Controllability of Neutral Differential Equation with Impulses on Time Scales

Malik, Muslim; Kumar, Vipin

doi:10.1007/s12591-019-00454-2

Cited by 16 publications

(8 citation statements)

References 16 publications

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“…Theorem 1 (Rothe's Fixed Point Theorem [1][2][3]6,17 ). Let E be a Banach space, and consider B ⊂ E a convex closed subset containing the zero of E in its interior.…”

Section: Characterization Of the Linear Systemmentioning

confidence: 99%

“…We begin with the definition of a natural Banach spaces where this type of problems can be set 2–7

\begin{array}{cc}\hfill \mathcal{P}{\mathcal{W}}_{t_1..{t}_p}&#x0003D;\left\{\right.&amp; z:\left[0,\tau \right]\to {\mathbb{R}}&#x0005E;n\kern3.0235pt \mid \kern3.0235pt z\kern0.5em \mathrm{is}\ \mathrm{continuous}\ \mathrm{except}\ \mathrm{at}\ \mathrm{the}\ \mathrm{points}\kern0.5em {t}_k,\kern3.0235pt k&#x0003D;1,\dots, p,\hfill \\ {}\hfill &amp; \left.\mathrm{where}\ \mathrm{the}\ \mathrm{one}\hbox{-} \mathrm{sided}\ \mathrm{limits}\kern0.5em z\left({t}_k&#x0005E;{-}\right),z\left({t}_k&#x0005E;{&#x0002B;}\right)\kern0.5em \mathrm{exist}\ \mathrm{and}\kern0.5em z\left({t}_k&#x0005E;{&#x0002B;}\right)&#x0003D;z\left({t}_k\right)\right\}\hfill \end{array}

equipped with the supremum norm, and

scriptP {scriptW}_{τ} = ({}, η : false[- r, τ false] \to ℝ^{n} false| η {false|}_{false[- r, 0 false]} \in scriptP {scriptW}_{r} and η {false|}_{false[0, τ false]} \in scriptP {scriptW}_{t_{1} . .<...})

…”

Section: Preliminariesmentioning

confidence: 99%

“…Under the above hypotheses, for all

\eta \in \mathcal{P}{\mathcal{W}}_r

and

u\in C\left(\left[0,\tau \right];{\mathbb{R}}&#x0005E;m\right)

the system () admits one solution

z(t)&#x0003D;z\left(t,\eta, u\right)

given by 3,5

z false(t false) = ({, \begin{matrix} array E (t, 0) [η (0) - h (z_{τ_{1}}, z_{τ_{2}}, \dots, z_{τ_{q}}) (0) - f_{- 1} (0, η - h (z_{τ_{1}}, z_{τ_{2}}, \dots, z_{τ_{q}}))] \\ array + f_{- 1} (t, z_{t}) + {true\int}_{0}^{t} E (t, θ) [A_{0} (θ) f_{- 1} (θ, z_{θ}) + f_{1} (θ, z_{θ}, u (θ))] d θ \\ arr... \end{matrix})

…”

Section: Main Theoremmentioning

confidence: 99%

“…It will confirm the conjecture that impulses or abrupt changes, delays, and nonlocal conditions on a system are intrinsic phenomena. This means that under certain conditions, they do not destroy some properties of the original system, such as controllability, [1][2][3] which is the objective of this work.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Controllability of semilinear neutral differential equations with impulses and nonlocal conditions

Camacho

Leiva

Riera-Segura

2022

Math Methods in App Sciences

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When a real‐life problem is mathematically modeled by differential equations or another type of equation, there are always intrinsic phenomena that are not taken into account and can affect the behavior of such a model. For example, external forces can abruptly change the model; impulses and delay can cause a breakdown of it. Considering these intrinsic phenomena in the mathematical model makes the difference between a simple differential equation and a differential equation with impulses, delay, and nonlocal conditions. So, in this work, we consider a semilinear nonautonomous neutral differential equation under the influence of impulses, delay, and nonlocal conditions. In this paper we study the controllability of these semilinear neutral differential equations with some of these intrinsic phenomena taking into consideration. Our aim is to prove that the controllability of the associated ordinary linear differential equation is preserved under certain conditions imposed on these new disturbances. In order to achieve our objective, we apply Rothe's fixed point Theorem to prove the exact controllability of the system. Finally, our method can be extended to the evolution equation in Hilbert spaces with applications to control systems governed by PDE's equations.

show abstract

“…Theorem 1 (Rothe's Fixed Point Theorem [1][2][3]6,17 ). Let E be a Banach space, and consider B ⊂ E a convex closed subset containing the zero of E in its interior.…”

Section: Characterization Of the Linear Systemmentioning

confidence: 99%

“…We begin with the definition of a natural Banach spaces where this type of problems can be set 2–7

\begin{array}{cc}\hfill \mathcal{P}{\mathcal{W}}_{t_1..{t}_p}&#x0003D;\left\{\right.&amp; z:\left[0,\tau \right]\to {\mathbb{R}}&#x0005E;n\kern3.0235pt \mid \kern3.0235pt z\kern0.5em \mathrm{is}\ \mathrm{continuous}\ \mathrm{except}\ \mathrm{at}\ \mathrm{the}\ \mathrm{points}\kern0.5em {t}_k,\kern3.0235pt k&#x0003D;1,\dots, p,\hfill \\ {}\hfill &amp; \left.\mathrm{where}\ \mathrm{the}\ \mathrm{one}\hbox{-} \mathrm{sided}\ \mathrm{limits}\kern0.5em z\left({t}_k&#x0005E;{-}\right),z\left({t}_k&#x0005E;{&#x0002B;}\right)\kern0.5em \mathrm{exist}\ \mathrm{and}\kern0.5em z\left({t}_k&#x0005E;{&#x0002B;}\right)&#x0003D;z\left({t}_k\right)\right\}\hfill \end{array}

equipped with the supremum norm, and

scriptP {scriptW}_{τ} = ({}, η : false[- r, τ false] \to ℝ^{n} false| η {false|}_{false[- r, 0 false]} \in scriptP {scriptW}_{r} and η {false|}_{false[0, τ false]} \in scriptP {scriptW}_{t_{1} . .<...})

…”

Section: Preliminariesmentioning

confidence: 99%

“…Under the above hypotheses, for all

\eta \in \mathcal{P}{\mathcal{W}}_r

and

u\in C\left(\left[0,\tau \right];{\mathbb{R}}&#x0005E;m\right)

the system () admits one solution

z(t)&#x0003D;z\left(t,\eta, u\right)

given by 3,5

z false(t false) = ({, \begin{matrix} array E (t, 0) [η (0) - h (z_{τ_{1}}, z_{τ_{2}}, \dots, z_{τ_{q}}) (0) - f_{- 1} (0, η - h (z_{τ_{1}}, z_{τ_{2}}, \dots, z_{τ_{q}}))] \\ array + f_{- 1} (t, z_{t}) + {true\int}_{0}^{t} E (t, θ) [A_{0} (θ) f_{- 1} (θ, z_{θ}) + f_{1} (θ, z_{θ}, u (θ))] d θ \\ arr... \end{matrix})

…”

Section: Main Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Controllability of semilinear neutral differential equations with impulses and nonlocal conditions

Camacho

Leiva

Riera-Segura

2022

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…In the last decades, the problem of controllability for various types of differential equations has been widely studied by many authors; see for instance previous studies [18][19][20][21][22][23]. Further, the controllability results for differential equations on time scales is a new area of research and only few authors have been established these results [24][25][26][27][28][29][30][31]. However, these results cannot be easily extended to the case of switched systems with non-instantaneous impulses.…”

Section: Introductionmentioning

confidence: 99%

Total controllability results for a class of time‐varying switched dynamical systems with impulses on time scales

Kumar

Djemaï

Defoort

et al. 2020

Asian Journal of Control

Self Cite

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In this paper, we establish some necessary and sufficient conditions of the controllability results for a class of impulsive switched systems with noninstantaneous jumps on time scales. Firstly, we define the solution of the considered problem by using the parameter variation method. We also define some Gramian matrices to establish the controllability results. Moreover, we give some conditions under which the time invariant impulsive switched system is totally controllable. At the end, we provide one numerical example for different time scales to validate the obtained analytical results.

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New discussion on the existence and controllability of fractional evolution inclusion of order 1 < r < 2 without compactness

Williams

Vijayakumar

2023

Math Methods in App Sciences

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In this article, we investigate the existence and controllability results for fractional evolution inclusions in Banach spaces. By eliminating compactness hypotheses on the semigroup produced by the linear component and any requirements on the multivalued nonlinearity defined in terms of measures of noncompactness, we apply a new method to arrive at the existence of mild solutions and controllability results. In fact, to satisfy the existence and controllability requirements, we propose a fixed-point strategy for weakly sequentially closed maps. Finally, a concrete example is provided to highlight our theoretical findings.

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Controllability of Neutral Differential Equation with Impulses on Time Scales

Cited by 16 publications

References 16 publications

Controllability of semilinear neutral differential equations with impulses and nonlocal conditions

Controllability of semilinear neutral differential equations with impulses and nonlocal conditions

Total controllability results for a class of time‐varying switched dynamical systems with impulses on time scales

New discussion on the existence and controllability of fractional evolution inclusion of order 1 < r < 2 without compactness

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