# Practical Stability with Respect to h-Manifolds for Impulsive Control Functional Differential Equations with Variable Impulsive Perturbations

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- (a)
- $(\lambda ,A)$-practically stable with respect to the function h, if given $(\lambda ,A)$ with $0<\lambda <A$, we have ${\phi}_{0}\in {M}_{t,\nu}(n-l)\left(\lambda \right)$, implying $x(t;{t}_{0},{\phi}_{0})\in {M}_{t}(n-l)\left(A\right),t\ge {t}_{0}$ for some ${t}_{0}\in {\mathbb{R}}_{+}$;
- (b)
- $(\lambda ,A)$-uniformly practically stable with respect to the function h, if (a) Definition 1 holds for every ${t}_{0}\in {\mathbb{R}}_{+}$;
- (c)
- $(\lambda ,A)$-globally practically exponentially stable with respect to the function h, if given $(\lambda ,A)$ with $0<\lambda <A$ and ${\phi}_{0}\in {M}_{t,\nu}(n-l)\left(\lambda \right)$, there exist positive constants $\gamma ,\mu $:$$x(t;{t}_{0},{\phi}_{0})\in {M}_{t}(n-l)\left(\right)open="("\; close=")">A+\gamma \left|\right|h({t}_{0},\phi ){\left|\right|}_{\nu}{e}^{-\mu (t-{t}_{0})}$$

**Remark**

**1.**

**Definition**

**2.**

**Definition**

**3.**

- (a)
- $(\lambda ,A)$-practically stable, if given $(\lambda ,A)$ with $0<\lambda <A$, we have ${u}_{0}<\lambda $, implying ${u}^{+}(t;{t}_{0},{u}_{0})<A,$ $t\ge {t}_{0}$ for some ${t}_{0}\in {\mathbb{R}}_{+}$;
- (b)
- $(\lambda ,A)$-uniformly practically stable, if (a) Definition 3 holds for every ${t}_{0}\in {\mathbb{R}}_{+}$;
- (c)
- $(\lambda ,A)$-globally practically exponentially stable, if for given $(\lambda ,A)$ with $0<\lambda <A$ and ${u}_{0}<\lambda $, there exist positive constants $\gamma ,\mu $:$${u}^{+}(t;{t}_{0},{u}_{0})<A+\gamma {u}_{0}{e}^{-\mu (t-{t}_{0})},$$

**Definition**

**4.**

- 1.
- V is continuous in $\mathcal{G}$ and locally Lipschitz continuous with respect to its second argument on each of the sets ${\mathcal{G}}_{k}$, $k=1,2,\dots $.
- 2.
- For each $k=1,2,\dots $ and $({t}_{0}^{*},{x}_{0}^{*})\in {\sigma}_{k}$, there exist the finite limits:$$V({t}_{0}^{*}{}^{-},{x}_{0}^{*})=\underset{\genfrac{}{}{0pt}{}{(t,x)\to ({t}_{0}^{*},{x}_{0}^{*})}{(t,x)\in {\mathcal{G}}_{k}}}{lim}V(t,x),V({t}_{0}^{*}{}^{+},{x}_{0}^{*})=\underset{\genfrac{}{}{0pt}{}{(t,x)\to ({t}_{0}^{*},{x}_{0}^{*})}{(t,x)\in {\mathcal{G}}_{k+1}}}{lim}V(t,x)$$

**Lemma**

**1.**

- 1.
- The function $g:[{t}_{0},\infty )\times {\mathbb{R}}_{+}\to \mathbb{R}$ is continuous in each of the sets $({t}_{k-1},{t}_{k}]\times {\mathbb{R}}_{+},k=1,2,\dots $.
- 2.
- ${J}_{k}\in C[{\mathbb{R}}_{+},\mathbb{R}]$ and ${\psi}_{k}\left(u\right)=u+{J}_{k}\left(u\right)\ge 0,k=1,2,\dots $ are non-decreasing with respect to u.
- 3.
- The maximal solution ${u}^{+}(t;{t}_{0},{u}_{0})$ of the comparison problem (3) is defined on $[{t}_{0},\infty )$.
- 4.
- The function $V:[{t}_{0},\infty )\times \mathsf{\Omega}\to {\mathbb{R}}_{+}$, $V\in {V}_{0}$, is such that for $t\in [{t}_{0},\infty )$, $\phi \in \mathcal{PC}$:$$V({t}^{+},\phi \left(0\right)+{I}_{k}\left(\phi \right))\le {\psi}_{k}\left(V(t,\phi \left(0\right))\right),t={t}_{k},k=1,2,\dots ,$$$${D}^{+}V(t,\phi \left(0\right))\le g(t,V(t,\phi \left(0\right))),t\ne {t}_{k},k=1,2,\dots $$

## 3. Main Results

**Theorem**

**1.**

- 1.
- $0<\lambda <A$ are given.
- 2.
- The assumptions of Lemma 1 are satisfied for $x\in {M}_{t}(n-l)\left(A\right)$.
- 3.
- There exists $r=r\left(A\right)>0$ such that $x\in {M}_{t}(n-l)\left(A\right)$ implies $x+{I}_{k}\left(x\right)\in {M}_{t}(n-l)\left(r\right)$ for all $k=1,2,\dots $.
- 4.
- The inequalities:$${w}_{1}\left(\right|\left|h(t,x)\right|\left|\right)\le V(t,x)\le \chi \left(t\right){w}_{2}\left(\right|\left|h(t,x)\right|\left|\right),$$
- 5.
- The inequality $\chi \left({t}_{0}\right){w}_{2}\left(\lambda \right)<{w}_{1}\left(A\right)$ holds.

- (a)
- If System (3) is $(\chi \left({t}_{0}\right){w}_{2}\left(\lambda \right),{w}_{1}\left(A\right))$-practically stable, then System (1) is $(\lambda ,A)$-practically stable with respect to the function h.
- (b)
- If System (3) is $(\chi \left({t}_{0}\right){w}_{2}\left(\lambda \right),{w}_{1}\left(A\right))$-uniformly practically stable, then System (1) is $(\lambda ,A)$-uniformly practically stable with respect to the function h.

**Proof.**

**Theorem**

**2.**

**Proof.**

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

## 4. Applications and Examples

**Hypothesis**

**1**

**(H1).**

**Hypothesis**

**2**

**(H2).**

**Hypothesis**

**3**

**(H3).**

**Hypothesis**

**4**

**(H4).**

**Hypothesis**

**5**

**(H5).**

**Theorem**

**3.**

- 1.
- $0<\lambda <A$ are given, and there exists a positive constant M such that $\underset{1\le i\le n}{max}{I}_{i}<M$.
- 2.
- Conditions H1–H5 hold.
- 3.
- There exists a positive constant μ such that:$$\underset{1\le i\le n}{min}({c}_{i}-{K}_{i}\sum _{j=1}^{n}\left|{a}_{ji}^{+}\right|)-\underset{1\le i\le n}{max}\left({L}_{i}\sum _{j=1}^{n}\left|{b}_{ji}^{+}\right|\right)\ge \mu ,$$

**Proof.**

**Example**

**1.**

**Example**

**2.**

**Remark**

**5.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(

**a**) The $(\lambda ,A)$-globally practically exponentially stable behavior of Model (14), (15) with respect to the function $h=\sqrt{{x}_{1}^{2}+{x}_{2}^{2}}$ for $\lambda =9$, $A=11$. (

**b**) The practically exponentially unstable with respect to the function $h=\sqrt{{x}_{1}^{2}+{x}_{2}^{2}}$ behavior of Model (14) with impulses (16).

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**MDPI and ACS Style**

Stamov, G.; Stamova, I.; Li, X.; Gospodinova, E.
Practical Stability with Respect to *h*-Manifolds for Impulsive Control Functional Differential Equations with Variable Impulsive Perturbations. *Mathematics* **2019**, *7*, 656.
https://doi.org/10.3390/math7070656

**AMA Style**

Stamov G, Stamova I, Li X, Gospodinova E.
Practical Stability with Respect to *h*-Manifolds for Impulsive Control Functional Differential Equations with Variable Impulsive Perturbations. *Mathematics*. 2019; 7(7):656.
https://doi.org/10.3390/math7070656

**Chicago/Turabian Style**

Stamov, Gani, Ivanka Stamova, Xiaodi Li, and Ekaterina Gospodinova.
2019. "Practical Stability with Respect to *h*-Manifolds for Impulsive Control Functional Differential Equations with Variable Impulsive Perturbations" *Mathematics* 7, no. 7: 656.
https://doi.org/10.3390/math7070656