1. Introduction
The concept of stability has been a central theme in the development of control theory and dynamical systems for centuries. Its roots trace back to classical control methodologies where ensuring that a system’s output remains bounded and behaves predictably over time became a fundamental objective. As the field evolved, the focus shifted toward more rigorous mathematical criteria to characterize system behavior, leading to the emergence of various stability concepts suited for different types of systems. Among these, Ulam–Hyers stability garnered particular interest, initially introduced in the context of nonlinear functional equations, especially within the scope of stability under small perturbations [
1].
Ulam–Hyers stability fundamentally addresses the question of how approximate solutions relate to exact solutions, emphasizing the robustness of solutions when subjected to minor inaccuracies or uncertainties. Its significance was magnified in controlling real-world systems where perfect models are rare, and imperfections are inevitable. Researchers recognized early on that this form of stability could be instrumental in guaranteeing system reliability amid unpredictable disturbances. Over time, the notion was extended to various classes of equations and systems, including fractional differential equations, which naturally model phenomena with memory and hereditary properties [
2,
3].
The extension of the Ulam–Hyers concept to fractional calculus significantly enhanced its relevance, especially in the context of engineering and applied sciences. Fractional models are renowned for their ability to accurately describe anomalous diffusion, viscoelasticity, and diverse biological processes. Their inherent complexity and non-local nature pose unique challenges in stability analysis. Consequently, the adaptation of Ulam–Hyers stability principles to fractional systems provided a powerful framework for understanding how solutions respond to perturbations or modeling inaccuracies within these non-integer-order systems [
4].
In particular, the application of Ulam–Hyers stability concepts has proven invaluable in nonlinear fractional systems, where classical stability criteria may fall short. Many real-world problems rely on fractional derivatives to incorporate memory effects, making the stability of solutions under perturbations a critical concern. This form of stability assures practitioners that approximate numerical solutions or models affected by uncertainties remain close to the true solutions, a property essential for both theoretical analysis and practical implementation [
5].
The relevance of Ulam–Hyers stability in fractional calculus extends well into the domain of fractional integral and differential equations commonly encountered in the calculus of variations, optimal control, and optimization. As these methods often involve inherently approximate computational techniques, the guarantees provided by Ulam–Hyers stability become crucial for ensuring the credibility and robustness of solutions. Consequently, stability analysis grounded in this concept underpins many modern approaches to solving fractional variational problems, facilitating the development of more reliable and efficient numerical schemes [
6].
Moreover, the importance of Ulam–Hyers stability extends beyond theoretical interest; it influences the design of control strategies and algorithms for systems modeled by fractional dynamics. Engineers and scientists rely heavily on such stability properties to develop controllers capable of maintaining desired system behavior despite uncertainties. The insights gained from this approach also foster advances in fractional optimal control, where balancing efficiency and stability often hinges on robustness assessments rooted in Ulam–Hyers principles [
7].
In summary, the evolution of stability concepts, particularly Ulam–Hyers stability, has played a transformative role in the advancement of fractional calculus and its applications. Its development reflects a deeper understanding of how systems respond to imperfections, uncertainties, and disturbances—factors that are unavoidable in practical scenarios. The importance of this stability notion is particularly pronounced in solving fractional problems in calculus of variations and fractional optimization, where the non-local, memory-dependent nature of these systems demands a more nuanced approach to stability analysis. As research continues to expand in this area, Ulam–Hyers stability remains a cornerstone, guiding the design of resilient, reliable, and accurate methods for tackling the complexities inherent in fractional systems [
8,
9].
Inspired by the insights from these foundational papers [
10,
11,
12,
13,
14,
15,
16,
17], this paper presents an innovative adaptation of the Gronwall inequality tailored to the Hilfer–Hadamard proportional fractional derivative. By applying Picard’s successive approximation method and leveraging the definition of Mittag–Leffler functions, we derive a formula that represents the solution of the Hilfer–Hadamard proportional fractional differential equation with constant coefficients, formulated in terms of the Mittag–Leffler kernel. We prove the uniqueness of this solution using Banach’s fixed-point theorem, drawing on various properties of the Mittag–Leffler kernel. Additionally, we investigate optimal stability results linked to certain special functions.
This paper aims to introduce a novel perspective on Ulam-type stability, termed optimal stability, which leverages classical special functions to establish a new framework. The approach focuses on deriving the most accurate approximation error bounds through an innovative form of perturbation-based stability analysis. This concept enables the development of various approximation schemes depending on the initially selected special functions, facilitating the assessment of maximal stability and minimal error. Consequently, it allows for the determination of a unique, optimal solution to functional equations, inequalities, and fractional equations. Although Ulam stability and its variants have attracted substantial scholarly interest, extending these concepts to more effective generalizations and optimizing controllability and stability criteria remain challenging, largely unexplored issues. The proposed notion of optimal stability not only encompasses existing stability concepts but also emphasizes the importance of optimization within the stability framework. It offers a comprehensive approach for improving the robustness of mathematical models applied in natural sciences and engineering, aiming to optimize their stability properties across various scenarios.
2. Preliminaries
Throughout this paper, consider a finite interval in which Let and where and Let be the Banach space of continuous functions on with the norm Let be the space of the n-times absolutely continuous functions on Also, we consider the Banach space of all Lebesgue measurable functions with the norm
2.1. Special Functions
Definition 1 - (D1)
For every and the one-parameter Mittag–Leffler function and two-parameter Mittag–Leffler function are, respectively, given by - (D2)
For every and the Wright function is defined as - (D3)
For every and the Hypergeometric function is given by
Note that the symbol represents the real part of the complex number .
Lemma 1. ([
18]).
Let Then, and are non-negative functions, and for and In addition, for and we have that 2.2. Fractional Calculus
Definition 2 ([
19,
20,
21]).
We note the following concepts:Let The Hadamard–Riemann–Liouville proportional fractional integral of order of a function is defined by where and
Let and be continuous such that, for we have and and for The proportional derivative operator of order with respect to the increasing function is given by Let The Hadamard–Riemann–Liouville proportional fractional derivative of order of a function is given by where and
Let The Hadamard–Caputo proportional fractional derivative of order is given as Let and The Hadamard–Hilfer proportional fractional derivative of order and type is given by
Lemma 2. ([
11,
19,
20,
21]).
We note the following results:Let and Then, Let such that If and then, Let and If then, Let and Then, for and we obtain Let Then, for with one has
2.3. Some New Fractional Calculus Connections Between Mittag–Leffler Functions
Lemma 3. ([
11,
18,
22]).
We have the following results:- (L1)
Let and Then, - (L2)
Let and Then, - (L3)
Let and Then, - (L4)
Let and Then,
Proof. - (L1)
Making use of Definition 2 and Lemmas 1 and 2, we have that
- (L2)
Applying Definition 2 and Lemmas 1 and 2, we have that
- (L3)
By Definition (2) and (L2), one has
- (L4)
Using (L2), Definition (2), and Lemma (2), we get
□
2.4. Extending the Gronwall Inequality via Proportional Fractional Operators
Theorem 1. - (T1)
Let and be two non-negative functions locally integrable on , and z be a non-negative, continuous and non-decreasing function on . If - (T2)
Let be non-negative functions locally integrable on and for . If - (T3)
Let the assumptions in (T1) hold and let y be a non-decreasing function on Then,
Proof. - (T1)
Consider for
,
and note that
We now claim that
and
as
for every
For
the inequality (
21) is true. Assume the formula is true for some
i.e.,
For
and since
for
, we have
Setting
and using the property of the beta function, we have that
Now, we claim
as
. Since
, there is a
such that
, and so, for
we have
which immediately (ratio test) guarantees the claim.
- (T2)
Set in (T1) and we obtain the desired result.
- (T3)
From (
14) and the fact that for
we have
,
Applying the fact that
for
we have
□
3. Description of Solutions for Hilfer–Hadamard–Cauchy-Type Problems
The Hilfer–Hadamard fractional derivative holds particular significance compared to other fractional derivatives such as Caputo, Hilfer, and Riemann–Liouville because it offers a versatile framework that interpolates between different types of fractional derivatives. Unlike the Caputo or Riemann–Liouville derivatives, which are often restricted to specific initial conditions, the Hilfer–Hadamard derivative incorporates a parameter that allows for a smooth transition between these traditional derivatives, enabling more flexible modeling of complex phenomena involving memory effects and non-local behaviors. Its logarithmic kernel makes it especially suitable for systems exhibiting multiplicative or scale-invariant properties, which are not as effectively captured by the other derivatives. Consequently, this derivative enhances the ability to describe processes with evolving memory characteristics more accurately, making it a valuable tool in both theoretical analysis and practical applications across various scientific disciplines [
12].
Below, we will examine and analyze the solutions of Hilfer–Hadamard–Cauchy- type problems.
3.1. On the Solutions of Proportional Fractional Systems with Constant Coefficients
Making use of Picard’s successive approximation method, we investigate the relationship between the following Hilfer–Hadamard proportional fractional differential equations with constant coefficients and the integral equation in the form of a Mittag–Leffler function. Consider
where
and
Lemma 4. Let and Then, the exact solution of (
28)
is given by Proof. Suppose
is a solution of (
28). From Lemma 2, we have that
where
for
The method of successive approximation is applied to obtain an exact form of the solution. Consider
Using Definition 2 and Lemma 2, for
we get
In a similar way, repeating the same procedure, for
one has
If we proceed inductively and let
we have
□
3.2. On the Solutions of Proportional Fractional Systems with Mixed Boundary Conditions
Here, we investigate the relationship between the following Hilfer–Hadamard proportional fractional differential system with mixed boundary conditions and the integral equation. Consider
where
and
Lemma 5. Consider the system (
30)
. Then, ϕ is a solution of (
30)
iff Proof. Let
be a solution of (
30). Making use of Lemma 4, (
30) is equivalent to
where
Setting
in (
33), with
we obtain
Now, taking
and
into (
33), we have that
and
Based on the mixed boundary conditions of (
30), one has
Inserting
into (
33), we obtain (
31). In addition, it is easy to show that
provided by (
31) satisfies (
30). □
4. The Uniqueness Property
Recall the Banach fixed-point theory [
12]: Let
U be a Banach space,
be closed, and
be a strict contraction. Then,
L has a fixed point in
From Lemma 5, consider the operator
(here,
) defined as
Theorem 2. Let Assume there is a such that where and If then (30) has a unique solution on Proof. Consider
, where
L is given in (
34). Let
and consider
, where
Note that is a closed, bounded, and convex subset of
Step 1.
For every
and
one has
Thus,
Step 2. is a contraction.
For every
and
we have that
Hence, Since then L is a contraction. Now, Banach’s fixed-point theory guarantees that L has a fixed point. □
5. Optimal Designs in Stability Studies
Here, set where and
Definition 3. We consider the following concepts:
- (D1)
Let and The proportional fractional system (
30)
is called maximal UH-stable, if there exists a such that, for every and for every of there is a of (
30)
with for every and
- (D2)
The proportional fractional system (
30)
is called maximal GUH-stable if there exists such that, for every and every of (
39)
, there is a of (
30)
with where and
- (D3)
The proportional fractional system (
30)
is called maximal UHR-stable with respect to φ if there exists a such that, for every and for every of there exists a of (
30)
with for every and
- (D4)
The proportional fractional system (
30)
is called maximal GUHR-stable with respect to φ if there exists a such that, for every and for every of there exists a of (
30)
with where and
Remark 1. We have that
- (N1)
is a solution of (
39)
iff there exists a such that, for every and where
- (N2)
is a solution of (
41)
iff there exists a such that, for every and where
5.1. The Maximal UH and GUH-Stability
Lemma 6. Let and If satisfies (
39)
, then, one has Proof. Let
be a solution of (
39). Applying Remark 1, we have that
Making use of Lemma
29, the solution of (
46) can be written as
Using Note
44 and Lemma 1, we get
Theorem 3. Let the assumptions of Theorem 2 hold. Then, (
30)
is maximal UH-stable and consequently maximal GUH-stable on Proof. Let
and
be a function satisfying (
39). Let
be the unique solution of
According to Lemma 5, one has
where
Note that
,
and
so,
Using Lemma 6 with
for every
we get
Making use of Theorem 1, we obtain
By setting
and
one has
Thus, (
30) is maximal UH-stable. In addition, by setting
with
Thus, (
30) is maximal GUH-stable. □
5.2. The Maximal UHR- and GUHR-Stability
Consider the assumption:
Suppose
is a non-decreasing function and there exists
such that, for
Lemma 7. Consider and If satisfies (
41)
, then, ψ satisfies Proof. Let
be a solution of (
41). Based on Remark 1, we have that
Making use of Lemma 5, the solution of (
50) can be written as
Using Remark 1 with Lemma 1, one has
Based on the assumption (A), we obtain
□
Theorem 4. Suppose and let the assumptions of Theorem 2 hold. Then, (30) is maximal UHR- and GUHR-stable on Proof. Let
and
be a function satisfying (
41). Suppose
is the unique solution of
Making use of Lemma 5, one has
for every
In addition, and yields
From Lemma 6, for every
we have that
Setting
and
we get
Hence, the proof is complete. □
6. Application
Consider the Hilfer–Hadamard proportional fractional differential system (
30) as follows:
where
Consider
For every
and
we have
where
Hence, by (
37), we obtain
in which
Thus, the assumption of Theorem 2 is satisfied. We now can conclude that (
53) has a unique solution on
Making use of (
43), we obtain
If we set
and
then, via Theorem 3, (
53) is maximal UH-stable. Plus, if we set
then, (
53) is also maximal GUH-stable on
Finally, by letting
in (
49), the assumption (A) is thus satisfied under
From (
44), we get
and
If we set
and
then, via Theorem 4, (
53) is maximal UHR stable on
In addition, if we consider
then, (
53) is also maximal GUHR stable on
Here, set
where all parameters have been introduced in the previous sections.
Contour plots of
for different
values are shown in
Figure 1. As can be seen, with increasing
from
to
, the resulting error generally increases. However, at certain points, the error values tend to converge and become nearly zero. Similarly, the error plots of
for
, obtained by applying the hypergeometric, Wright, and Mittag–Leffler control functions, respectively, are presented in
Figure 2. As observed, the resulting error magnitude is notably lower when the hypergeometric controller is employed compared to the other configurations. This highlights that the choice of special control function plays a crucial role in optimizing stability problems, underscoring the significant impact of controller selection on the overall performance.
Special functions play a fundamental role in fractional calculus and the precise modeling of complex systems. Their unique properties enable an accurate representation of fractional-order dynamics, which are increasingly prevalent in stability analysis and control design. In optimization tasks, these functions facilitate the development of more effective controllers by capturing intricate system behaviors that conventional methods may overlook. Incorporating special functions allows for enhanced computational efficiency and higher solution accuracy in fractional calculus applications. Consequently, their utilization becomes essential for advancing stability and optimization strategies, ultimately leading to more robust and reliable control systems. Their significance is underscored by their ability to bridge theoretical models with practical, real-world system behaviors.
The numerical results presented in
Table 1 provide a comprehensive validation of the aforementioned insights. Based on the data, it can be concluded that the previous claims are indeed accurate and thoroughly substantiated, confirming the effectiveness of the approaches discussed.
7. Conclusions
In this study, we introduced a novel version of the Gronwall inequality tailored to the Hilfer–Hadamard proportional fractional derivative, establishing new theoretical foundations for such equations. Through the application of Picard’s successive approximations and properties of Mittag–Leffler functions, we derived a robust representation formula expressed in terms of the Mittag–Leffler kernel. Additionally, leveraging Banach’s fixed-point theorem, we demonstrated the uniqueness of the solution under specific conditions. Furthermore, we explored optimal stability results associated with special functions. Our exploration of stability results further solidifies the mathematical framework, offering deeper insights into the behavior of fractional systems governed by this class of equations. These findings contribute meaningfully to the growing body of research in fractional differential equations and their applications. For future research, we recommend exploring other well-known special functions, including the Fox H-function, the G-function, and the Meijer G-function, as control functions [
12,
24,
25]. The results obtained from these functions will be compared with the findings of this study to thoroughly evaluate their effectiveness and relevance in the context of our proposed methodology.