Invariant set theorems for non-autonomous time-fractional systems

Abstract

Though the commonly used fractional Lyapunov stability and LaSalle theorems have been beneficial in the development of the theory and applications of fractional derivatives, their proofs hold certain flaws which can make their applicability questionable. From this point, we established new invariant set-based stability theorems for fractional order Caputo systems. As a consequence, we have obtained the fractional version of the Lyapunov stability theorem. In addition, sufficient conditions for the uniform asymptotic stability of Caputo systems are derived. Finally, two illustrative applications from population dynamics are presented to validate the effectiveness of the theoretical results.

Appendix A

• $\omega (X_{t_0})$ is non-empty: The sequence $\{X(t_n, X_{t_0})\}$ is bounded, hence contains a convergent subsequence. Let u the limit of this subsequence, then $u \in \omega (X_{t_0})$ which proves the non-emptiness of $\omega (X_{t_0})$.

• $\omega (X_{t_0})$ is closed: Let $(u_n)$ be a convergent sequence in $\omega (X_{t_0})$ such that $u_n \rightarrow u$. We shall show that $u \in \omega (X_{t_0})$. By the definition of the $\omega$-limit set, for any n, there exists a sequence $t_{n,m}\underset{m\rightarrow \infty }{\longrightarrow }\ \infty$ with $X\left( t_{n, m}, X_{t_0}\right) \underset{m\rightarrow \infty }{\longrightarrow }\ u_n$. Therefore

  $$\begin{aligned} \forall n, \,\, \exists M(n),\,\, \forall m\ge M(n),\quad \left\| X\left( t_{n, m}, X_{t_0}\right) -u_n\right\| <\frac{1}{n}. \end{aligned}$$

  This implies that

  $$\begin{aligned}{} & {} \left\| X\left( t_{n, {\mathcal {M}}(n)}, X_{t_0} \right) -u\right\| \\{} & {} \quad \le \left\| X\left( t_{n, {\mathcal {M}}(n)}, X_{t_0}\right) -u_n\right\| +\left\| u_n-u\right\| ,\\{} & {} \quad \le \frac{1}{n}+\left\| u_n-u\right\| , \end{aligned}$$

  where, ${\mathcal {M}}(n)=M(n)+n$. Since $u_n\underset{n\rightarrow \infty }{\longrightarrow }\ u$, we infer that $X\left( t_{n, {\mathcal {M}}(n)}, X_{t_0} \right) \underset{n\rightarrow \infty }{\longrightarrow }\ u$. Adding the fact that $t_{n,{\mathcal {M}}(n)} \underset{n\rightarrow \infty }{\longrightarrow } \infty ,$ we conclude that u is indeed an $\omega$-limit point, that is, $u\in \omega (X_{t_0})$. Thus, $\omega (X_{t_0})$ is closed.

• $\omega (X_{t_0})$ is compact: Since the orbit is bounded, so is the set of its limit points. Hence $\omega (X_{t_0})$ is bounded. We showed previously that it is closed. This implies its compactness.

• $X\left( t,X_{t_0}\right) \underset{t \rightarrow \infty }{\longrightarrow }\ \omega \left( X_{t_0}\right)$: Suppose the convergence (2.2) fails, there exists then a sequence $t_n \rightarrow \infty$ such that

  $$\begin{aligned} \liminf _{n\rightarrow \infty }d\left( X\left( t_n, X_{t_0}\right) , \omega (X_{t_0})\right) >0. \end{aligned}$$

  (A.1)

  Besides, the boundedness of the orbit guarantees the existence of a subsequence, also denoted by $t_{n}$, such that $X\left( t_{n}, X_{t_0}\right) \underset{n\rightarrow \infty }{\longrightarrow }u$. Then

  $$\begin{aligned} u \in \omega (X_{t_0}),{} & {} \end{aligned}$$

  (A.2)
  $$\begin{aligned} d\left( X\left( t_n, X_{t_0}\right) , \omega (X_{t_0})\right)\underset{n\rightarrow \infty }{\longrightarrow } & {} d\left( u, \omega (X_{t_0})\right) , \end{aligned}$$

  (A.3)

  where the limit (A.3) is obtained using the continuity of the function $x\mapsto d\left( x, \omega (X_{t_0})\right)$. Combining (A.1) and (A.3), yields that $d\left( u, \omega (X_{t_0})\right) >0$. This obviously contradicts (A.2).

• $\omega (X_{t_0})$ is connected: : Suppose $\omega (X_{t_0})$ is disconnected. Then there exist two disjoint open subsets ${\mathcal {A}}$ and ${\mathcal {B}}$ of ${\textbf{R}}^d$ such that ${\mathcal {A}} \cap \omega (X_{t_0})$ and ${\mathcal {B}} \cap \omega (X_{t_0})$ are nonempty, and $\omega (X_{t_0})$ is contained in ${\mathcal {A}} \cup {\mathcal {B}}$. There is some $t_1>0$ such that $X(t_1,X_{t_0}) \in {\mathcal {A}}$ and some $t_2>t_1$ such that $X(t_2,X_{t_0}) \in {\mathcal {B}}$. However, the set $K=\left\{ X(t,X_{t_0}): t_1 \le t \le \right.$ $\left. t_2\right\}$ is the continuous image of the interval $[t_1,t_2]$, thus a connected set. Hence, K cannot be contained in ${\mathcal {A}} \cup {\mathcal {B}}$. In fact, there exists at least one $\tau _0>0$ such that $X(\tau _0,X_{t_0}) \notin {\mathcal {A}} \cup {\mathcal {B}}$. In a similar way, we construct an increasing unbounded sequence $(\tau _i)$ such that $X(\tau _i, X_{t_0}) \in ({\mathcal {A}} \cup {\mathcal {B}})^{c}$ for each i. On another hand, by the compactness, the $(X(\tau _i,X_{t_0}))_i$ has a subsequence that converges to u. Thus, $u \in \omega (X_{t_0})$, and $u \in ({\mathcal {A}} \cup {\mathcal {B}})^{c}$since $({\mathcal {A}} \cup {\mathcal {B}})^{c}$ is closed. This is a Contradiction because $({\mathcal {A}} \cup {\mathcal {B}})^{c}\subset \left( \omega (X_{t_0})\right) ^{c}$.