistic-like population dynamics, we get conditions which improve the ones previously known, obtained by means of the Lyapunov direct method.     

About the stability of high dimensional systems of functional differential equations

Summary In this work, using comparison techniques we obtain sufficient conditions for the stability of the trivial solution of a composite system of neutral type. Moreover, we establish necessary and sufficient conditions for the stability of the null solution of a system of retarded type in hierarchical form. Practical applications are also considered.     

The stability of generalized Volterra equations

In this paper, a particular type of a system of generalized Volterra equations [1], whose solutions are assured to be nonnegative for arbitrary nonnegative initial values, is considered. The extended stability theorem of LaSalle is used for deriving conditions for a nonnegative equilibrium point to be stable with respect to a certain subset of the Euclidean space. The obtained stability theorem has a close relation with Lyapunov's stability condition for linear systems with constant coefficients and is generally less restrictive than conditions known so far.