The Lyapunov-Razumikhin theorem for the conformable fractional system with delay

1.   Introduction

Fractional differential systems have been widely investigated due to their applications in science and engineering, including solving nonlinear equations, associative memory, data analysis, intelligent control, and optimization ^{[3,4,6,8,11,12,13,14,17,18]}. The advantages of fractional-order calculus are that it can increase the flexibility of a system with infinite memory and genetic characteristics. There have been studied and developments in the theoretic aspects such as controllability, periodicity, asymptotic behavior etc.

In ^[9], R. Khalil et al. defined the conformable fractional derivative. Recently, many researchers have studied definitions and properties of conformable fractional derivatives other than the Caputo, the Grunwald–Letnikov and the Riemann–Liouville fractional derivatives for all of them do not satisfy the rules

where

for all $t > t_{0}$ and $\alpha\in(0, 1]$.

The aim of this paper is to construct Razumikhin-type uniform stability and a uniform asymptotic stability theorem for the conformable fractional system with delay. Moreover, a numerical example is given to show that our theorem can be applied in an uncomplicated way.

2.   Problem formulation and preliminaries

In this section, we approach some preliminary definitions and necessary lemmas.

Definition 2.1. ^[10] For a function $\mu:[t_{0}, \infty)\rightarrow \mathbb{R}$, the conformable fractional derivative of $\mu$ of order $\alpha$ is defined by

for all $t > t_{0}$ and $\alpha\in(0, 1]$.

If the conformable fractional derivative of $\mu(t)$ of order $\alpha$ exists on $(t_{0}, \infty)$, then the function $\mu(t)$ is said to be $\alpha$-differentiable on the interval $(t_{0}, \infty)$.

Definition 2.2. ^[10] Given a function $\mu:[t_{0}, \infty)\rightarrow \mathbb{R}$, the conformable fractional integral starting from $t_{0}$ of $\mu$ of order $\alpha$, where $0 < \alpha\leq 1$ is defined by

Lemma 2.3. ^[10] Given $\alpha\in(0, 1)$ and a continuous function $\mu:[t_{0}, \infty)\rightarrow \mathbb{R}$, we have

for all $t > t_{0}$.

Lemma 2.4. ^[10] Given a $\alpha$-differentiable function $\mu:[t_{0}, \infty)\rightarrow \mathbb{R}$ with $\alpha\in(0, 1]$, we have

for all $t > t_{0}$.

Lemma 2.5. ^[10] Given a symmetric positive definite matrix $P$ and a $\alpha$-differentiable function $\mu:[t_{0}, \infty)\rightarrow \mathbb{R}$ with $\alpha\in(0, 1]$. Then $T_{t_{0}}^{\alpha} \mu^{T}(t)P\mu(t)$ exists on $[t_{0}, \infty)$ and

for all $t > t_{0}$.

Consider the conformable fractional system with delay

where $0 < \alpha\leq 1$, $\mu(t)\in \mathbb{R}^n$ is the state vector, and $g:\mathbb{R}\times C([-\eta, 0], \mathbb{R}^{n}) \rightarrow \mathbb{R}^{n}$. For each solution $\mu(t)$ of (2.6), we assume the initial condition

where $\phi\in C([-\eta, 0], \mathbb{R}^{n})$.

3.   Main results

Theorem 3.1. Suppose that $\kappa_{1}, \kappa_{2}, \kappa_{3}:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ are continuous non-decreasing functions, $\kappa_{1}(s)$ and $\kappa_{2}(s)$ are positive for $s > 0, \kappa_{1}(0) = \kappa_{2}(0) = 0$, and $\kappa_{2}$ is strictly increasing. If thereexists a differentiable functional $\vee :\mathbb{R}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{+}$ such that

for $t\in\mathbb{R}, \mu\in\mathbb{R}^{n}$, and for any given $t_{0}\in\mathbb{R}$ the conformable fractional derivative of $\vee$ along the solution $\mu(t)$ of conformable system (2.6) satisfies

whenever $\vee(t+\theta, \mu(t+\theta))\leq \vee(t, \mu(t))$ for all $\theta\in [-\eta, 0]$, then conformable system (2.6) is uniformly stable. If $\kappa_{3}(s) > 0$ for $s > 0$ and there exists a continuous non-decreasing function $\zeta(s) > s$ for $s > 0$ such that

whenever $\vee(t+\theta, \mu(t+\theta))\leq \zeta(\vee(t, \mu(t)))$ for all $\theta\in [-\eta, 0]$, then conformable system (2.6) is uniformly asymptotically stable.

Proof of Theorem 1. Suppose that $\mu(t) = \mu(t, t_{0}, \phi)$, $\vee(t) = \vee(t, \mu(t))$, and

There exists $\hat \theta\in [-\eta, 0]$ such that $\vee^{*}(t) = \vee(t+\hat \theta, \mu(t+\hat \theta))$, and either $\hat{\theta} = 0$ or $\hat{\theta} < 0$ and

for $\hat{\theta}\leq\theta\leq 0.$

Next, we show that

For $\hat{\theta} < 0$, we have $\vee^{*}(t+\Delta t, \mu(t+\Delta t)) = \vee^{*}(t, \mu(t))$ for sufficiently small $\Delta t > 0$, and thus $T_{t_{0}}^{\alpha} \vee^{*}(t, \mu(t)) = 0$.

For $\hat{\theta} = 0$, we have $\vee^{*}(t) = \vee(t, \mu(t))$ and $T_{t_{0}}^{\alpha} \vee^{*}(t, \mu(t)) = T_{t_{0}}^{\alpha} \vee(t, \mu(t))\leq 0$ by (3.2), So (3.5) holds. Moreover, we have

Given $\epsilon > 0$, we can choose a sufficiently small $\delta > 0$ with $\kappa_{2}(\delta) < \kappa_{1}(\epsilon)$.

Assume that $\|\mu_{t_{0}}\| < \delta$. Then from (3.6), it follows that

which implies that $\|\mu(t)\| < \epsilon$. This shows that conformable system (2.6) is uniformly stable.

Suppose $\delta > 0$ and $H > 0$ are such that $\varrho(\delta) = u(H)$. Since $\|\phi\|\leq \delta$, we have $\|\mu_{t_{0}}\|\leq H$ and $\vee(t, \mu(t)) < \varrho(\delta)$ for $t\geq t_{0}-\eta$.

Suppose that $\beta$ with $0 < \beta\leq H$ is arbitrary. From the properties of the function $\zeta(s)$, there exists $\iota > 0$ such that $\zeta(s)-s > \iota$ for $u(\beta)\leq s \leq \varrho(\delta)$. Let $M$ be the smallest integer such that $u(\beta)+M\iota\geq \varrho(\delta)$ and let $T = \frac {M\varrho(\delta)}{\gamma}$ when $\gamma = \inf\limits_{\beta\leq s < H}\kappa_{3}(s)$.

Next, we will show that $\vee(t, \mu(t))\leq u(\beta)+(M-1)\iota$ for $t\geq t_{0}+\varrho(\delta)/\gamma$. If $u(\beta)+(M-1)\iota < \vee(t, \mu(t))$ for $t_{0}-\eta\leq t < t_{0}+\varrho(\delta)/\gamma$, then the fact that $\vee(t, \mu(t))\leq \varrho(\delta)$ for all $t\geq t_{0}-\eta$ yields

for $t_{0}-\eta\geq t\geq t_{0}+\varrho(\delta)/\gamma$ and $\zeta\in[-\eta, 0].$ Thus $T_{t_{0}}^{\alpha}\varrho(t, \mu(t))\leq -\kappa_{3}(|\mu(t)|)\leq -\gamma$ for $t_{0}\leq t < t_{0}+\varrho(\delta)/\gamma.$ Consequently, we have

Then $\vee(t, \mu(t))\leq u(\beta)+(M-1)\iota$ at $t_{1} = t_{0}+\varrho(\delta)/\gamma$. This implies that $\vee(t, \mu(t))\leq u(\beta)+(M-1)\iota$ for all $t\geq t_{0}+\varrho(\delta)/\gamma$, since $T_{t_{0}}^{\alpha}\vee(t, \mu(t))$ is negative when $\vee(t, \mu(t)) = u(\beta)+(M-1)\iota$.

Now, let $\bar{t}_{j} = j \varrho(\delta)/\gamma$ for $j = 1, 2, \ldots, M,$ and let $\bar{t}_{0} = 0.$ Assume that, for some integer $k\geq 1$, in the interval $\bar{t}_{k-1}-r\leq t-t_{0}\leq \bar{t}_{k}$, we have

Then

and

when $t-t_{0}-\bar{t}_{k-1}\geq \varrho(\delta)/\gamma.$ Consequently, we have

which implies that $\vee(t, \mu(t))\leq u(\beta)+(M-k)\iota$ for all $t\geq t_{0}+\bar{t}_{k-1}$.

Finally, we have $\vee(t, \mu(t))\leq u(\beta)$ for all $t\geq t_{0}+M\varrho(\delta)/\gamma$. This shows that conformable system (2.6) is uniformly asymptotically stable.

Consider the conformable fractional linear system with delay

where $0 < \alpha\leq 1, \mu(t)\in \mathbb{R}^n$ is the state vector, $A, B$ are known real constant matrices and $\eta$ is a positive real constant. For each solution $\mu(t)$ of (3.8), we assume the initial condition

where $\phi\in C([-\eta, 0];\mathbb{R}^n)$. The uncertainty $f(\cdot)$ represents the nonlinear parameter perturbation with respect to the state $x(t)$ and is bounded in magnitude of the form

where $\delta$ is a given constant.

Theorem 3.2. Given a positive scalar $\delta$, system (3.8) is uniformly stable if there exists a symmetric positive definite matrix $K$ such that the following symmetric linear matrix inequality holds:

Proof of Theorem 2. Let $K$ be a symmetric positive definite matrices. Consider the Lyapunov-Razumikhin functional of the form

Taking the conformable fractional derivative of $\vee(t)$ along the trajectory of system (3.8), we have

Next, from (3.9), we obtain

for $\epsilon > 0$. When $\vee(t+\theta, \mu(t+\theta))\leq \vee(t, \mu(t))$ for all $\theta\in [-\eta, 0],$ we obtain

According to (3.11) and (3.13), it is straightforward to see that

where $\xi(t) = col\{\mu(t), \mu(t-\eta), f(\mu(t-\eta))\}$. Note that if condition (3.10) holds, then system (3.8) is uniformly stable.

4.   A numerical example

In this section, a numerical example is given in order to present the effectiveness of our main results by showing the maximum upper bound of the parameter $\delta$.

Example 4.1. Consider the conformable linear system

Solving LMI (3.10) with $A = \begin{bmatrix} 2 & 0\\ 0 & 0.9\\ \end{bmatrix}, B = \begin{bmatrix} -1 & 0\\ -1 & -1\\ \end{bmatrix}, \eta = 0.3$ and $\alpha = 0.8$, we obtain the parameters $\epsilon = 0.9033$ and $K = \begin{bmatrix} 0.4522 & -0.0649\\ -0.0649 & 0.4652\\ \end{bmatrix}$, which guarantee asymptotic stability of system (4.1) when $\delta = 0.3$.

Moreover, the maximum upper bound of the parameter $\delta$ which guarantees the asymptotical stability of system (4.1) is $0.4131$ in Table 1. The permissible upper bounds $\delta$ for various $\eta$ and $\alpha$ are shown in Table 1.

We let $A = \begin{bmatrix} 2 & 0\\ 0 & 0.9\\ \end{bmatrix}, B = \begin{bmatrix} -1 & 0\\ -1 & -1\\ \end{bmatrix}, \eta = 0.5$, $\alpha = 1$, $f(t) = 0.01t$ and $\phi(t) = \begin{bmatrix} 2 & -4\\ \end{bmatrix}^{T}$, $\forall$ $t\in [-0.5, 0]$. Figure 1. shows the trajectories of solutions $\mu(t)$ of system (4.1).

5.   Conclusions

In this paper, an approach using the Lyapunov-Razumikhin theorem for the uniform stability and uniform asymptotic stability of the conformable fractional system with a delay has been presented. Some inequalities are adopted along with a Lyapunov-Razumikhin functional. Then we show a new delay-dependent asymptotic stability criterion of a conformable fractional linear system with delay. Finally, we give a numerical example to illustrate some advantages and applicability of our result. It will be important that future research investigate the asymptotic stability of the conformable fractional system with time-varying delay.

Acknowledgments

This research was financially supported by National Research Council of Thailand (NRCT) and Khon Kaen University (Mid-Career Research Grant NRCT5-RSA63003-07).

Conflict of interest

The authors declare no conflict of interest.