On the stability of certain perturbed systems of differential equations and the relationship with the magnitude of the perturbation (original) (raw)
2010, Revista de Matemática: Teoría y Aplicaciones
In this work we consider a class of polytopes of third order square matrices, studied early. We obtain a condition to guarantee Hurwitz stability of each of elements of the polytope. This condition is more simples than one obtained before. Taking into account that to the considered set of matrices correspond a family of perturbed systems of differential equations, we study the relationship between the stability condition and the magnitude of the class of perturbations considered for this family. Resumen En el presente trabajo consideramos una clase de politopos de matrices cuadradas de tercer orden, estudiada anteriormente. Obten-emos una condición para garantizar la estabilidad, según Hurwitz, de cada uno de los elementos del politopo. Dicha condición es más simple que la obtenida con anterioridad. Teniendo en cuenta que al con-junto considerado de matrices corresponde una familia de ecuaciones diferenciales perturbada, estudiamos la relación entre la condición de estabilidad y la magnitud de la clase de perturbaciones considerada para esta familia.
Sign up for access to the world's latest research.
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Related papers
On the practical stability of dependent parameter perturbed systems
Publicationes Mathematicae Debrecen, 2010
We investigate in this paper the global uniform practical asymptotic stability for a class of dependent parameter perturbed systems where the associated linear system is globally exponentially stable and the perturbation term is subject to two conditions. An example is given to illustrate the result of this paper formulated in the form of Linear matrix inequalities.
On the stability of convex symmetric polytopes of matrices
Electronic Journal of …, 2000
In this article we investigate the stability properties of a convex symmetric time-varying second-order matrix's polytope depending on a real positive parameter. We apply the results obtained to the calculation of the stability radius of a second order matrix under affine general perturbations, and under linear structured multiple perturbations.
Note on the stability for linear systems of differential equations
In this paper, by applying the fixed point alternative method, we give a necessary and sufficient condition in order that the first order linear system of differential equationsż(t) + A(t)z(t) + B(t) = 0 has the Hyers-Ulam-Rassias stability and find Hyers-Ulam stability constant under those conditions. In addition to that we apply this result to a second order differential equationÿ(t) + f (t)ẏ(t) + g(t)y(t) + h(t) = 0.
A general framework for the perturbation theory of matrix equations
2021
A general framework is presented for the local and non-local perturbation analysis of general real and complex matrix equations in the form F(P,X)=0F(P,X) = 0F(P,X)=0, where FFF is a continuous, matrix valued function, PPP is a collection of matrix parameters and XXX is the unknown matrix. The local perturbation analysis produces condition numbers and improved first order homogeneous perturbation bounds for the norm ∣deX∣\|\de X\|∣deX∣ or the absolute value ∣deX∣|\de X|∣deX∣ of deX\de XdeX. The non-local perturbation analysis is based on the method of Lyapunov majorants and fixed point principles. % for the operator pi(p,cdot)\pi(p,\cdot)pi(p,cdot). It gives rigorous non-local perturbation bounds as well as conditions for solvability of the perturbed equation. The general framework can be applied to various matrix perturbation problems in science and engineering. We illustrate the procedure with several simple examples. Furhermore, as a model problem for the new framework we derive a new perturbation theory for continuous-time algebr...
2009
Introduction and notation. The sensitivity of computational problems is a major factor determining the accuracy of computations in machine arithmetic. It may be revealed and taken into account by the methods of perturbation analysis [14, 6]. Below we consider the technique of Lyapunov majorants for perturbation analysis of algebraic matrix equations F (A, X) = 0 arising in science and engineering, where A is a matrix parameter and X is the solution. We shall use the following notations: i := √ −1 – the imaginary unit; R and C – the spaces of m × n matrices over the field of real R and complex C numbers; R = R, In – the identity n×n matrix; A, A and A = A > – the complex conjugate, the transpose and the complex conjugate transpose of the matrix A, respectively; vec(A) – the column–wise vectorization of the matrix A; A ⊗B – the Kronecker product of the matrices A and B; ‖ ·‖ – a vector or a matrix norm; ‖ ·‖F and ‖ ·‖2 – the Frobenius norm and the 2–norm of a matrix or a vector, re...
Stability of polytopic polynomial matrices
Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148), 2001
This paper gives a necessary and sufficient condition for robust D-stability of Polytopic Polynomial Matrices.
Robust stability of linear systems described by higher-order dynamic equations
IEEE Transactions on Automatic Control, 2000
In this note we study the stability radius of higher order di erential and difference systems with respect to various classes of complex a ne perturbations of the coe cient matrices. Di erent perturbation norms are considered. The aim is to derive robustness criteria which are expressed directly in terms of the original data. Previous results on robust stability of Hurwitz and Schur polynomials 13] are extended to monic matrix polynomials. For disturbances acting via a uniform input matrix, computable formulae are obtained whereas for perturbations with multiple input matrices structured singular values are involved.
The Stability of Linear Systems
2018
This paper examines the relation of the exponential dichotomy and the stability concepts for systems of linear differential equations. We are going to show some relationship between the studied concepts, more precisely we are presenting how the stability of a linear non-autonomous system is investigated with the help of the exponential dichotomy. Furthermore we are going to show how the stable and unstable subspace of an exponentially dichotomic system can be specified using the definition of the exponential dichotomy.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.