1. Introduction
Recently, the differential calculus of fractional order has garnered significant attention from scientists because of its growing range of applications. Fractional calculus offers a versatile framework for modeling complex phenomena in various domains. Fractional differential equations are utilized to model control systems in various disciplines, including mechanics, physics, biology, and others. Comprehensive analyses of these systems with real-life applications are presented in several monographs, including [
1,
2,
3,
4,
5,
6,
7].
The Hilfer fractional derivative is a differential operator that is increasingly utilized in modeling various phenomena, including such complex systems as diffusion and wave processes, economic crises, viscoelastic problems, and many others. Several models have been validated through experimental studies, demonstrating concordance with observed real-world behaviors, such as the spectra of relaxation in glass-forming systems [
8]. The process of system modeling involves solving differential equations, both ordinary and partial, which describe these phenomena. Finding solutions to these equations corresponds to studying the controllability of the systems.
Many research articles and books have addressed the controllability of fractional-order systems, employing various types of fractional derivatives. The controllability of fractional systems utilizing the Caputo differential operator has been investigated in works such as [
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20]. Specifically, Ref. [
9] discusses the controllability of positive linear fractional systems using the Metzler matrix, while [
10] explores the applications of Duhamel’s formula in the analysis of linear fractional control systems. The approximate controllability of semilinear systems is addressed in [
11], where semigroup theory and fixed-point methods are applied. In [
12], new criteria for cone-type constrained controllability of semilinear systems with delays are presented, whereas [
13] examines the controllability of semilinear systems with delays using Rothe’s fixed point theorem. The controllability of nonlinear fractional systems with the Caputo derivative has been studied in a limited number of publications. Ref. [
14] applies Schauder’s fixed-point theorem to establish controllability criteria for nonlinear system without delays, while [
15] employs the same theorem for nonlinear system with delays. Sufficient conditions for nonlinear implicit fractional systems are derived in [
16] using Darbo’s fixed-point theorem. Ref. [
17] focuses on controlling the dynamic behavior of a fractional stochastic model around the rumor-demise equilibrium, whereas [
18] utilizes Schaefer’s and Banach’s fixed-point theorems to establish conditions for the existence and uniqueness of solutions for the nonlinear fractional systems satisfying the integral boundary conditions. Additionally, the controllability of descriptor systems is examined in [
19], while regional controllability of sub-diffusion processes in discussed in [
20]. The Riemann–Liouville operator has been examined, among others, in [
21,
22,
23]. Several researchers have studied the Hilfer fractional systems, including [
24,
25,
26,
27,
28,
29,
30,
31]. Especially, their applications in physics are discussed in [
24], for modelling nonlinear fractional diffusion processes in [
25], methods of solving various type differential equations with Hilfer derivative are presented in [
26,
27,
28,
30], boundary value problems for fractional differential inclusions are explored in [
29], and for fractional sequential coupled systems of integro-differential equations in [
31]. However, the controllability of Hilfer fractional systems has been addressed in only a limited number of publications, including [
32,
33,
34,
35,
36].
Among the methods for studying the controllability of systems, symmetry-based approaches deserve particular attention. These methods enable a refined decomposition of functions or trajectories, simplifying the process of solving fractional differential equations and enhancing the understanding of physical systems. In particular, mapping one solution curve onto another or employing symmetry-based decomposition of solutions can facilitate the resolution of differential equations while providing analytical clarity. Symmetry methods have been infrequently applied to differential equations of the fractional order. For example, symmetry properties for diffusion equations of the fractional order are explored in [
37], reflection symmetries in fractional mechanics are analyzed in [
38], and nonlocal symmetries are investigated in [
39]. Furthermore, symmetry in time-fractional diffusion problems with the Caputo derivative is addressed in [
40], and various aspects of Lie symmetries and their associated conservation laws for fractional differential equations are comprehensively discussed in [
41]. However, none of these studies address the Hilfer fractional derivative.
The main purpose of this research is to establish and prove new controllability criteria for symmetrized semilinear control systems with the Hilfer operator, utilizing symmetry properties. The application of projection methods with respect to the Hilfer derivative involves the use of a reflection operator, defined within the framework of fractional calculus, which incorporates the integro-differential structure of Hilfer derivatives.
The content of the paper is organized as follows.
Section 2 presents some preliminary concepts and formulas related to fractional calculus.
Section 3 introduces the mathematical model of the Hilfer control systems under consideration and derives the solution equation for these systems using the Laplace transformation.
Section 4 presents the main results of the study, including criteria for the controllability of the symmetrized semilinear Hilfer fractional system.
Section 5 presents two examples to illustrate the computational and practical aspects of the theoretical results. Finally, concluding remarks and information on future work are provided in
Section 6.
2. Preliminaries
In this section, some basic definitions, formulas, and notations concerning fractional calculus used in this paper are provided [
1,
2,
3,
24].
Suppose that is a function of the real variable that is integrable over for . Also, let denote the nth order differentiation operator.
The right-sided and left-sided integral operators
and
are, respectively, defined by the following formulas:
and are referred to as the
Riemann–Liouville fractional integral (right-sided and left-sided, respectively) of order
for the function
f, where
is the Gamma function.
The right-sided operator is usually denoted by .
The Hilfer operator is a generalization of the Caputo and the Riemann–Liouville operators, parameterized by two values: an order
and a type
. It can be defined separately for left-sided and right-sided derivatives. Following [
24], we formulate both definitions of the Hilfer fractional derivative.
Definition 1. Let and for . The right-sided and left-sided integro-differential operators and are, respectively, defined by the following formulas:and are called the Hilfer fractional derivative (right-sided and left-sided, respectively) of order α and type β for the function f, provided that the right-side expressions exist. The right-sided Hilfer fractional derivative is usually denoted by .
The Hilfer fractional derivative is a nonlocal operator because it is based on definite integrals. This means that the current state depends both on the current time and on previous states. The derivative depends on the function f over an interval, not just at a single point, due to the integral terms. Moreover, the Hilfer derivatives incorporate past states, which makes them useful for modeling physical systems with memory effects.
The two-parameter class of Hilfer derivatives generalizes both the Riemann–Liouville and the Caputo derivatives (definitions of both are included, among others, in [
2]). The Hilfer derivative allows interpolation between these fractional derivatives. Notably, for
and
the Hilfer derivative becomes the Riemann–Liouville derivative. In particular, we have
For
and
, it reduces to the Caputo derivative, i.e.,
Symmetric operators are used when the problem or system exhibits spatial symmetry or when balanced influence from both directions is necessary, while antisymmetric operators are useful for capturing directional imbalances or processes where opposite sides contribute differently. The Hilfer fractional derivative (whether left- or right-sided), of course, is not inherently a symmetric operator (nor antisymmetric), meaning it does not treat positive and negative directions equally (or oppositely), unlike, for example, the Riesz fractional derivative. Therefore, we introduce the definitions and properties of the symmetric and antisymmetric Hilfer fractional derivatives and integrals. Corresponding facts in cases of the Riemann–Liouville and the Caputo derivatives of the order
are discussed in [
38].
Definition 2. Let and for . The symmetric Hilfer fractional derivative over the finite interval is defined by the following formula: Since the symmetric Riemann–Liouville fractional integrals over the finite interval
are defined as follows:
it follows that
and the following result is straightforward to prove.
Proposition 1. Let and for . For the symmetric Hilfer fractional derivative over the finite interval , the following holds: Additional definitions that will be used in this paper are formulated below.
The formula for the Laplace transform of the Hilfer derivative, as provided in [
24,
42], is the following:
where
is the Riemann–Liouville integral of order
evaluated as
. An important fact is that t
is constant for all
t, even though the function
f varies.
Let us also recall, according to [
2,
43], the definitions of Mittag–Leffler functions.
Definition 3. The two-parameter Mittag–Leffler function is defined by the formulawhere . If
is a bounded linear operator, the above definition can be extended to linear operators, i.e.,
The Mittag–Leffler is a function of exponential order.
Additional notations used in this paper include the following:
denotes the Hilbert space of square-integrable functions with values in .
represents the linear space of locally square-integrable functions with values in .
is the space of continuous
-valued functions defined on
, with the following norm:
Moreover, refers to the transposition of the matrix B.
3. Hilfer System Description
We now present the fractional-order model of symmetrized semilinear fractional control systems that will be analyzed in this paper.
Let
denote the symmetric on the
Hilfer fractional differential operator as defined in (
5). Assume that
represents the state vector (in the case of fractional systems, also referred to as the pseudo-state vector) and that
is the control (input function). Let
A be an
matrix with real entries,
an
matrix with continuous bounded entries, and
f a continuous nonlinear function
depending on time and state. Then, the system is described by the following:
with zero initial conditions:
where
and
. This constitutes a fractional-order Cauchy problem.
One approach to solving the Cauchy problem (
9) and (
10) relies on the Laplace transformation. To begin, we apply the Laplace transform to the right-sided component of (
9), considering the conditions (
10). This yields
for any fixed
. Substituting the initial condition (
10), we obtain the following:
Rearranging, we find the following:
Applying the convolution formula, we obtain the following:
Finally, by applying the inverse Laplace transform and using the convolution formula, the solution to the right-sided component of (
9) and (
10) is given by the following:
Since the Laplace operator is unique, the obtained solution is also unique.
Based on the above derivation and the definition of the symmetric fractional derivative (
5), we can state a theorem regarding the existence and uniqueness of the solution to (
9) and (
10).
Theorem 1. Assume zero initial conditions, and let be a control. Then, there exists a unique solution to the symmetrized semilinear Hilfer fractional system (9) and (10), given by the following: It is worth mentioning that a set of trajectories satisfying the relation (
11) is called the set of reachable states or the attainable set. Hence, for (
9) and (
10) the attainable set is as follows.
Definition 4. The set of reachable states on for the symmetrized Hilfer fractional system (9) and (10) is as follows:for and . Next, based on the definitions presented in [
36], we formulate several definitions of controllability for the fractional system (
9) and (
10) on
.
Definition 5. The symmetrized Hilfer control system (9) and (10) is said to be controllable on to a nonempty set if for each , there exists a control such that If
, the system (
9) and (
10) is said to be null controllable on
.
Definition 6. The symmetrized Hilfer control system (9) and (10) is said to be locally controllable on if , defined by (4), includes a neighborhood of zero in Definition 7. The symmetrized Hilfer control system (9) and (10) is said to be globally controllable on if for each , there exists a control such that It follows that (
9) and (
10) is globally controllable on the close interval
if
.
4. Main Contribiution
In the following section, we establish new criteria for the controllability of symmetrized Hilfer fractional systems. Additionally, we propose a decomposition of the symmetric derivative in terms of projection and reflection functions to identify the relationship between the system’s trajectory on the interval and its behavior in the subintervals and . We will also demonstrate that the controllability of the system depends on its behavior over the right subintervals.
Assume that
is a function of
, which is integrable over
, and let
be a vector whose components are either 0 or 1. We define a reflection function
on
as follows:
and a projection
of the function
f as follows:
For the reflection operator (
13) and the projection (
14) it holds
The following theorem specifies the representation of the symmetric Hilfer derivative in terms of the projection and reflection functions.
Theorem 2. Let , where and , be the projection of f defined by the formula (14), and for . The corresponding symmetric Hilfer fractional derivative in the interval is then represented as follows: Proof. Using the definition of Hilfer derivative and the integration and reflection properties, we see that
□
It is worth noting that the above theorem can subsequently be applied to the derivative and so on. Therefore, the symmetrization process can be extended.
Next, we adopt the notation from [
5] and define the following:
for the Mittag–Leffler function
Since
is of exponential order and for
, it holds that
we conclude that for
there exist constants
and
such that
The positive constant
is called the growth exponent of a given function of exponential order.
A linear fractional system is said to be controllable on a closed interval if its Grammian matrix (also known as the controllability matrix) is nonsingular. In our case, the linear Hilfer system corresponding to (
9) and (
10) is controllable on
if the following controllability matrix is nonsingular:
Assume that the given continuous function
f meets the assumptions:
for
and any positive constants
, provided that
.
Remark 1. The assumptions (19) and (20) imposed on the function f are not very restrictive. In the mathematical modeling of dynamical systems, such constraints are often satisfied. The first one ensures that the force f remains within certain limits, which could correspond to physical constraints, such as maximum motor torque or the elastic limits of materials. The second ensures that small changes in the input result in correspondingly small changes in the output, reflecting stability and predictability in control performance. Of course, these assumptions do not capture the complexities of all real-world systems, particularly highly nonlinear or chaotic systems. Now we may formulate criteria for controllability of symmetric Hilfer system (
9) and (
10).
Theorem 3. Assume that the controllability matrix , defined by (18), is nonsingular and that f satisfies conditions (19) and (20). Then, the symmetrized Hilfer system (9) and (10) is globally controllable on . Proof. Suppose that the Gramian matrix
, defined by (
18), is nonsingular. We conclude that the corresponding (
9) and (
10) linear system is globally controllable on
. Furthermore, under the assumption of bounded entries in the matrix
, let us suppose that
on
. Consider the following set:
where
is defined as follows:
with
where
is represented as (
17) and
is given by (
18).
S is closed, convex, and bounded on
.
Then, we introduce a nonlinear transformation
F that maps
S into
S as follows:
The function
is continuous in the Banach space
.
It follows that for each
, the following inequality holds:
where
represents the modulus of continuity function [
44], and
The modulus of continuity of the function
is uniformly bounded, since neither
B nor
depend on the choice of
y in
S, and are, therefore, equicontinuous. The modules of continuity of the functions
are as follows:
where the terms have the following bands:
The first two modules to the right are bounded by , where and takes a nonnegative values,
The third term is bounded by
due to assumption (
20),
The fourth term is bounded by , another nonnegative function satisfying .
Finally,
where
. This implies the following:
Hence, for any set
:
where
is the natural projection of
onto the space
Since the measure of noncompactness is given by
, we obtain the following:
By applying the generalized Darbo fixed-point theorem [
45], it can be concluded that
has a fixed point. Therefore, there exists the control
that steers the semilinear system from the zero initial state to a final state
. Therefore, it follows that the system (
9) and (
10) is globally controllable on the closed interval
. □
Theorem 4. Assume that is compact and convex, with 0
as its interior point, and that represents an admissible control. Let the nonlinear function f satisfy conditions (19) and (20). If the controllability matrix , defined by (18), is nonsingular and holds for all the eigenvalues of A and then the symmetrized semilinear Hilfer system (9) and (10) is null controllable on the closed interval . Proof. Under the assumptions that conditions (
19) and (
20) hold and
, given by (
18), are nonsingular, Theorem 4 implies that the symmetrized Hilfer system (
9) and (
10) is controllable on
assuming the controls are unconstrained. Now, suppose that the set
is compact and convex, with 0 as its interior point, and
. If
holds for all eigenvalues
of
A and
then the system (
9) and (
10) is asymptotically stable [
46]. Therefore, for the control
the solution is
, and any trajectory
satisfies the following:
for any
and a neighborhood
of
. Given the convexity and compactness of
, any state
can reach zero in finite time. Thus, the semilinear Hilfer system (
9) and (
10) is null controllable on the closed interval
with constrained controls
. □
Theorem 5. Let . If the system of projectionswith zero initial conditions is globally controllable on for then the symmetrized Hilfer control system described by (9) and (10) is globally controllable on the closed interval . Proof. The result follows directly from Theorems 2 and 3. □
This theorem highlights that even though the Hilfer derivative is non-local, in the case of the symmetric derivative, the influence of the time interval can be confined to subintervals. These subintervals can be made sufficiently small, as the process of partitioning the interval into halves, and , can be recursively applied to the second subinterval.
Examples of semilinear or nonlinear models that can be symmetrized over certain intervals include those involving sine or cosine functions, which describe periodic phenomena such as ocean waves, light waves, sound waves, and similar real-world patterns.
5. Examples
To demonstrate the computational and practical significance of the theoretical results presented in the paper, two illustrative examples are provided in this section. The first example, a numerical one, demonstrates how to apply the formulated controllability criterion to verify the controllability of the considered semilinear Hilfer systems. The second example, a practical application, illustrates how the described symmetry methods can be used with the Euler–Lagrange equation of motion.
Example 1. This example presents a numerical illustration of the computational complexity involved in verifying the controllability of the considered systems using the results presented earlier.
Suppose we have the following symmetric Hilfer fractional system:for , , and . We examine its controllability using Theorem 3.
It can be easily calculated that for and , , defined by (17), has the following form: Therefore, the controllability matrix , given by (18), for the Hilfer system (23) on the interval is as follows:Thus, is nonsingular. Also, conditions (19) and (20) hold for the given function f, because both the sine and cosine functions are bounded. It follows that the symmetrized semilinear Hilfer system (23) is globally controllable on Example 2. The presented symmetry methods can be applied to the Euler–Lagrange equation of motion. The Euler–Lagrange equation was formulated by Euler and Lagrange during their investigation of the tautochrone problem. The tautochrone problem involves determining the curve along which a point mass, under the influence of a constant gravitational force, slides to the lowest point of the curve in the same amount of time, regardless of its starting position. Niels Abel was the first to apply the theory of fractional-order derivatives and integrals to solve this problem. Foundations for fractional Euler–Lagrange equations are provided in [47]. First, we recall a useful property of fractional-order derivatives. If for , thenThis property enables the application of the Hilfer fractional derivative of order to fractional problems of any order. Let us adopt the simpler notation . We consider the symmetrized Hilfer fractional dynamical system described by the Euler–Lagrange equation of motion as follows:where , y is a differentiable real-valued function, and the action functional S is defined as follows:indicating that it depends only on the Hilfer derivatives and is independent of the Lagrangian L. Assume that for and is the projection of defined by the formula (14), and for . Then, considering (26), the Euler–Lagrange equation on takes the following form: Calculating the -th components of the projection, we have the following:Hence, the system as follows:for . Applying Theorem 2, we obtain the following system of Hilfer fractional Equations (28) in the subinterval : Finally, we transform the system (29) into an equivalent system of fractional integral equations:where P is a polynomial of the form for arbitrary . In this way, we obtain the general solution of Equation (25) in the subinterval . 6. Conclusions and Future Work
In this paper, semilinear Hilfer fractional systems have been investigated. The symmetric fractional derivative and its properties have been analyzed. A symmetrized model for these systems has been proposed and examined with the nonlinear function , that depends on the time as well as on the state. The solution formula for the systems under consideration has been derived using the Laplace transform.
New controllability criteria for symmetrized Hilfer fractional systems have been formulated and proved (Theorems 3–5). The concepts of projection and reflection functions in the context of the symmetric Hilfer derivative have been introduced. A decomposition of the symmetric Hilfer fractional derivative, utilizing the projection function and reflection operator, has been presented. This decomposition not only provides the controllability criterion for symmetrized Hilfer systems (Theorem 5) but also clarifies the relationship between the system’s trajectory across subintervals.
The paper proposes a deterministic model with time-varying parameters. The only element of the system that is not time-varying is the main matrix of the system, matrix A. For continuous systems of the fractional order, the accurate analytical determination of the transition matrix becomes very challenging when the elements of the main matrix of the system vary with time. Numerical integration methods can potentially approximate the transition matrix, but such an approach does not provide a solution to the fractional-order differential equation in a form even close to that discussed in this study. This is undoubtedly a challenge for future research.
The findings provide a robust framework for analyzing and controlling symmetric fractional systems, with potential applications in modeling complex real-world phenomena governed by fractional dynamics. Symmetric Hilfer fractional derivatives can be useful in modeling symmetric processes, such as those found in physical models requiring symmetric non-local operators, including diffusion and wave equations. The structure of the proposed model strikes a balance between capturing complex behaviors and maintaining analytical and numerical tractability. Modeling a specific system requires carefully defining its elements. However, mathematical models often approximate real-world processes to some extent. In this paper, the presented symmetry methods are applied to the Euler–Lagrange equations of motion, enhancing the practicality of the theoretical results. In future work, the obtained results will be applied in a simulation environment to electrical circuits with mirror symmetry, such as differential amplifiers, mirror transformers, and impedance-matching systems.
Moreover, the obtained results can be further extended to study the antisymmetric Hilfer fractional derivative and its properties, offering a broader perspective on the dynamics of fractional systems.