A robust uniform practical stability approach for Caputo fractional hybrid systems

Abbreviations

r-d: right dense; l-d: left dense; r-s: right scattered; l-s: left scattered; UPS: uniform practical stability; SUPS: strongly uniform practical stability; LF: Lyapunov function; CFr$\Delta$D: Caputo fractional delta derivative; CFr$\Delta$DiD: Caputo fractional delta Dini derivative; GLFr$\Delta$D: Grunwald–Letnikov fractional delta derivative; GLFr$\Delta$DiD: Grunwald–Letnikov fractional delta Dini derivative; CFrDy$\mathbb{T}$: Caputo fractional dynamic equations on time scales; CFrD: Caputo fractional derivative

1.   Introduction

In the field of dynamic systems, stability analysis is essential for understanding system responses to various initial conditions and disturbances. Traditional Lyapunov stability theory has made significant contributions here, focusing on convergence to the equilibrium ^[1,2]. However, this strict convergence requirement can be overly restrictive in real-world applications where slight deviations are often tolerable. UPS offers a more flexible framework, allowing systems to operate within an acceptable range around the equilibrium rather than requiring exact convergence (see ^[3,4]).

The recent growth of fractional calculus has introduced valuable tools for stability analysis. By extending differentiation and integration to arbitrary order, fractional calculus has proven to be highly effective in modeling complex systems with memory effects and non-local interactions ^[5,6,7]. Among these tools, the Caputo fractional derivative (CFrD) has shown practical relevance for various applications (see ^[8,9]). However, most studies have relied on scalar Lyapunov functions (LFs), which evaluate stability on an individual-variable basis ^[10,11]. It is widely recognized that as the complexity of a dynamical system increases, identifying a suitable LF becomes more challenging. This difficulty often leads to the use of multiple LFs, forming a vector LF, where each component reveals insights into different parts of the system's behavior. By this approach, certain complex elements can be simplified into interconnected subsystems, allowing for more relaxed conditions. Thus, the method of vector LFs provides a highly adaptable framework for analysis ^[12,13,14].

Previous studies, including ^[15,16,17], have concentrated largely on stability aspects like uniform, asymptotic, and variational stability within delay and impulsive settings, primarily for continuous-time systems. In contrast, research such as ^[] has examined stability in discrete settings. This paper introduces an innovative approach using vector LFs for Caputo fractional dynamic equations on time scales (CFrDy$\mathbb{T}$), establishing UPS in terms of two measures $(m_0, m)$, and presenting a versatile framework well-suited for real-world applications. This approach ensures that systems remain within specific bounds despite small disturbances, reflecting a balance between rigorous stability and practical adaptability.

With the introduction of time scale calculus in ^[19], a unified approach for analyzing systems evolving in both discrete and continuous time domains has emerged, extending traditional continuous analysis techniques to discrete cases. Integrating this calculus with fractional dynamics enables a more sophisticated framework for understanding systems that undergo both gradual and abrupt changes, common in hybrid systems. When combined with fractional calculus, time scale calculus provides a versatile platform for analyzing dynamic behaviors in both continuous and discrete settings.

Building on these advancements, this paper extends the results in ^[20,21], presenting a novel methodology for examining the $(m_0, m)$-uniform practical stability (UPS) and $(m_0, m)$-strongly uniform practical stability (SUPS) of CFrDy$\mathbb{T}$. To further extend the scope of practical stability analysis ^[22], this work applies the novel derivatives introduced in ^[23], the Caputo fractional delta derivative (CFr$\Delta$D) and Caputo fractional delta Dini derivative (CFr$\Delta$DiD) of order $\zeta\in(0, 1)$, which enables unified stability analysis, creating a robust framework for hybrid systems exhibiting both gradual and abrupt changes.

Let us examine the system

Here,

is a function satisfying

and $^{C}\Delta^\zeta \mathfrak{y}$ denotes the $CFr\Delta D$ of $\mathfrak{y} \in \mathbb{R}^N$ of order $\zeta$. Let

represent a solution of (1.1). Assuming that the function $\mathfrak{D}$ possesses sufficient smoothness to ensure the existence, uniqueness, and continuous dependence of solutions (see ^[24,25]), this paper examines the $(m_0, m)$-UPS and $(m_0, m)$-SUPS of the system (1.1).

To achieve this, we use the comparison system of the form

where

The system (1.2) is a simpler system of lesser dimension than (1.1), whose qualitative properties including the existence and uniqueness of its solution

and practical stability are already known or easy to find, see ^[26].

The structure of this research is outlined as follows: In Section 2, necessary terminologies, key definitions, remarks, and foundational lemmas that support the later developments are given. In Section 3, we develop and give details of core findings and theoretical contributions of our research. In Section 4, we present practical examples to illustrate the relevance and applicability of our approach. Lastly, in Section 5, we present a summary of the main findings and discuss their implications.

2.   Preliminaries and definitions

Definition 2.1. ^[27] If $t\in \mathbb{T}$, then

is called the forward jump operator and

is called the backward jump operator. So that $t\in \mathbb{T}$ is said to be: right dense (r-d) if

right scattered (r-s) if $\varpi(t) > t$, left dense (l-d) if $\rho(t) = t$, and left scattered if $\rho(t) < t$. Also the function

is called the graininess function.

Definition 2.2. ^[28] A function

is said to be r-d continuous represented by $C_{rd}$, if it remains continuous at every r-d point in $\mathbb{T}$, and has finite left-sided limits at each l-d point of $\mathbb{T}$.

Definition 2.3. ^[28] We define the following class of functions:

$\mathcal{K} = \{\psi\in[[0, r], [0, \infty)]\}$: $\psi(t)$ is strictly increasing on $[0, r]$ and $\psi(0) = 0$;

$\mathcal{CK} = \{\mathfrak{a}\in C_{rd}[\mathbb{T}\times\mathbb{R}_+, \mathbb{R}_+]:\mathfrak{a}(t, s)\in \mathcal{K}$ for each $t\}$;

$\mathfrak{M} = \{m\in C_{rd}[\mathbb{T}\times\mathbb{R}^n, \mathbb{R}_+]:\inf\limits_{(t, x)}m(t, x) = 0\}$.

We now define the derivative of LF utilizing the CFr$\Delta$DiD as presented in ^[23].

Definition 2.4. Let

where $T > t_0$. Then a Lyapunov-like function

is defined as the generalized CFr$\Delta$DiD relative to (1.1) as follows: given $\epsilon > 0$, there exists a neighborhood $P(\epsilon)$ of $t\in\mathbb{T}$ such that

for each $s\in P(\epsilon)$ and $s > t$, and

is any solution of (1.1), where $0 < \zeta < 1$, $\varpi$ is the forward jump operator as defined in Definition 2.1,

and $[\frac{(t- t_0)}{\tau}]$ denotes the integer part of the fraction $\frac{(t-t_0)}{\tau}$.

Definition 2.5. ^[23] Using Definition 2.6, we state the CFr$\Delta$DiD of the LF $\gimel(t, \mathfrak{y})$ as

which is equivalent to

and

where $t\in \mathbb{T}$, and $\mathfrak{y}, \mathfrak{y}_0\in \mathbb{R}^n$, and

For discrete times, Eq (2.1) becomes

and for continuous times ($\mathbb{T} = \mathbb{R}$), Eq (2.1) becomes

Notice that (2.3) is the same in ^[5], where $\kappa > 0$.

Definition 2.6. Let

be any solution of (1.1), then system (1.1) is said to be:

($UP_1$) $(m_0, m)$-uniformly practically stable if, given $(\lambda, A)\in\mathbb{R}_+$ with $0 < \lambda < A$, we have $m_0(t_0, \mathfrak{y}_0) < \lambda$ implies $m(\mathfrak{y}(t)) < A$, $\forall$ $t_0\in\mathbb{T}$;

($UP_2$) $(m_0, m)$-uniformly practically quasi-stable if given $(\lambda, B, T) > 0$ and $t_0 \in\mathbb{T}$, we have $m_0(t_0, \mathfrak{y}_0) < \lambda$ $\implies$ $m(\mathfrak{y}(t)) < B$, for $t\geq t_0+T$;

($UP_3$) $(m_0, m)$-SUPS if $(UP_1)$ and $(UP_2)$ hold together.

Definition 2.7. Corresponding to Definition 2.6, the solution

of (1.2), is called uniformly practically stable if given $0 < \lambda < A$, we have

for some $t_0\in\mathbb{T}\cap\mathbb{R}_+$.

Lemma 2.1. ^[23] Let

Assume $\exists$ $t_1 > t_0$, with

and

for $t_0 \leq t < t_1$. If the CFr$\Delta$DiD of $\mathcal{R}$ and $\mathcal{B}$ are defined at $t_1$, then the inequality

holds.

Lemma 2.2. (Comparison theorem) Let

$(i)$ $\Theta\in C_{rd}[\mathbb{T}\times\mathbb{R}_+^n, \mathbb{R}_+^n]$ and $\Theta(t, \kappa)\tau$ is non-decreasing in $\kappa$.

$(ii)$ $\gimel(t, \mathfrak{y})\in C_{rd}[\mathbb{T}\times\mathbb{R}^N, \mathbb{R}_+^N]$ be locally Lipschitzian in $\mathfrak{y}$ and

$(iii)$ $\varrho(t) = \varrho(t; t_0, \kappa_0)$ be the maximal solution of (1.2) existing on $\mathbb{T}$.

Then

provided that

where

is any solution of (1.1), $t\in\mathbb{T}$, $t\ge t_0$.

Proof. We shall make this proof by the principle of induction. For an assertion

the following holds:

$({\rm{i}})$ $\mathfrak{S(t_0)}$ is true since $\gimel(t_0, \mathfrak{y}_0)\leq \kappa_0$.

$({\rm{ii}})$ Assuming $t$ is r-s and $\mathfrak{S(t)}$ is true. Then we prove that $\mathfrak{S}(\varpi(t))$ is true; i.e.,

set

and then

then from Definition 2.6, we obtain

also,

then,

then,

so that,

By the non-decreasing property of $\Theta(t, \kappa)\tau$ and since $\mathfrak{S(t)}$ is true, then

so Eq (2.6) holds.

$({\rm{iii}})$ For r-d points $t^*\in\mathcal{U}$, where $\mathcal{U}$ is a right neighborhood $t\in\mathbb{T}$. We can clearly see that $\mathfrak{S}(t^*)$ is true immediately from [5, Lemma 3] since at every r-d point of $\mathbb{T}$, $\gimel(t, \mathfrak{y}(t))$ is continuous and the domain is $\mathbb{R}$.

$({\rm{iv}})$ Let $t$ be l-d, and let $\mathfrak{S}(s)$ be true for all $s > t$. We want to show that $\mathfrak{S}(t)$ is true. This immediately follows r-d continuity of $\gimel(t, \mathfrak{y}(t))$, $\mathfrak{y}(t)$, and the maximal solution $\varrho(t)$. Thus, by the principle of induction, the assertion $\mathfrak{S}(t)$ holds for all $t \in \mathbb{T}$, thereby concluding the proof.

The proof is completed. □

3.   Main results

Now, we present the $(m_0, m)$-UPS and $(m_0, m)$-SUPS results for the fractional dynamic system (1.1).

Theorem 3.1. ($(m, m_0)$-UPS) Assume that

$(\mathfrak{m}_1)$ $0 < \lambda < A$;

$(\mathfrak{m}_2)$ $m_0, m\in\mathfrak{M}$, and $m_0$ is uniformly finer than $m$, implying $m(t, \mathfrak{y})\leq\phi(m_0(t, \mathfrak{y}))$, $\phi\in\mathcal{K}$, whenever $m_0(t, \mathfrak{y}) < \lambda$;

$(\mathfrak{m}_3)$ There exists

and

such that if

$\mathfrak{V}(t, \mathfrak{y})$ is locally Lipschitz in $\mathfrak{y}$, and for

then,

$(\mathfrak{m}_4)$ for $(t, \mathfrak{y})\in\mathfrak{S}(m, A)$

where

$\Theta(t, \kappa)$ is quasimonotone nondecreasing in $\kappa$ with

$\mathfrak{a}\in\mathcal{CK}$, and for each $i$, $1\leq i\leq n$, $\Theta_i(t, \kappa)\tau(t)+\kappa_i$ is nondecreasing in $\kappa$ for all $t\in\mathbb{T}$;

$(\mathfrak{m}_5)$ $\phi(\lambda) < A$ and $\mathfrak{a}(\lambda) < \mathfrak{b}(A)$ hold;

Then the UPS properties of (1.2) imply the corresponding $(m_0, m)$-UPS properties of (1.1).

Proof. Since (1.2) is UPS, then we deduce from (2.4) that $\forall$ $t_0\in\mathbb{T}$ and $0 < \lambda < A$,

We can easily assert that (1.1) is $(m, m_0)$-UPS with respect to $(\lambda, A)$.

However, if the assertion were false, then for any solution

of (1.1) with

there would be a point $t_1 > t_0$ such that

Setting

and from (2.5), we have

where $\varrho(t)$ is the maximal solution of (1.2).

Also, from assumptions $(\mathfrak{m}_2)$ and $(\mathfrak{m}_5)$, it is clear to see that

Then, we deduce that,

Combining (3.3)–(3.6), we obtain

Equation (3.7) is a contradiction, so the statement that (1.1) is $(m_0, m)$-UPS is true. □

Theorem 3.2. ($(m, m_0)$-SUPS) Assume that conditions $(\mathfrak{m}_1)$–$(\mathfrak{m}_4)$ of Theorem 3.1 is satisfied, and Eq (1.2) is SUPS; then, Eq (1.1) is $(m, m_0)$-SUPS.

Proof. To make this proof, we need (1.1) to be $(m_0, m)$-UPS and $(m_0, m)$-uniformly practically quasi-stable together. Now, (1.1) is $(m_0, m)$-UPS by Theorem 3.1, so we need to prove only $(m_0, m)$-uniform practical quasi-stability. By the assumption of the theorem, Eq (1.2) is $(m_0, m)$-SUPS for

so that for all $t_0\in\mathbb{T}$,

where $\kappa_i(t, t_0, \kappa_0)$ is any solution of (1.2). By the UPS of (1.2), we can comfortably make the assumption that

so that

for $t\geq t_0$. Setting

and by Lemma 2.2, we have the estimate

where $\varrho(t)$ is the maximal solution of (1.2). Following this argument closely, and from (3.8) and (3.9), we obtain

So that

whenever

for $t\geq t_0+T$. Therefore, we conclude that (1.1) is $(m_0, m)$-SUPS. □

4.   Illustrations

4.1.   Illustration 1

Given the following system

with initial conditions

and

for

Choose the LF for (4.1) to be

where

and

Using (2.5), the CFr$\Delta$DiD for

is computed as

Applying (2.12) and (2.14) in ^[23], we obtain

When

then

Similarly, compute the CFr$\Delta$DiD for

Applying (2.12) and (2.14) in ^[23], we obtain

When

then,

Therefore,

From (4.2) and (4.3), it follows that

Next, we choose a comparison system

with

Clearly, Figure 1 below shows the stability of system (4.5). The vector inequality (4.4), along with all the other requirements of Theorem 3.1, holds when the matrix $M$ has eigenvalues with negative real parts. Given that the eigenvalues of $M$ are both $-2$, it can be concluded that (4.1) is not only $(m, m_0)$-UPS but also $(m, m_0)$-SUPS.

4.2.   Illustration 2

Given the following system

with initial conditions

for

Choose a vector LF

and

Computing the CFr$\Delta$DiD for

we obtain:

From (2.12) and (2.14) in ^[23], we obtain

When

so that we obtain

Using

we obtain

If $\mathbb{T} = \mathbb{R}$ then $\tau = 0$, reducing (4.7) to the form

When $\mathbb{T} = \mathbb{N}_0$ then, $\tau = 1$, reducing (4.7) to

resulting in the same outcome as (4.8). Evidently, this applies to any other discrete time as well.

Also, computing the CFr$\Delta$DiD for

we obtain:

From (2.12) and (2.14) in ^[23], we obtain

When

so that

using

we obtain

If $\mathbb{T} = \mathbb{R}$, then $\tau = 0$, reducing (4.9) to

When $\mathbb{T} = \mathbb{N}_0$, then $\tau = 1$, and (4.9) reduces to:

resulting in (4.10).

Finally, for the CFr$\Delta$DiD of

we obtain

From (2.12) and (2.14) in ^[23], we obtain

As

then

Using

we obtain

If $\mathbb{T}$ is continuous, then $\tau = 0$, and (4.11) reduces to:

Otherwise, for discrete $\mathbb{T}$ $\tau = 1$, (4.11) reduces to:

then we also obtain (4.12).

From (4.8), (4.10), and (4.12), we obtain

Next, we choose a comparison system of the form

with

The vector inequality (4.13), along with all the conditions outlined in Theorems 3.1 and 3.2, hold true if the matrix $M$ has eigenvalues with negative real parts. Given that the eigenvalues of $M$ are

it follows that (4.6) is uniformly practically stable. Thus, we conclude that system (4.6) exhibits UPS.

5.   Conclusions

In conclusion, this paper develops a unified approach for UPS analysis of CFrDy$\mathbb{T}$, utilizing vector LFs and a two-measure framework. By applying the CFr$\Delta$D and the CFr$\Delta$DiD, this study extends stability analysis methods across both continuous and discrete domains, making it versatile for hybrid systems exhibiting both gradual and abrupt changes. The use of vector LFs over scalar LFs enables a broader and more robust stability analysis, accommodating complex system dynamics with improved precision and adaptability. Illustrative examples, Eqs (4.1) and (4.6), demonstrate the practical application of the proposed framework, underscoring its effectiveness and relevance in capturing the essential stability characteristics of fractional dynamic systems. This work offers significant contributions to stability theory by enhancing the tools available for fractional dynamic systems, with implications for fields such as engineering, control theory, and applied sciences.

Author contributions

Michael Ineh: conceptualization, methodology, software, formal analysis, writing—original draft preparation, writing—review and editing; Umar Ishtiaq: conceptualization, validation, investigation, resources, visualization, funding acquisition; Jackson Ante: conceptualization, methodology, supervision; Mubariz Garayev: validation, investigation, resources, visualization, funding acquisition; Ioan-Lucian Popa: conceptualization, software, validation, investigation, project administration. All authors have read and agreed to the published version of the manuscript.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors extend their appreciation to King Saud University for funding this work through Researchers Supporting Project number (RSPD2025R1056), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare that they have no conflicts of interest.