A spectral characterization of exponential stability for linear time-invariant systems on time scales

Pötzsche, Christian; Siegmund, Stefan; Wirth, Fabian

doi:10.3934/dcds.2003.9.1223

Cited by 113 publications

(57 citation statements)

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“…, which together with equation (8) agrees with the form given by (6). Based on the results of Pötzsche et al [31], we make an analysis of the exponential stability of equation (5) in case of a constant function u(·). In this case, in the formula of solution (7), it disappears the part u(t) − u(0).…”

Section: Solutions Of Linear Equations With Cf Fractional Delta Derivsupporting

confidence: 55%

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Solutions of systems with the Caputo–Fabrizio fractional delta derivative on time scales

Mozyrska

Torres

Wyrwas

2019

Nonlinear Analysis: Hybrid Systems

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Caputo-Fabrizio fractional delta derivatives on an arbitrary time scale are presented. When the time scale is chosen to be the set of real numbers, then the Caputo-Fabrizio fractional derivative is recovered. For isolated or partly continuous and partly discrete, i.e., hybrid time scales, one gets new fractional operators. We concentrate on the behavior of solutions to initial value problems with the Caputo-Fabrizio fractional delta derivative on an arbitrary time scale. In particular, the exponential stability of linear systems is studied. A necessary and sufficient condition for the exponential stability of linear systems with the Caputo-Fabrizio fractional delta derivative on time scales is presented. By considering a suitable fractional dynamic equation and the Laplace transform on time scales, we also propose a proper definition of Caputo-Fabrizio fractional integral on time scales. Finally, by using the Banach fixed point theorem, we prove existence and uniqueness of solution to a nonlinear initial value problem with the Caputo-Fabrizio fractional delta derivative on time scales.

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Section: Solutions Of Linear Equations With Cf Fractional Delta Derivsupporting

confidence: 55%

“…For simplicity, we take t 0 = 0 ∈ T. Proof. The result is obtained following the proof of [31,Theorem 21]. h .…”

Section: Solutions Of Linear Equations With Cf Fractional Delta Derivmentioning

confidence: 99%

Solutions of systems with the Caputo–Fabrizio fractional delta derivative on time scales

Mozyrska

Torres

Wyrwas

2019

Nonlinear Analysis: Hybrid Systems

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“…Let us show that condition (iv) of this theorem guarantees that inequality (28) holds. For this purpose, we will prove that Sp( 22 (t) −  T 12 (t) −1 11 (t) 12 (t)) ⊂ ℝ + , for all t ∈ .…”

Section: Lemmamentioning

confidence: 99%

“…Later, the time scale theory has been developed in several directions. [25][26][27][28][29] In particular, the monograph of Martynyuk 29 has presented the direct Lyapunov method, integral inequalities, and the comparison method for the stability analysis of the solutions of dynamic equations on time scale.…”

Section: Introductionmentioning

confidence: 99%

Distributed leader‐follower consensus for a class of semilinear second‐order multiagent systems using time scale theory

Babenko

Defoort

Djemaï

et al. 2018

Intl J Robust & Nonlinear

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The problem of distributed leader-follower consensus for second-order linear multiagent systems with unknown nonlinear inherent dynamics is investigated in this paper. It is assumed that the dynamic of each agent is described by a semilinear second-order dynamic equation on an arbitrary time scale. Using calculus on time scales and direct Lyapunov's method, some sufficient conditions are derived to ensure that the tracking errors exponentially converge to zero. Some numerical results show the effectiveness of the proposed scheme.

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“…(which is positive) for λ ∈ (0, 1) R . To this end, we will prove that e −λA (σ(t), t) ≤ e ⊖λ 1 (σ(t), t) for t ∈ T and λ ∈ (0, 1) R (43) or equivalently…”

Section: −2λbmentioning

confidence: 99%

On Different Types of Stability for Linear Delay Dynamic Equations

Braverman¹

2017

Z. Anal. Anwend.

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A spectral characterization of exponential stability for linear time-invariant systems on time scales

Cited by 113 publications

References 0 publications

Solutions of systems with the Caputo–Fabrizio fractional delta derivative on time scales

Solutions of systems with the Caputo–Fabrizio fractional delta derivative on time scales

Distributed leader‐follower consensus for a class of semilinear second‐order multiagent systems using time scale theory

On Different Types of Stability for Linear Delay Dynamic Equations

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