Weighted exponential stability for generalized delay functional differential equations with bounded delays
 Snezhana Hristova^{1}Email author and
 Valentina Proytcheva^{2}
https://doi.org/10.1186/168718472014185
© Hristova and Proytcheva; licensee Springer. 2014
Received: 10 May 2014
Accepted: 4 June 2014
Published: 22 July 2014
Abstract
In the paper functional differential equations with several bounded delays are considered. The delays are presented in the form of delay operators and they generalize many wellknown types of delay differential equations in the literature, such as differential equations with constant delays, with variable delays, with distributed delays, differential equations with maxima, etc. A special type of stability, known as ψexponential stability, is studied and several sufficient conditions are obtained. The used function ψ plays the role of a weight in the norm and, additionally, it gives the rate of increase of the solutions which are not exponentially stable in the regular sense. The modified Razumikhin method as well as comparison results have been applied. Several examples illustrate the results obtained.
MSC:34K20.
Keywords
1 Introduction
One of the main qualitative questions in the theory of differential equations is stability. The problems of the stability of the solutions of differential equations via Lyapunov functions have been successfully investigated and various types of stability have been introduced. One of the useful types of stability is the socalled exponential stability. In [1, 2] the authors have investigated the exponential stability of impulsive delay differential equations by using the method of Lyapunov functions and by Razumikhin techniques.
The exponential stability gives information as regards the rate of decay of the solutions only in the case when the solutions are stable. But often the solutions are not bounded at all, or they are bounded but not exponentially stable. In this case we could apply a weight such that the solution could become exponentially stable in a different sense. We could introduce and use the socalled ψstability. The notion of ψstability of degree k for ordinary differential equations has been introduced by Akinyele [3]. Also various types of ψstability have been studied for nonlinear systems of ordinary differential equations [4], for nonlinear Volterra integrodifferential systems [5], for impulsive differential equations [6]. Meanwhile, in [6] points in ${\mathbb{R}}^{n}$ are incorrectly mixed with values of functions with a range ${\mathbb{R}}^{n}$ and the weights in the examples are useless.
In the mathematical simulation in various important branches of control theory, pharmacokinetics, economics, etc., one has to analyze the influence of the deviation of the regulated quantity. Such a kind of problems could be adequately modeled by differential equations that contain a delay operator. In this paper differential equations with a special type of delay operators are considered. These delay equations generalize many types of equations well known in the literature. Several sufficient conditions for the ψexponential stability by scalar Lyapunov functions are obtained. The modified Razumikhin method and comparison results have been applied. Several examples, solved and graphed by Wolfram Mathematica, are given to illustrate the main concepts of the weighted exponential stability.
2 Preliminary notes and definitions
Let $r>0$ be a given number. Define the operators ${G}_{k}:C([r,\mathrm{\infty}),{\mathbb{R}}^{n})\to {\mathbb{R}}^{n}$, $k=1,2,\dots ,m$, such that for any function $x\in C([r,\mathrm{\infty}),{\mathbb{R}}^{n})$, any point $t\in {\mathbb{R}}_{+}$, and $k=1,2,\dots ,m$, there exists a point $\xi \in [tr,t]$, depending on x, t, k, such that ${G}_{k}(x)(t)=Cx(\xi )$, where C is a constant.
where $x\in {\mathbb{R}}^{n}$, $f:{\mathbb{R}}_{+}\times {\mathbb{R}}^{n}\times {\mathbb{R}}^{nm}\to {\mathbb{R}}^{n}$, ${t}_{0}\in {\mathbb{R}}_{+}$, $\phi :[r,0]\to {\mathbb{R}}^{n}$.

if for any point $t\in {\mathbb{R}}_{+}$ the point ${\xi}_{k}\in \mathbb{R}$ is such that ${\xi}_{k}=t{h}_{k}$, $k=1,2,\dots ,m$, where ${h}_{k}=\mathrm{const}\in [0,r]$, $k=1,2,\dots ,m$, then ${G}_{k}(x)(t)=x(t{h}_{k})$ for $t\in {\mathbb{R}}_{+}$ and (1) is reduced to the delay differential equations with constant delays, well known and studied in the literature, ${x}^{\prime}=f(t,x(t),x(t{h}_{1}),x(t{h}_{2}),\dots ,x(t{h}_{m}))$;

if for any point $t\in {\mathbb{R}}_{+}$ the point ${\xi}_{k}\in \mathbb{R}$ is such that ${max}_{s\in [t{h}_{k},t]}x(s)=x({\xi}_{k})$, ${h}_{k}=\mathrm{const}\in [0,r]$, $k=1,2,\dots ,m$, then ${G}_{k}(x)(t)={max}_{s\in [t{h}_{k},t]}x(s)$ for $t\in {\mathbb{R}}_{+}$ and (1) is reduced to differential equations with maxima, which are partially studied in [7–13], see also the monograph [14] and the references cited therein:${x}^{\prime}=f(t,x(t),\underset{s\in [t{h}_{1},t]}{max}x(s),\underset{s\in [t{h}_{2},t]}{max}x(s),\dots ,\underset{s\in [t{h}_{m},t]}{max}x(s));$

if for any point $t\in {\mathbb{R}}_{+}$ the point ${\xi}_{k}\in \mathbb{R}$ is such that ${\xi}_{k}=t{\tau}_{k}(t)$, where the functions ${\tau}_{k}:{\mathbb{R}}_{+}\to [0,r]$, then ${G}_{k}(x)(t)=x(t{\tau}_{k}(t))$ for $t\in {\mathbb{R}}_{+}$ and (1) is reduced to the delay differential equations with variable bounded delays, well known and studied in the literature: ${x}^{\prime}=f(t,x(t),x(t{\tau}_{1}(t)),x(t{\tau}_{2}(t)),\dots ,x(t{\tau}_{m}(t)))$ (for example, $\tau (t)=rsin(t)$ or $\tau (t)=\frac{rt}{t+1}$ for $t\in {\mathbb{R}}_{+}$).
In our work we will assume that IVP (1), (2) (initial value problem) has a solution $x(t;{t}_{0},\phi )$ defined on $[{t}_{0}r,\mathrm{\infty})$ for any ${t}_{0}\in {\mathbb{R}}_{+}$ and $\phi \in C([r,0],{\mathbb{R}}^{n})$.
In our investigations we will use a special function, which will play the role of the weight in the regular norm in ${\mathbb{R}}^{n}$ and it will help us to generalize the wellknown exponential stability. Let ${\psi}_{k}:[r,\mathrm{\infty})\to (0,\mathrm{\infty})$, $k=1,2,\dots ,n$, be given continuous functions and let ψ be a $n\times n$dimensional matrix defined by $\psi =diag[{\psi}_{1},{\psi}_{2},\dots ,{\psi}_{n}]$.
We will define exponential stability with a weight for generalized nonlinear delay functional differential equations (1).
 (1)ψexponentially stable, if for any initial point ${t}_{0}\in {\mathbb{R}}_{+}$ and any initial function $\phi \in C([r,0],{\mathbb{R}}^{n})$ there exist $\beta =\beta ({t}_{0})>0$ and a constant $\delta >0$ such that the solution $x(t;{t}_{0},\phi )$ of IVP (1), (2) satisfies$\parallel \psi (t)x(t;{t}_{0},\phi )\parallel \le \beta ({t}_{0}){\parallel \psi \phi \parallel}_{{t}_{0}}{e}^{\delta (t{t}_{0})},\phantom{\rule{1em}{0ex}}t\ge {t}_{0},$
 (2)
ψuniformly exponentially stable, if β in (1) does not depend on ${t}_{0}$.
Remark 1 The definition of stability given above generalizes the exponential stability, well known in the literature (if ${\psi}_{k}(t)\equiv 1$ for $k=1,2,\dots ,n$).
Remark 2 If the weight function ψ is onetoone, then we could consider the substitution $y(t)=\psi (t)x(t)$, where $x(t)$ is a solution of (1), and study the exponential stability of $y(t)$ which is equivalent to the ψexponential stability of $x(t)$. But if the function $\psi (t)$ is not a onetoone function, then we have to study directly the behavior of the solution (1).
The ψexponential stability defined above is often used in connection with the rate of increase/decrease of unbounded solutions.
Definition 2 The rate of increase of a function $u:[a,\mathrm{\infty})\to {\mathbb{R}}^{n}$ is smaller than the function $\eta \in C([a,\mathrm{\infty}),(0,\mathrm{\infty}))$ if there exists $C=C(a,u(a))>0$ such that $\parallel u(t)\parallel <C\eta (t)$ for $t\ge a$.
then any solution of (1) has a rate of increase smaller than ${e}^{\delta t}$, where $C=\beta ({t}_{0}){\parallel \phi \parallel}_{{t}_{0}}{e}^{\delta {t}_{0}}$.
Remark 4 If the solutions of (1) are unbounded but the zero solution is weighted exponentially stable, then with the help of the weight we could easily obtain the rate of increase of the solutions.
Remark 5 If the weight function is ψ and we consider the substitution $y(t)=\psi (t)x(t)$, where $x(t)$ is a solution of (1), then the function $y(t)$ could satisfy a more complicated equation than the given one. Therefore, the study of the exponential stability of the zero solution of the equation could be more difficult than the study of the ψexponential stability of the given one. That is why it requires directly studying of the ψexponential stability and obtaining of appropriate sufficient conditions.
where $x\in \mathbb{R}$, $\phi \in C([1,0],\mathbb{R})$.
Let $\phi (t)={t}^{2}$ for $t\in [1,0]$. Then the solution $x(t)$ is unbounded.
where ${\psi}^{\prime}(t)=diag[{\psi}_{1}^{\prime}(t),{\psi}_{2}^{\prime}(t),\dots ,{\psi}_{n}^{\prime}(t)]$.
3 Main results
We will obtain some sufficient conditions for the weighted exponential stability of nonlinear generalized delay functional differential equations.
 (i)
$a(t){\parallel \psi (t)x\parallel}^{p}\le V(t,x)\le b(t){\parallel \psi (t)x\parallel}^{p}$ for $t\in [r,\mathrm{\infty})$, $x\in {\mathbb{R}}^{n}$;
 (ii)for any number $t\in {\mathbb{R}}_{+}$ and any function $\varphi \in C([r,0],{\mathbb{R}}^{n})$ such that $V(t,\varphi (0))\ge {e}^{{\int}_{tr}^{t}w(s)\phantom{\rule{0.2em}{0ex}}ds}V(t+s,\varphi (s))$ for $s\in [r,0)$ the inequality${D}_{\text{(1)}}V(t,\varphi (0))<w(t)V(t,\varphi (0))$(5)
holds.
Then the zero solution of the generalized delay functional differential equation (1) is ψexponentially stable.
Proof Consider a solution $x(t)=x(t;{t}_{0},\phi )$ of IVP (1), (2) for $\phi \in C([r,0],{\mathbb{R}}^{n})$, ${t}_{0}\in {\mathbb{R}}_{+}$.
We will prove the function $Q(t)$ to be nonpositive.
Let $t\in [{t}_{0}r,{t}_{0}]$. Then from condition (i) we get $Q(t)=V(t,\phi (t{t}_{0}))M({t}_{0}){({\parallel \psi \phi \parallel}_{{t}_{0}})}^{p}\le b(t){\parallel \psi (t)\phi (t{t}_{0})\parallel}^{p}M({t}_{0}){({\parallel \psi \phi \parallel}_{{t}_{0}})}^{p}\le 0$. Therefore, the function $Q(t)\le 0$ on $[{t}_{0}r,{t}_{0}]$.
Case 1. Let ${t}^{\ast}r<{t}_{0}$.
Case 2. Let ${t}^{\ast}r\ge {t}_{0}$.
Inequality (9) contradicts (8).
The proof of Theorem 1 is completed. □
Remark 6 Note that the function $w(t)$ in inequality (5) could be replaced by a positive constant C.
Corollary 1 Let there exist positive constants A, B, C, p such that $a(t)\equiv A$, $b(t)\equiv B$, $w(t)\equiv C$ and the conditions of Theorem 1 are satisfied.
Then the zero solution of the generalized delay functional differential equation (1) is ψuniformly exponentially stable.
In the case when the weight is involved in the derivative of the Lyapunov function we obtain the following sufficient conditions for the ψexponential stability.
 (i)
$a(t){\parallel x\parallel}^{p}\le V(t,x)\le b(t){\parallel x\parallel}^{p}$ for $t\in [r,\mathrm{\infty})$, $x\in {\mathbb{R}}^{n}$;
 (ii)for any number $t\in {\mathbb{R}}_{+}$ and any function $\varphi \in C([r,0],{\mathbb{R}}^{n})$ such that $V(t,\psi (t)\varphi (0))\ge {e}^{{\int}_{tr}^{t}w(s)\phantom{\rule{0.2em}{0ex}}ds}V(t+s,\psi (t+s)\varphi (s))$ for $s\in [r,0)$ the inequality${D}_{\text{(1)}}V(t,\psi (t)\varphi (0))<w(t)V(t,\psi (t)\varphi (0))$
holds.
Then the zero solution of the generalized delay functional differential equation (1) is ψexponentially stable.
□
Remark 7 The function $w(t)$ under the conditions of Theorem 1 and Theorem 2 could not vanish, so in the case when the derivative of the Lyapunov function is strongly negative, we need another type of sufficient condition.
 (i)
$a(t){\parallel x\parallel}^{p}\le V(t,x)\le b(t){\parallel x\parallel}^{p}$ for $x\in {\mathbb{R}}^{n}$;
 (ii)for any number $t\in {\mathbb{R}}_{+}$ and any function $\varphi \in C([r,0],{\mathbb{R}}^{n})$ such that $V(t,\psi (t)\varphi (0))>\gamma V(t+s,\psi (t+s)\varphi (s))$ for $s\in [r,0)$ the inequality${D}_{\text{(1)}}V(t,\psi (t)\varphi (0))<0$
holds.
Then the zero solution of the generalized delay functional differential equation (1) is ψexponentially stable.
Proof Let $x(t)=x(t;{t}_{0},\phi )$ be a solution of IVP (1), (2).
Choose a positive number $\lambda :\lambda <\frac{ln\gamma}{r}$. Then $\gamma {e}^{\lambda r}<1$.
where the function $M(t)$ is defined by (6).
Let $t\in [{t}_{0}r,{t}_{0}]$. From condition (i) it follows that $v(t)=V(t,\psi (t)x(t))\le b(t){(\parallel \psi (t)\phi (t{t}_{0})\parallel )}^{p}\le M({t}_{0}){\parallel \psi \phi \parallel}_{{t}_{0}}^{p}\le {e}^{\lambda (t{t}_{0})}M({t}_{0}){\parallel \psi \phi \parallel}_{{t}_{0}}^{p}$.
Therefore, from (ii) we get ${D}_{\text{(1)}}V({t}^{\ast},\psi ({t}^{\ast})x({t}^{\ast}))<0$, which contradicts (12).
The proof of Theorem 3 is completed. □
Corollary 2 Let conditions of Theorem 2/Theorem 3 be satisfied for $a(t)\equiv A>0$ and $b(t)\equiv b>0$.
Then the zero solution of the generalized delay functional differential equation (1) is ψuniformly exponentially stable.
4 Applications
Now we will give some examples to illustrate the theoretical result obtained.
Example 2 Consider again the delay differential equation (3). We will prove theoretically that the zero solution of (3) is ψuniformly exponentially stable, where $\psi (t)={2}^{1.22t}$.
Let $V(x)=0.5{x}^{2}$. It is easy to check the validity of condition (i) of Theorem 3 for $p=1$, $a(t)\equiv 1$, $b(t)\equiv 1$.
According to Theorem 3 the zero solution of (3) is ψuniformly exponentially stable, i.e. the inequality ${2}^{1.22t}x(t)\le ({{2}^{1.22t}\phi }_{{t}_{0}}){e}^{\lambda (t{t}_{0})}$ holds, where $\lambda =0.22<ln0.8$.
Also, $x(t)\le {2}^{1.22}{2}^{1.22t}{e}^{0.22t}({2}^{1.22{t}_{0}}{\phi }_{{t}_{0}}{e}^{0.22{t}_{0}})$, which proves that the rate of increase of any solution of (3) is smaller than ${2}^{1.22t}{e}^{0.22t}$.
Now we will consider the particular case of (1) for $G(x)(t)={max}_{s\in [tr,t]}x(s)$, i.e. we will consider differential equations with ‘maxima’ [14].
Note that it is not possible to obtain the solutions of the system (14) in an explicit form. So, we will use the results obtained above to draw a conclusion about the behavior of the solutions.
Let $V(t,x,y)=0.5({x}^{2}+{y}^{2})$.
where $u=(x,y)$.
Inequality (15) proves that the zero solution of (14) is also exponentially stable.
Now we will consider the particular case of $G(x)(t)=x(t1)$, i.e. a nonlinear differential equation with a constant delay.
where $x,y\in \mathbb{R}$.
Applying the above results we will prove the ψexponential stability of the zero solution of (16), where $\psi (t)=({\psi}_{1}(t),{\psi}_{2}(t))$, ${\psi}_{1}(t)={\psi}_{2}(t)={e}^{6t}$.
Consider $V(t,x,y)=0.5({x}^{2}+{y}^{2})$. Then the condition (i) of Theorem 3 is satisfied for $p=2$, $a(t)\equiv 1$ and $b(t)=1$.
where $u=(x,y)$, $x(t)$, $y(t)$ is the solution of (16), $\lambda =0.5$.
From inequality (19) it follows that $\parallel u(t)\parallel \le {e}^{5.75{t}_{0}+6}{\parallel \phi \parallel}_{{t}_{0}}{e}^{5.75t}$, i.e. the rate of increase of any solution of (16) is smaller than ${e}^{5.75t}$.
Declarations
Acknowledgements
The research was partially supported by Project BG051PO001/3.305 001 Science and Business, financed by the Operative Program, Development of Human Resources, European Social Fund and Fund Scientific Research MU13FMI002, Plovdiv University.
Authors’ Affiliations
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