Stability of impulsive delay differential equations

Math Methods in App Sciences

O’Regan

et al. 2018

Self Cite

MSC Classification: 34A37; 93B05This paper investigates the relative controllability of delay differential systems with linear impulses and linear parts defined by permutable matrices. We use the impulsive delay Grammian matrix to discuss the relatively controllability of impulsive linear delay controlled systems and we use the Krasnoselskii's fixed point theorem to discuss the relatively controllability of impulsive semilinear delay controlled systems. Finally, two examples are presented to illustrate our theoretical results. KEYWORDS impulsive delay differential equations, impulsive delay Grammian matrix, relative controllability INTRODUCTIONMathematical control theory is an important area of research and classical differential controlled systems in finite and infinite dimensional spaces have been discussed in the literature. However, very little is known for delay differential controlled systems and impulsive delay controlled systems. 954

“…Let

V false(\cdot, \cdot false) : double-struckR \times double-struckR \to R^{n \times n}

(see) be given by

V false(t, s false) = e^{A false(t - s false)} X false(t, s + ϑ false), t > s,

and

X false(t, s false) = e_{ϑ}^{\overset{B}{true} false(t - s false)} + \sum_{s - ϑ < t_{j} \leq t} C_{j} e_{ϑ}^{\overset{B}{true} false(t - ϑ - t_{j} false)} X false(t_{j}, s false), \overset{B}{true} = e^{- A ϑ} B .

…”

Section: Preliminariesmentioning

confidence: 99%

“…For any solution

ν \in C^{1} false(false[- ϑ, 0 false], R^{n} false) \cup ([], P C false(normalΩ, R^{n} false) \cap false(\cup_{i = 0}^{k} C^{1} false(t_{i}, t_{i + 1} false] false))

(here t 0 = 0 and t k + 1 = τ 1 ) of , from,, Corollary 3.3 we obtain the following:

\begin{matrix} right ν (t) & left = V (t, - ϑ) ψ (- ϑ) + \int_{- ϑ}^{0} V (t, s) [ψ^{'} (s) - A ψ (s)] d s & right \\ right & left + \int_{0}^{t} V (t, s) [f (s, ν (s)) + D u (s)] d s . \end{matrix}

…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…First, we investigate the relative controllability of the linear impulsive delay controlled system as follows:

({, \begin{matrix} array ν^{'} (t) = A ν (t) + B ν (t - ϑ) + D u (t), t \in Ω, t \notin 𝒯, ϑ > 0, \\ array Δ ν (t_{i}) = C_{i} ν (t_{i}), t_{i} \in 𝒯, \\ array ν (t) = ψ (t), - ϑ \leq t \leq 0, \end{matrix})

using the impulsive delay Cauchy matrix introduced in . Next, we construct a suitable control function for , which means that we give a condition necessary and sufficient for

u \in L^{2} false(normalΩ, R^{n} false)

to lead the solution of to ν 1 at the time τ 1 .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Relative controllability of delay differential systems with impulses and linear parts defined by permutable matrices

You

Math Methods in App Sciences

O’Regan

et al. 2018

Self Cite

“…For the theory of impulsive differential equations, we refer the reader to the monographs of Bainov and Simeonov (1993) and Samoilenko et al (1995) and for applications to the works of Benchohra et al (2006) and Ivanov and Slyn’ko (2009) and the references therein. You and Wang (2016) introduced the notation of an impulsive delayed matrix function and used the variation of constants method to obtain the representation of solutions to linear impulsive delay differential equations. However, there are only a few papers in the literature seeking the explicit solution of linear impulsive multi-delay differential equations with permutable matrices.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic stability of solutions of impulsive multi-delay differential equations

You

Transactions of the Institute of Measurement and Control

O’Regan

2018

Self Cite

In this paper, we consider the asymptotic stability of solutions to impulsive multi-delayed differential equations with linear parts defined by pairwise permutable matrices. First, we introduce the concept for an impulsive multi-delayed Cauchy matrix and then use it to obtain the representation of solutions to linear impulsive Cauchy problems via the variation of constants principle. Next, we give a norm estimate of the impulsive multi-delayed Cauchy matrix and establish sufficient conditions to guarantee that the trivial solutions are asymptotically stable when the nonlinear terms satisfy appropriate conditions. Finally, two numerical examples are given to illustrate the effectiveness of the results.

Bifurcation type phenomena for positive solutions of a class of impulsive differential equations

Math Methods in App Sciences

Zuo

2023