Algebraic Conditions For Stability Research Articles

Algebraic conditions for stability in Runge-Kutta methods and their certification via semidefinite programming

In this work, we present approaches to rigorously certify A- and A(α)-stability in Runge-Kutta methods through the solution of convex feasibility problems defined by linear matrix inequalities. We adopt two approaches. The first is based on sum-of-squares programming applied to the Runge-Kutta E-polynomial and is applicable to both A- and A(α)-stability. In the second, we sharpen the algebraic conditions for A-stability of Cooper, Scherer, Türke, and Wendler to incorporate the Runge-Kutta order conditions. We demonstrate how the theoretical improvement enables the practical use of these conditions for certification of A-stability within a computational framework. We then use both approaches to obtain rigorous certificates of stability for several diagonally implicit schemes devised in the literature.

An approach to the stability of reset switched systems using dynamics without reset

In this paper, we consider switched linear systems that have jumps in the state (determined by a reset rule) at each switching instant. Those systems are called switched systems with state reset. We show that it is possible to analyze the state trajectories of a switched system with state reset by considering suitable switched systems without state resets, that is, with continuous state trajectories. We establish sufficient conditions for the stability of switched systems with state reset using the associated systems without reset. Moreover, introducing some restrictions, we also derive simple algebraic stability conditions. The obtained results are used in numerical examples to show the effectiveness of the proposed approach.

The space of stability conditions on the projective plane

The space of Bridgeland stability conditions on the bounded derived category of coherent sheaves on mathbf {P}^2 has a principal connected component hbox {Stab}^dag (mathbf{P }^2). We show that hbox {Stab}^dag (mathbf{P }^2) is the union of geometric and algebraic stability conditions. As a consequence, we give a cell decomposition for hbox {Stab}^dag (mathbf{P }^2) and show that hbox {Stab}^dag (mathbf{P }^2) is contractible.

Sufficient algebraic conditions for stability of cones of polynomials

In this paper a sufficient condition for a cone of polynomials to be Hurwitz is established. Such condition is a matrix inequality, which gives a simple algebraic test for the stability of rays of polynomials. As an application to stable open-loop systems, a cone of gains c such that the function u=− kc T x is a stabilizing control feedback for all k>0 is shown to exist.

On absolute stability analysis by polyhedral Lyapunov functions

This paper addresses the problem of computing a polyhedral Lyapunov function (PLF) for a linear differential inclusion (LDI). A numerically efficient algorithm is presented to compute a PLF for an LDI. It is also pointed out that for the case of a polyhedron defined by vertices (rather than by faces) one can state algebraic stability conditions analogous to those given by Molchanov and Pyatnitskiy (1986).

On the stability of arrays of precipitates

The problem of the stability of an array of elastically interacting plate-shaped precipitates is considered. Algebraic conditions for stability against coarsening are developed, and the results of detailed numerical tests of the stability against displacement and of the stability against coarsening, of a particularly simple and symmetric array are reported.

Stable Matching of Difference Schemes

Approximations that result from the natural matching of two stable dissipative difference schemes across a coordinate line are shown to be stable. The basic idea is to reformulate the matching scheme consistent to an equivalent initial boundary value problem and to verify the algebraic conditions for stability of such systems. An interesting comparison to the above result is the case of redefinition of a scheme at a single point. In particular, we show that some unstable perturbations do not upset the stability of the Lax–Wendroff scheme.

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.