On the exponential stability of discrete volterra systems

Crisci, M. R.; Kolimanovskll, V.B.; Russo, E.; Vecchio, Antonia

doi:10.1080/10236190008808251

Cited by 14 publications

(18 citation statements)

References 5 publications

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“…As Vecchio [35] remarked, the theory of this type of equations is still undeveloped and only recently a book on difference equations [19] includes few results on the linear summation equation (1.5). Contributions by Elaydi and his collaborators towards a solid foundation for Volterra difference equations include [16][17][18][19][20][21][22][23][24] and those by Kolmanovskii and his collaborators include [9][10][11][12][13][14], to mention a few; see also the recently published work [29]. The latter results include interesting applications to numerical methods of Volterra integral equations.…”

Section: Related Results; Additional Commentsmentioning

confidence: 97%

Linearized stability analysis of discrete Volterra equations

Song¹,

Baker

2004

Journal of Mathematical Analysis and Applications

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Various different types of stability are defined, in a unified framework, for discrete Volterra equations of the type x(n) = f (n) + n j =0 K(n, j, x(n)) (n 0). Under appropriate assumptions, stability results are obtainable from those valid in the linear case (K(n, j, x(n)) = B(n, j )x(j )), and a linearized stability theory is studied here by using the fundamental and resolvent matrices. Several necessary and sufficient conditions for stability are obtained for solutions of the linear equation by considering the equations in various choices of Banach space B, the elements of which are sequences of vectors (x(n), f (n) ∈ E d , B(n, j ) : E d → E d , n, j 0, etc.). We show that the theory, including a number of new results as well as results already known, can be presented in a systematic framework, in which results parallel corresponding results for classical Volterra integral equations of the second kind.

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Section: Related Results; Additional Commentsmentioning

confidence: 97%

Linearized stability analysis of discrete Volterra equations

Song¹,

Baker

2004

Journal of Mathematical Analysis and Applications

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“…the phase space B, and UE stability in the sense of resolvent matrix. The definitions are given in Section 2.2 in accordance with [10,11,4,9]. Note that usually exponential stability for Volterra difference systems (1.2) is understood in the following way.…”

Section: Introductionmentioning

confidence: 99%

“…It seems that at least some of these tests are new (related problems were actively discussed e.g. in [11,20,9]).…”

Section: Introductionmentioning

confidence: 99%

Structured stability radii and exponential stability tests for Volterra difference systems

Braverman

Karabash

2013

Computers & Mathematics with Applications

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Uniform exponential (UE) stability of linear difference equations with infinite delay is studied using the notions of a stability radius and a phase space. The state space X is supposed to be an abstract Banach space. We work both with non-fading phase spaces c 0 (Z − , X ) and ℓ ∞ (Z − , X ) and with exponentially fading phase spaces of the ℓ p and c 0 types. For equations of the convolution type, several criteria of UE stability are obtained in terms of the Z-transform K(ζ) of the convolution kernel K(·), in terms of the input-state operator and of the resolvent (fundamental) matrix. These criteria do not impose additional positivity or compactness assumptions on coefficients K(j). Time-varying (non-convolution) difference equations are studied via structured UE stability radii r t of convolution equations. These radii correspond to a feedback scheme with delayed output and time-varying disturbances. We also consider stability radii r c associated with a time-invariant disturbance operator, unstructured stability radii, and stability radii corresponding to delayed feedback. For all these types of stability radii two-sided estimates are obtained. The estimates from above are given in terms of the Z-transform K(ζ), the estimate from below via the norm of the input-output operator. These estimates turn into explicit formulae if the state space X is Hilbert or if disturbances are time-invariant. The results on stability radii are applied to obtain various exponential stability tests for non-convolution equations. Several examples are provided.

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“…For example, the boundedness of solutions of discrete Volterra equations was studied in [2,5,10] or [13]- [18], the periodicity was investigated in papers [6,8,15,18]. A survey of the fundamental results on the stability of linear Volterra difference equations, of both convolution and non-convolution type, can be found in [7], see also [3,4,11,12,17] or [19]. In [3] and [4] the authors study the exponential stability of equation…”

Section: A(t S)x(s)ds + F (T)mentioning

confidence: 99%

“…A survey of the fundamental results on the stability of linear Volterra difference equations, of both convolution and non-convolution type, can be found in [7], see also [3,4,11,12,17] or [19]. In [3] and [4] the authors study the exponential stability of equation…”

Section: A(t S)x(s)ds + F (T)mentioning

confidence: 99%