Mittag-Leffler stability of random-order fractional nonlinear uncertain dynamic systems with impulsive effects

Abstract

This paper investigates the Mittag-Leffler stability (MLS) of nonlinear uncertain dynamic systems (NUDSs) with impulsive effects involving the random-order fractional derivative (ROFD) under the fuzzy concept. The major tool used in this paper is Lyapunov’s direct method, which brings high efficiency in surveying the stability theory of dynamic systems. Some algebraic inequalities on the ROFD are established, which is necessary to study the MLS of NUDSs. Examples and simulations are also provided to demonstrate the effectiveness of the derived theoretical results.

Appendix

The manner of the proofs of Lemma 1, Lemma 2, and Lemma 3 is similar to the proofs of Lemma 3.1, Theorem 3.2, and Lemma 3.2 in [6], respectively. Therefore, in the following, we will briefly present the above assertions.

1.1 A. Proof of Lemma 1

Assume that $0<q_1 \le (q_0+n_x) \le q_2 < 1$, $x \in [c,d]$ and $\psi \in \mathcal K_{\psi }$ provided that $\psi _*(d,c) \in (0,1)$. Observe that

$$\begin{aligned} \psi ^{-q_1}_*(d,c) \le \psi ^{-(q_0+n_x)}_*(d,c) \le \psi ^{-q_2}_*(d,c). \end{aligned}$$

(5.1)

First, it follows from (i) that ${}^C_{gr}\mathcal D_{c^+}^{(q_0+n_x);\psi } w$ is a positive fuzzy function. Then, by utilizing the formula (2.5) of Definition 4, we receive from (5.1) and the property of Gamma function, $\Gamma (r_2) \le \Gamma (r_1)$ if $0<r_1 \le r_2 \le 1$, that

$$\begin{aligned}&\mathbb H({}^C_{gr}\mathcal D_{c^+}^{(q_0+n_x);\psi } w(x)) \\&= \frac{1}{\Gamma (1-(q_0+n_x))} \int \limits _{c}^{x}{ (\psi (x)-\psi (t))^{-(q_0+n_x)} \mathbb H(w'_{gr}(t)) dt} \\&\ge \frac{\psi ^{-(q_0+n_x)}_*(d,c)}{\Gamma (1-q_2)} \int \limits _{c}^{x}{ \dfrac{(\psi (x)-\psi (t))^{-(q_0+n_x)}}{\psi _*^{-(q_0+n_x)}(d,c)} \mathbb H(w'_{gr}(t)) dt} \\&\ge \frac{(\psi (d)-\psi (c))^{-q_1}}{\Gamma (1-q_2)} \int \limits _{c}^{x}{ \left( \dfrac{\psi (x) -\psi (t)}{\psi (d)-\psi (c)}\right) ^{-q_1} \mathbb H(w'_{gr}(t)) dt} \\&= \frac{\Gamma (1-q_1)}{\Gamma (1-q_2)\Gamma (1-q_1)} \int \limits _{c}^{x}{ (\psi (x) -\psi (t))^{-q_1} \mathbb H(w'_{gr}(t)) dt} \\&= \mathcal M \, \mathbb H ({}^C_{gr}\mathcal D_{c^+}^{q_1;\psi } w(x)), \end{aligned}$$

where $\mathcal M:= \Gamma (1-q_1)/\Gamma (1-q_2)$. Therefore, we deduce there exists a $\mathcal M$ such that the inequality (2.9) holds.

Next, from (ii), we see the fuzzy function ${}^C_{gr}\mathcal D_{c^+}^{(q_0+n_x);\psi } w$ is negative. With the same manner, we also receive from the formula (2.5) and the inequality (5.1) that

$$\begin{aligned}&\mathbb H({}^C_{gr}\mathcal D_{c^+}^{(q_0+n_x);\psi } w(x)) \\&= \frac{1}{\Gamma (1-(q_0+n_x))} \int \limits _{c}^{x}{ (\psi (x)-\psi (t))^{-(q_0+n_x)} \mathbb H(w'_{gr}(t)) dt} \\&\ge \frac{\psi ^{-(q_0+n_x)}_*(d,c)}{\Gamma (1-q_1)} \int \limits _{c}^{x}{ \dfrac{(\psi (x)-\psi (t))^{-(q_0+n_x)}}{\psi _*^{-(q_0+n_x)}(d,c)} \mathbb H(w'_{gr}(t)) dt} \\&\ge \frac{(\psi (d)-\psi (c))^{-q_2}}{\Gamma (1-q_1)} \int \limits _{c}^{x}{ \left( \dfrac{\psi (x) -\psi (t)}{\psi (d)-\psi (c)}\right) ^{-q_2} \mathbb H(w'_{gr}(t)) dt} \\&= \frac{\Gamma (1-q_2)}{\Gamma (1-q_2)\Gamma (1-q_1)} \int \limits _{c}^{x}{ (\psi (x) -\psi (t))^{-q_2} \mathbb H(w'_{gr}(t)) dt} \\&= \mathcal M \, \mathbb H ({}^C_{gr}\mathcal D_{c^+}^{q_2;\psi } w(x)), \end{aligned}$$

where $\mathcal M:= \Gamma (1-q_2)/\Gamma (1-q_1)$, which leads to the inequality (2.10) holds.

1.2 B. Proof of Lemma 2

It follows from Definition 2 that in order to demonstrate the inequality (2.11), we will show that

$$\begin{aligned}&\mathbb H\big ({}^C_{gr}\mathcal D_{a^+}^{(q_0+n_x);\psi } w^2(x)\big ) \nonumber \\&\qquad - \mathbb H\big (2w(x){}^C_{gr}\mathcal D_{a^+}^{(q_0+n_x);\psi } w(x)\big ) \le 0. \end{aligned}$$

(5.2)

Through the formula (2.5) in Definition 4 and by using the technique of integration by parts, the left-hand side of (5.2) (LHS-(5.2) for short) becomes

$$\begin{aligned}&\text {LHS}-(5.2) \nonumber \\&= \dfrac{1}{\Gamma (1-(q_0+n_x))} \int \limits _{a}^{x}{} (\psi (x)-\psi (t))^{-(q_0+n_x)} \nonumber \\&\quad \cdot \left( \frac{\partial [w^{gr}(t)]^2}{\partial t} -2w^{gr}(x)\frac{\partial w^{gr}(t)}{\partial t} \right) dt \nonumber \\&=\dfrac{1}{\Gamma (1-(q_0+n_x))} \int \limits _{a}^{x}{ (\psi (x)-\psi (t))^{-(q_0+n_x)} } \nonumber \\&\qquad \cdot \dfrac{\partial }{\partial t}\left( w^{gr}(t) -w^{gr}(x)\right) ^2 dt \nonumber \\&=\dfrac{1}{\Gamma (1-(q_0+n_x))}\dfrac{\left( w^{gr}(t) -w^{gr}(x)\right) ^2}{\left( \psi (x)-\psi (t)\right) ^{(q_0+n_x)}}\Bigg |_{a}^{x} \nonumber \\&- \dfrac{(q_0+n_x)}{\Gamma (1-(q_0+n_x))} \int \limits _{a}^{x}{\psi '(t) \dfrac{\left( w^{gr}(t) -w^{gr}(x)\right) ^2}{(\psi (x)-\psi (t))^{(q_0+n_x)+1}} dt} \end{aligned}$$

(5.3)

where $w^{gr}(t):=w^{gr}(\alpha ,t, \gamma _w)$. It follows from L’Hopital rule that, for each $(\alpha ,\gamma _w) \in [0,1]\times [0,1],$

$$\begin{aligned}&\mathop {\lim }\limits _{t \rightarrow x } \dfrac{\left( w^{gr}(t) -w^{gr}(x)\right) ^2}{\left( \psi (x)-\psi (t)\right) ^{(q_0+n_x)}} \\&= 2\mathop {\lim }\limits _{t \rightarrow x } \frac{\partial w^{gr}(t)}{\partial t} \dfrac{\left( w^{gr}(x) -w^{gr}(t)\right) }{(q_0+n_x) \psi '(t) (\psi (x)-\psi (t))^{(q_0+n_x)-1}} \\&= 0. \end{aligned}$$

Then, we imply from (5.3) that

$$\begin{aligned}&\text {LHS}-(5.2) \\&=-\dfrac{1}{\Gamma (1-(q_0+n_x))} \dfrac{\left( w^{gr}(a) -w^{gr}(x)\right) ^2}{\left( \psi (x)-\psi (a)\right) ^{(q_0+n_x)}} \\&-\dfrac{(q_0+n_x)}{\Gamma (1-(q_0+n_x))} \int \limits _{a}^{x}{\psi '(t) \dfrac{\left( w^{gr}(t) -w^{gr}(x)\right) ^2}{(\psi (x)-\psi (t))^{(q_0+n_x)+1}} dt}\\&\le 0. \end{aligned}$$

Therefore, we deduce the inequality on the ROFD (5.2) holds.

1.3 C. Proof of Lemma 3

It follows from Definition 2 that in order to demonstrate the inequality (2.12), we will show that

$$\begin{aligned}&\mathbb H \left( {}^C_{gr}\mathcal D_{a^+}^{(q_0+n_x);\psi } W^T Q W \right) \nonumber \\&\le 2 [W^{gr}(\alpha ,x,\gamma _{W})]^T Q \mathbb H \left( {}^C_{gr}\mathcal D_{a^+}^{(q_0+n_x);\psi } W(x)\right) . \end{aligned}$$

(5.4)

Because Q is symmetric, the matrix Q can be represented as $Q = P \Lambda P^T,$ where $P \in \mathbb R^{n \times n}$ is orthogonal and $\Lambda \in \mathbb R^{n \times n}$ is diagonal. Hence, we notice

$$\begin{aligned}{}[W^{gr}(x)]^T Q W^{gr}(x)&= [W^{gr}(x)]^T P \Lambda P^T W^{gr}(x) \nonumber \\&= (P^T W^{gr}(x))^T \Lambda (P^T W^{gr}(x) ) \nonumber \\&= [Y^{gr}(x)]^T \Lambda Y^{gr}(x), \end{aligned}$$

(5.5)

where one puts $W^{gr}(x):=W^{gr}(\alpha ,x,\gamma _{W})=(w^{gr}_1(x),\ldots ,w^{gr}_n(x))$ and $Y^{gr}(x) =P^T W^{gr}(x)$. Because $\Lambda$ is diagonal, one notices that

$$\begin{aligned}{}[W^{gr}(x)]^T \Lambda W^{gr}(x) = \sum \limits _{i = 1}^n { \beta _{ii} [w_{i}^{gr}(x)]^2}. \end{aligned}$$

By utilizing the inequality (5.2) in Lemma 2 and by the formula (2.5), we imply from (5.5) that

$$\begin{aligned}&\mathbb H \left( {}^C_{gr}\mathcal D_{a^+}^{(q_0+n_x);\psi } W^T(x) Q W (x) \right) \\&= \dfrac{1}{\Gamma (1-(q_0+n_x))} \int \limits _{a}^{x}{ (\psi (x)-\psi (t))^{-(q_0+n_x)} } \\&\qquad \qquad \qquad \cdot \frac{\partial \left( [W^{gr}(t)]^T Q W^{gr}(t)\right) }{\partial t} dt \\&= \sum \limits _{i = 1}^n { \beta _{ii} \mathbb H \left( {}^C_{gr}\mathcal D_{a^+}^{(q_0+n_x);\psi } w^2_i(x)\right) }\\&\le 2\sum \limits _{i = 1}^n { \beta _{ii} \mathbb H \left( w_i(x) {}^C_{gr}\mathcal D_{a^+}^{(q_0+n_x);\psi } w_i(x)\right) } \\&= 2 \mathbb H \left( \sum \limits _{i = 1}^n { \beta _{ii} w_i(x) {}^C_{gr}\mathcal D_{a^+}^{(q_0+n_x);\psi } w_i(x) } \right) \\&= 2 [Y^{gr}(x)]^T \Lambda \mathbb H \left( {}^C_{gr}\mathcal D_{a^+}^{(q_0+n_x);\psi } Y(x) \right) \\&\le 2 [W^{gr}(x)]^T P \Lambda P^T \mathbb H \left( {}^C_{gr}\mathcal D_{a^+}^{(q_0+n_x);\psi } W(x) \right) \\&= 2 [W^{gr}(x)]^T Q \mathbb H \left( {}^C_{gr}\mathcal D_{a^+}^{(q_0+n_x);\psi } W(x) \right) , \end{aligned}$$

which implies that the inequality (5.4) holds.