Symmetry Properties and Their Application to Hilfer Fractional Systems

Abstract

The paper investigates semilinear Hilfer fractional systems. A symmetric fractional derivative and its properties are discussed. A symmetrized model for these systems is proposed and examined. A bounded nonlinear function $f$ is applied, depending on the time as well as on the state. The Laplace transformation is used to derive the solution formula for the systems under consideration. The primary contribution of the paper is the formulation and proof of controllability criteria for symmetrized Hilfer systems. To deepen the understanding of the dynamics of such systems, the concept of reflection symmetries is introduced with a detailed analysis of their essential features, including projection functions and a reflection operator. Furthermore, a decomposition of the symmetric Hilfer fractional derivative is presented, utilizing the projection function and reflection operator. This decomposition not only provides a controllability condition for symmetrized Hilfer systems but also clarifies the relationship between the system’s trajectory across subintervals. Two illustrative examples are presented to demonstrate the computational and practical significance of the theoretical results.

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 $f:{\mathbb{R}}^{+}\to \mathbb{R}$ is a function of the real variable $t\in {\mathbb{R}}_{+}$ that is integrable over $[0,t]$ for $t>0$. Also, let $D^n={\displaystyle \frac{d^n}{d{t}^n}}$ denote the nth order differentiation operator.

The right-sided and left-sided integral operators $I_{a^{+}}^{\alpha }$ and $I_{b^{-}}^{\alpha }$ are, respectively, defined by the following formulas:

$$I_{a^{+}}^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{(t-\sigma )}^{\alpha -1}f\left(\sigma \right)d\sigma ,$$

(1)

$$I_{b^{-}}^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_t^b{(\sigma -t)}^{\alpha -1}f\left(\sigma \right)d\sigma ,$$

(2)

and are referred to as the Riemann–Liouville fractional integral (right-sided and left-sided, respectively) of order $\alpha \in {\mathbb{R}}_{+}$ for the function f, where $\Gamma$ is the Gamma function.

The right-sided operator $I_{0^{+}}^{\alpha }$ is usually denoted by $I^{\alpha }$.

The Hilfer operator is a generalization of the Caputo and the Riemann–Liouville operators, parameterized by two values: an order $\alpha$ and a type $\beta$. 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 $n-1<\alpha <n$ and $0⩽\beta ⩽1,$ for $n\in \mathbb{N}$. The right-sided and left-sided integro-differential operators $D_{a^{+}}^{\alpha ,\beta }$ and $D_{b^{-}}^{\alpha ,\beta }$ are, respectively, defined by the following formulas:

$$D_{a^{+}}^{\alpha ,\beta }f\left(t\right)=I_{a^{+}}^{\beta (1-\alpha )}{D}^n{I}_{a^{+}}^{(1-\alpha )(1-\beta )}f\left(t\right),$$

(3)

$$D_{b^{-}}^{\alpha ,\beta }f\left(t\right)=I_{b^{-}}^{\beta (1-\alpha )}(-D^n)I_{b^{-}}^{(1-\alpha )(1-\beta )}f\left(t\right),$$

(4)

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 $D_{0^{+}}^{\alpha ,\beta }$ is usually denoted by $D^{\alpha ,\beta }$.

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 $0<\alpha <1$ and $\beta =0,$ the Hilfer derivative becomes the Riemann–Liouville derivative. In particular, we have $D^{\alpha ,0}f\left(t\right)=D^{\alpha }f\left(t\right)=D{I}^{1-\alpha }f\left(t\right).$ For $0<\alpha <1$ and $\beta =1$, it reduces to the Caputo derivative, i.e., $D^{\alpha ,1}f\left(t\right)={}_{ }{}^{C}D_{ }^{\alpha }f\left(t\right)=I^{1-\alpha }Df\left(t\right).$

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 $\alpha \in (1,2)$ are discussed in [38].

Definition 2. 

Let $n-1<\alpha <n$ and $0⩽\beta ⩽1,$ for $n\in \mathbb{N}$. The symmetric Hilfer fractional derivative over the finite interval $[0,T]$ is defined by the following formula:

$${}_{ }{}^{s}D_{[0,T]}^{\alpha ,\beta }f\left(t\right)={\displaystyle \frac{1}{2}}\left[D^{\alpha ,\beta }f\left(t\right)+{(-1)}^{n+1}{D}_{T^{-}}^{\alpha ,\beta }f\left(t\right)\right].$$

(5)

Since the symmetric Riemann–Liouville fractional integrals over the finite interval $[0,T]$ are defined as follows:

$${}_{ }{}^{s}I_{[0,T]}^{\alpha }f\left(t\right)={\displaystyle \frac{1}{2\Gamma \left(\alpha \right)}}{\int }_0^T{|t-\sigma |}^{\alpha -1}f\left(\sigma \right)d\sigma ,$$

(6)

it follows that

$${}_{ }{}^{s}I_{[0,T]}^{\alpha }f\left(t\right)={\displaystyle \frac{1}{2}}\left[I^{\alpha }f\left(t\right)+I_{T^{-}}^{\alpha }f\left(t\right)\right],$$

(7)

and the following result is straightforward to prove.

Proposition 1. 

Let $n-1<\alpha <n$ and $0⩽\beta ⩽1,$ for $n\in \mathbb{N}$. For the symmetric Hilfer fractional derivative over the finite interval $[0,T]$, the following holds:

$${}_{ }{}^{s}D_{[0,T]}^{\alpha ,\beta }f\left(t\right)={}_{ }{}^{s}I_{ }^{\beta (1-\alpha )}{D}^n{}_{ }{}^{s}I_{ }^{(1-\alpha )(1-\beta )}f\left(t\right).$$

(8)

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:

$$\mathcal{L}\left[D^{\alpha ,\beta }f\left(t\right)\right]=s^{\alpha }\mathcal{L}\left[f\left(t\right)\right]-s^{(1-\alpha )\beta }{I}^{(1-\alpha )(1-\beta )}f(0+),$$

where $I^{(1-\alpha )(1-\beta )}f(0+)$ is the Riemann–Liouville integral of order $(1-\alpha )(1-\beta )$ evaluated as $t\to 0^{+}$. An important fact is that t$I^{(1-\alpha )(1-\beta )}f(0+)$ 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 formula

$$E_{\alpha ,\beta }\left(z\right)=\sum _{k=0}^{+\infty }{\displaystyle \frac{z^k}{\Gamma (\alpha k+\beta )}}$$

where $\alpha ,\beta \in {\mathbb{R}}_{+},z\in \mathbb{C}$.

If $A:X\to X$ is a bounded linear operator, the above definition can be extended to linear operators, i.e.,

$$E_{\alpha ,\beta }\left(A\right)=\sum _{k=0}^{+\infty }{\displaystyle \frac{A^k}{\Gamma (\alpha k+\beta )}}.$$

The Mittag–Leffler is a function of exponential order.

Additional notations used in this paper include the following:

• $L^2([0,+\infty ),$ ${\mathbb{R}}^m)$ denotes the Hilbert space of square-integrable functions with values in ${\mathbb{R}}^m$.
• $L_{loc}^2([0,+\infty ),$ ${\mathbb{R}}^m)$ represents the linear space of locally square-integrable functions with values in ${\mathbb{R}}^m$.
• $C_n[0,T]$ is the space of continuous ${\mathbb{R}}^n$-valued functions defined on $[0,T]$, with the following norm:

  $${\displaystyle \left|\right|y\left|\right|=\underset{1⩽i⩽n}{max}\left|y_i\left(t\right)\right|,t\in [0,T].}$$

Moreover, $B^T$ 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 ${}_{ }{}^{s}D_{[0,T]}^{\alpha ,\beta }$ denote the symmetric on the $[0,T]$ Hilfer fractional differential operator as defined in (5). Assume that $y\left(t\right)\in {\mathbb{R}}^n$ represents the state vector (in the case of fractional systems, also referred to as the pseudo-state vector) and that $v\in L_{\mathit{loc}}^2([0,+\infty ),{\mathbb{R}}^m)$ is the control (input function). Let A be an $n\times n$ matrix with real entries, $B\left(t\right)$ an $n\times m$ matrix with continuous bounded entries, and f a continuous nonlinear function $f:[0,T]\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ depending on time and state. Then, the system is described by the following:

$${}_{ }{}^{s}D_{ }^{\alpha ,\beta }y\left(t\right)=Ay\left(t\right)+B\left(t\right)v\left(t\right)+f(t,y\left(t\right)),t\in [0,T],$$

(9)

with zero initial conditions:

$$I^{(1-\alpha )(1-\beta )}y(0+)=0,$$

(10)

where $0<\alpha <1$ and $0⩽\beta ⩽1$. 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

$$s^{\alpha }\mathcal{L}\left[y\left(t\right)\right]-s^{(1-\alpha )\beta }{I}^{(1-\alpha )(1-\beta )}y(0+)=A\mathcal{L}\left[y\left(t\right)\right]+\mathcal{L}[B\left(t\right)v\left(t\right)+f(t,y\left(t\right))],$$

for any fixed $t⩾0$. Substituting the initial condition (10), we obtain the following:

$$s^{\alpha }\mathcal{L}\left[y\left(t\right)\right]-A\mathcal{L}\left[y\left(t\right)\right]=\mathcal{L}[B\left(t\right)v\left(t\right)+f(t,y\left(t\right))].$$

Rearranging, we find the following:

$$\mathcal{L}\left[y\left(t\right)\right]={(s^{\alpha }I-A)}^{-1}\mathcal{L}[B\left(t\right)v\left(t\right)+f(t,y\left(t\right))]=\mathcal{L}\left[t^{\alpha -1}{E}_{\alpha ,\alpha }\left(A{t}^{\alpha }\right)\right]\mathcal{L}[B\left(t\right)v\left(t\right)+f(t,y\left(t\right))].$$

Applying the convolution formula, we obtain the following:

$$\mathcal{L}\left[y\left(t\right)\right]=\mathcal{L}[\left(t^{\alpha -1}{E}_{\alpha ,\alpha }\left(A{t}^{\alpha }\right)\right)\ast (B\left(t\right)v\left(t\right)+f(t,y\left(t\right))].$$

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:

$$y\left(t\right)={\int }_0^t{(t-\sigma )}^{\alpha -1}{E}_{\alpha ,\alpha }\left(A{(t-\sigma )}^{\alpha }\right)[B\left(t\right)v\left(\sigma \right)+f(\sigma ,y\left(\sigma \right))]d\sigma .$$

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 $v\in L_{\mathit{loc}}^2([0,+\infty ),{\mathbb{R}}^m)$ be a control. Then, there exists a unique solution $y\left(T\right)\in {\mathbb{R}}^n$ to the symmetrized semilinear Hilfer fractional system (9) and (10), given by the following:

$$y\left(T\right)={\displaystyle \frac{1}{2}}\left({\int }_0^{{\displaystyle \frac{T}{2}}}{(T-\sigma )}^{\alpha -1}{E}_{\alpha ,\alpha }\left(A{(T-\sigma )}^{\alpha }\right)[B\left(\sigma \right)v\left(\sigma \right)+f(\sigma ,y\left(\sigma \right))]d\sigma \right.$$

(11)

$$\left.+{\int }_{{\displaystyle \frac{T}{2}}}^T{(\sigma -T)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(A{(\sigma -T)}^{\alpha }\right)[B\left(\sigma \right)v\left(\sigma \right)+f(\sigma ,y\left(\sigma \right))]d\sigma \right).$$

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 $[0,T]$ for the symmetrized Hilfer fractional system (9) and (10) is as follows:

$$\begin{array}{cc}K\left(T\right)=\{y\left(t\right)\in {\mathbb{R}}^n:y\left(T\right)=& {\displaystyle \frac{1}{2}}\left({\int }_0^{{\displaystyle \frac{T}{2}}}{(T-\sigma )}^{\alpha -1}{E}_{\alpha ,\alpha }\left(A{(T-\sigma )}^{\alpha }\right)[B\left(\sigma \right)v\left(\sigma \right)+f(\sigma ,y\left(\sigma \right))]d\sigma \right.\hfill \\ & \left.+{\int }_{{\displaystyle \frac{T}{2}}}^T{(\sigma -T)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(A{(\sigma -T)}^{\alpha }\right)[B\left(\sigma \right)v\left(\sigma \right)+f(\sigma ,y\left(\sigma \right))]d\sigma \right)\},\hfill \end{array}$$

(12)

for $T⩾0$ and $v\left(t\right)\in {\mathbb{R}}^m$.

Next, based on the definitions presented in [36], we formulate several definitions of controllability for the fractional system (9) and (10) on $[0,T]$.

Definition 5. 

The symmetrized Hilfer control system (9) and (10) is said to be controllable on $[0,T]$ to a nonempty set $S\subseteq {\mathbb{R}}^n$ if for each $\tilde{y}\in S$, there exists a control $\tilde{v}\in L^2([0,T],{\mathbb{R}}^m)$ such that $y\left(T\right)=\tilde{y}.$

If $S=\left\{0\right\}$, the system (9) and (10) is said to be null controllable on $[0,T]$.

Definition 6. 

The symmetrized Hilfer control system (9) and (10) is said to be locally controllable on $[0,T]$ if $K\left(T\right)$, defined by (4), includes a neighborhood of zero in ${\mathbb{R}}^n.$

Definition 7. 

The symmetrized Hilfer control system (9) and (10) is said to be globally controllable on $[0,T]$ if for each $\tilde{y}\in {\mathbb{R}}^n$, there exists a control $\tilde{v}\in L^2([0,T],{\mathbb{R}}^m)$ such that $y\left(T\right)=\tilde{y}.$

It follows that (9) and (10) is globally controllable on the close interval $[0,T]$ if $K\left(T\right)={\mathbb{R}}^n$.

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 $[0,T]$ and its behavior in the subintervals $[0,{\displaystyle \frac{T}{2}}]$ and $[{\displaystyle \frac{T}{2}},T]$. We will also demonstrate that the controllability of the system depends on its behavior over the right subintervals.

Assume that $f:{\mathbb{R}}_{+}\to \mathbb{R}$ is a function of $t\in {\mathbb{R}}_{+}$, which is integrable over $[0,T]$, and let $\left[\nu \right]=[{\nu }_1,\cdots ,{\nu }_n]$ be a vector whose components are either 0 or 1. We define a reflection function $\mathcal{R}$ on $[0,T]$ as follows:

$${\mathcal{R}}_{[0,T]}f\left(t\right)=f(T-t),$$

(13)

and a projection $f_{\left[\nu \right]}$ of the function f as follows:

$$f_{\left[\nu \right]}\left(t\right)={\displaystyle \frac{1}{2}}(1+{(-1)}^{\nu }{\mathcal{R}}_{[0,T]})f\left(t\right).$$

(14)

For the reflection operator (13) and the projection (14) it holds

$${\mathcal{R}}_{[0,T]}{f}_{\left[\nu \right]}\left(t\right)=f_{\left[\nu \right]}(T-t)={(-1)}^{\nu }{f}_{\left[\nu \right]}\left(t\right).$$

(15)

The following theorem specifies the representation of the symmetric Hilfer derivative in terms of the projection $f_{\left[\nu \right]}$ and reflection ${\mathcal{R}}_{[0,T]}$ functions.

Theorem 2. 

Let $f_{\left[\nu \right]}$, where $\left[\nu \right]=[{\nu }_1,\cdots ,{\nu }_n]$ and ${\nu }_i\in \{0,1\},i=1\dots ,n$, be the projection of f defined by the formula (14), $0<\alpha <1,$ and $0⩽\beta ⩽1,$ for $n\in \mathbb{N}$. The corresponding symmetric Hilfer fractional derivative in the interval $[0,T]$ is then represented as follows:

$${}_{ }{}^{s}D_{[0,T]}^{\alpha ,\beta }{f}_{\left[\nu \right]}\left(t\right)={(1+{(-1)}^{\nu }{\mathcal{R}}_{[0,T]})}^2{}_{ }{}^{s}D_{[{\displaystyle \frac{T}{2}},T]}^{\alpha ,\beta }{f}_{\left[\nu \right]}\left(t\right).$$

(16)

Proof. 

Using the definition of Hilfer derivative and the integration and reflection properties, we see that

$${}_{ }{}^{s}D_{[0,T]}^{\alpha ,\beta }{f}_{\left[\nu \right]}\left(t\right)={}_{ }{}^{s}I_{[0,T]}^{\beta (1-\alpha )}D{}_{ }{}^{s}I_{[0,T]}^{(1-\alpha )(1-\beta )}{f}_{\left[\nu \right]}\left(t\right)$$

$$={\displaystyle \frac{1}{2\Gamma \left(\beta \right(1-\alpha \left)\right)}}{\int }_0^T{|t-\sigma |}^{\beta (1-\alpha )-1}{f}_{\left[\nu \right]}\left(\sigma \right)d\sigma$$

$$·{\displaystyle \frac{d}{d\sigma }}{\displaystyle \frac{1}{2\Gamma \left(\right(1-\alpha \left)\right(1-\beta \left)\right)}}{\int }_0^T{|t-\sigma |}^{(1-\alpha )(1-\beta )-1}{f}_{\left[\nu \right]}\left(\sigma \right)d\sigma$$

$$={\displaystyle \frac{1}{2\Gamma \left(\beta \right(1-\alpha \left)\right)}}\left({\int }_0^{{\displaystyle \frac{T}{2}}}{|t-\sigma |}^{\beta (1-\alpha )-1}{f}_{\left[\nu \right]}\left(\sigma \right)d\sigma +{\int }_{{\displaystyle \frac{T}{2}}}^T{|t-\sigma |}^{\beta (1-\alpha )-1}{f}_{\left[\nu \right]}\left(\sigma \right)d\sigma \right)$$

$$·{\displaystyle \frac{1}{2\Gamma \left(\right(1-\alpha \left)\right(1-\beta \left)\right)}}{\displaystyle \frac{d}{d\sigma }}\left({\int }_0^{{\displaystyle \frac{T}{2}}}{|t-\sigma |}^{(1-\alpha )(1-\beta )-1}{f}_{\left[\nu \right]}\left(\sigma \right)d\sigma +{\int }_{{\displaystyle \frac{T}{2}}}^T{|t-\sigma |}^{(1-\alpha )(1-\beta )-1}{f}_{\left[\nu \right]}\left(\sigma \right)d\sigma \right)$$

$$=\left|\begin{array}{c}T-\sigma =\omega \hfill \\ d\sigma =-d\omega \hfill \\ f_{\left[\nu \right]}(t-\sigma )={(-1)}^{\nu }{f}_{\left[\nu \right]}\left(\omega \right)\hfill \\ \begin{array}{ccc}\sigma & 0& \frac{T}{2}\\ \omega & \frac{T}{2}& T\end{array}\hfill \end{array}\right|$$

$$={\displaystyle \frac{1}{2\Gamma \left(\beta \right(1-\alpha \left)\right)}}\left({(-1)}^{\nu }\left[{\int }_{{\displaystyle \frac{T}{2}}}^T{|T-t-\omega |}^{\beta (1-\alpha )-1}{f}_{\left[\nu \right]}\left(\omega \right)d\omega \right]+{\int }_{{\displaystyle \frac{T}{2}}}^T{|t-\sigma |}^{\beta (1-\alpha )-1}{f}_{\left[\nu \right]}\left(\sigma \right)d\sigma \right)$$

$$·{\displaystyle \frac{1}{2\Gamma \left(\right(1-\alpha \left)\right(1-\beta \left)\right)}}{\displaystyle \frac{d}{d\sigma }}\left({(-1)}^{\nu }\left[{\int }_{{\displaystyle \frac{T}{2}}}^T{|T-t-\omega |}^{(1-\alpha )(1-\beta )-1}{f}_{\left[\nu \right]}\left(\omega \right)d\omega \right]\right.$$

$$\left.+{\int }_{{\displaystyle \frac{T}{2}}}^T{|t-\sigma |}^{(1-\alpha )(1-\beta )-1}{f}_{\left[\nu \right]}\left(\sigma \right)d\sigma \right)$$

$$=(1+{(-1)}^{\nu }{\mathcal{R}}_{[0,T]}){}_{ }{}^{s}I_{[{\displaystyle \frac{T}{2}},T]}^{\beta (1-\alpha )}·(1+{(-1)}^{\nu }{\mathcal{R}}_{[0,T]})D{}_{ }{}^{s}I_{[{\displaystyle \frac{T}{2}},T]}^{(1-\alpha )(1-\beta )}{f}_{\left[\nu \right]}\left(t\right)$$

$$={(1+{(-1)}^{\nu }{\mathcal{R}}_{[0,T]})}^2{}_{ }{}^{s}D_{[{\displaystyle \frac{T}{2}},T]}^{\alpha ,\beta }{f}_{\left[\nu \right]}\left(t\right).$$

□

It is worth noting that the above theorem can subsequently be applied to the derivative ${}_{ }{}^{s}D_{[{\displaystyle \frac{T}{2}},T]}^{\alpha ,\beta }{f}_{\left[\nu \right]}\left(t\right),$ and so on. Therefore, the symmetrization process can be extended.

Next, we adopt the notation from [5] and define the following:

$$\Phi \left(t\right)=t^{\alpha -1}{E}_{\alpha ,\alpha }\left(A{t}^{\alpha }\right),$$

(17)

for the Mittag–Leffler function $E_{\alpha ,\alpha }\left(A{t}^{\alpha }\right)={\sum }_{k=0}^{+\infty }{\displaystyle \frac{A^k{t}^{\alpha k}}{\Gamma \left(\alpha \right(k+1\left)\right)}}.$ Since $E_{\alpha ,\alpha }$ is of exponential order and for $0<\alpha <1$, it holds that

$${\displaystyle \underset{t\to +\infty }{lim}}{t}^{\alpha -1}=0,$$

we conclude that for $t>0$ there exist constants $M>0$ and $\varrho ⩾0$ such that

$$\left|\right|\Phi \left(t\right)\left|\right|⩽M{e}^{\varrho t}.$$

The positive constant $\varrho$ 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 $[0,T]$ if the following controllability matrix is nonsingular:

$$W\left(T\right)=\underset{0}{\overset{{\displaystyle \frac{T}{2}}}{\int }}\Phi (T-\sigma )B\left(\sigma \right)B^T\left(\sigma \right){\Phi }^T(T-\sigma )d\sigma +\underset{{\displaystyle \frac{T}{2}}}{\overset{T}{\int }}\Phi (\sigma -T)B\left(\sigma \right)B^T\left(\sigma \right){\Phi }^T(\sigma -T)d\sigma .$$

(18)

Assume that the given continuous function f meets the assumptions:

$$\left|\right|f(t,y)\left|\right|⩽{\kappa }_1,$$

(19)

$$\left|\right|f(t,y)-f(t,\overline{y})\left|\right|⩽{\kappa }_2\left|\right|y-\overline{y}\left|\right|$$

(20)

for $y,\overline{y}\in {\mathbb{R}}^n$ and any positive constants ${\kappa }_1,{\kappa }_2$, provided that $0⩽{\kappa }_2<1$.

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 $W\left(T\right)$, 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 $[0,T]$.

Proof. 

Suppose that the Gramian matrix $W\left(T\right)$, defined by (18), is nonsingular. We conclude that the corresponding (9) and (10) linear system is globally controllable on $[0,T]$. Furthermore, under the assumption of bounded entries in the matrix $B\left(t\right)$, let us suppose that $|b_{ij}\left(t\right)|⩽N$ on $[0,T]$. Consider the following set:

$$S=\{y:|\left|y\right||⩽\epsilon \},$$

where $ϵ\in {\mathbb{R}}_{+}$ is defined as follows:

$$\epsilon =N_{\epsilon }\left[\right|y_T\left|\right||\Phi \left(T\right)||+{\kappa }_{1}T\left|\right|\Phi \left(T\right)\left|\right|]$$

with

$$N_{\epsilon }=\left|\right|W^{-1}\left(T\right)\left|\right|nN\left|\right|\Phi \left(T\right)\left|\right|,$$

where $\Phi \left(T\right)$ is represented as (17) and $W\left(T\right)$ is given by (18). S is closed, convex, and bounded on $C_n^{\alpha ,\beta }[0,T]$.

Then, we introduce a nonlinear transformation F that maps S into S as follows:

$$F_y\left(t\right)=B^T\left(t\right){\Phi }^T(T-t)W^{-1}\left(T\right)\left(\tilde{y}-{\int }_0^T\Phi (T-\sigma )f(\sigma ,y\left(\sigma \right))d\sigma \right).$$

The function $F_y$ is continuous in the Banach space $C_n^{\alpha ,\beta }[0,T]$.

It follows that for each $y\in S$, the following inequality holds:

$$\omega (F_y,h)⩽\omega (B^T{\Phi }^T,h)q,$$

where $\omega$ represents the modulus of continuity function [44], and

$$q=sup\{W^{-1}\left(T\right)(\tilde{y}-{\int }_0^T\Phi (T-\sigma )f(\sigma ,y\left(\sigma \right))d\sigma ):y\in S\}.$$

The modulus of continuity of the function $F_y$ is uniformly bounded, since neither B nor $\Phi$ depend on the choice of y in S, and are, therefore, equicontinuous. The modules of continuity of the functions ${}_{ }{}^{s}D_{ }^{\alpha ,\beta }{F}_y$ are as follows:

$$\begin{gathered} {|}^s{D}^{\alpha ,\beta }{F}_y\left(t\right){-}^s{D}^{\alpha ,\beta }{F}_y\left(s\right)|⩽|A{F}_y\left(t\right)-A{F}_y\left(s\right)| \\ +|B\left(t\right)F_y\left(t\right)-B\left(t\right)F_y\left(s\right)|+|f(t,F_y\left(t\right))-f(s,F_y\left(s\right))| \\ ⩽|A{F}_y\left(t\right)-A{F}_y\left(s\right)|+|B\left(t\right)F_y\left(t\right)-B\left(t\right)F_y\left(s\right)| \\ +|f(t,F_y\left(t\right))-f(t,F_y\left(t\right))|+|f(t,F_y\left(t\right))-f(s,F_y\left(s\right))| \\ ⩽{\eta }_1\left(\right|t-s\left|\right)+{\kappa }_2|y\left(t\right)-y\left(s\right)|+{\eta }_2\left(\right|t-s\left|\right), \end{gathered}$$

where the terms have the following bands:

• The first two modules to the right are bounded by ${\eta }_1\left(\right|t-s\left|\right)$, where ${\displaystyle \underset{h\to 0^{+}}{lim}{\eta }_1\left(h\right)=0}$ and ${\eta }_1$ takes a nonnegative values,
• The third term is bounded by ${\kappa }_2|y\left(t\right)-y\left(s\right)|$ due to assumption (20),
• The fourth term is bounded by ${\eta }_2\left(\right|t-s\left|\right)$, another nonnegative function satisfying ${\displaystyle \underset{h\to 0^{+}}{lim}{\eta }_2\left(h\right)=0}$.

Finally,

$${|}^s{D}^{\alpha ,\beta }{F}_y\left(t\right){-}^s{D}^{\alpha ,\beta }{F}_y\left(s\right)|⩽\eta (|t-s|)+{\kappa }_2|y\left(t\right)-y\left(s\right)|,$$

where $\eta ={\eta }_1+{\eta }_2$. This implies the following:

$$\omega {(}^s{D}^{\alpha ,\beta }{F}_y,h)⩽{\kappa }_2\omega {(}^s{D}^{\alpha ,\beta }y,h)+\eta \left(h\right).$$

Hence, for any set $S_1\subset S$:

$${\omega }_0\left(F_y{S}_1\right)=0\mathrm{and}{\omega }_0{(}^s{D}^{\alpha ,\beta }{F}_y{S}_1)⩽k_2{\omega }_0{(}^s{D}^{\alpha ,\beta }{S}_2),$$

where $S_2$ is the natural projection of $S_1$ onto the space $C_n^{\alpha ,\beta }[0,T].$ Since the measure of noncompactness is given by $\mu \left(S_1\right)={\displaystyle \frac{1}{2}}{\omega }_0\left(S_1\right)$, we obtain the following:

$$\mu \left(F_y{S}_1\right)⩽{\kappa }_2\mu \left(S_1\right).$$

By applying the generalized Darbo fixed-point theorem [45], it can be concluded that $F_y$ has a fixed point. Therefore, there exists the control

$$\tilde{v}\left(t\right)=\left\{\begin{array}{ccc}{B}^T\left(\sigma \right){\Phi }^T(T-t)W^{-1}\left(T\right)\left(\tilde{y}-{\int }_0^{{\displaystyle \frac{T}{2}}}\Phi (T-\sigma )f(\sigma ,y\left(\sigma \right))d\sigma \right)\hfill & for\hfill & t\in [0,{\displaystyle \frac{T}{2}}]\hfill \\ B^T\left(\sigma \right){\Phi }^T(t-T)W^{-1}\left(T\right)\left(\tilde{y}-{\int }_{{\displaystyle \frac{T}{2}}}^T\Phi (\sigma -T)f(\sigma ,y\left(\sigma \right))d\sigma )\right)\hfill & for\hfill & t\in [{\displaystyle \frac{T}{2}},T]\hfill \end{array}\right.$$

(21)

that steers the semilinear system from the zero initial state to a final state $\tilde{y}\in C_n^{\alpha ,\beta }[0,T]$. Therefore, it follows that the system (9) and (10) is globally controllable on the closed interval $[0,T]$. □

Theorem 4. 

Assume that $U\subset {\mathbb{R}}^m$ is compact and convex, with 0 as its interior point, and that $u\left(t\right)\in U$ represents an admissible control. Let the nonlinear function f satisfy conditions (19) and (20). If the controllability matrix $W\left(T\right)$, defined by (18), is nonsingular and $|arg\left({\lambda }_k\right)|>{\displaystyle \frac{\alpha \pi }{2}}$ holds for all the eigenvalues ${\lambda }_k$ of A and $1⩽k⩽n,$ then the symmetrized semilinear Hilfer system (9) and (10) is null controllable on the closed interval $[0,T]$.

Proof. 

Under the assumptions that conditions (19) and (20) hold and $W\left(T\right)$, given by (18), are nonsingular, Theorem 4 implies that the symmetrized Hilfer system (9) and (10) is controllable on $[0,T]$ assuming the controls are unconstrained. Now, suppose that the set $U\subset {\mathbb{R}}^m$ is compact and convex, with 0 as its interior point, and $u\left(t\right)\in U$. If $|arg\left({\lambda }_k\right)|>{\displaystyle \frac{\alpha \pi }{2}}$ holds for all eigenvalues ${\lambda }_k$ of A and $1⩽k⩽n,$ then the system (9) and (10) is asymptotically stable [46]. Therefore, for the control $u\left(t\right)=0$ the solution is $y=0$, and any trajectory $y\left(t\right)\in {\mathbb{R}}^n$ satisfies the following:

$$\underset{t\to +\infty }{lim}y\left(t\right)=0\mathrm{and}y\left(T\right)\in N\left(0\right),$$

for any $0<T<+\infty$ and a neighborhood $N\left(0\right)$ of $0\in {\mathbb{R}}^n$. Given the convexity and compactness of $U\subset {\mathbb{R}}^m$, any state $y\left(T\right)$ can reach zero in finite time. Thus, the semilinear Hilfer system (9) and (10) is null controllable on the closed interval $[0,T]$ with constrained controls $u\left(t\right)\in U$. □

Theorem 5. 

Let $0<\alpha <1,0⩽\beta ⩽1$. If the system of projections

$${}_{ }{}^{s}D_{ }^{\alpha ,\beta }{y}_{\left[\nu \right]}\left(t\right)=A{y}_{\left[\nu \right]}\left(t\right)+B\left(t\right)v\left(t\right)+f(t,y_{\left[\nu \right]}\left(t\right)),$$

(22)

with zero initial conditions is globally controllable on $[{\displaystyle \frac{T}{2}},T]$ for $T⩾0,$ then the symmetrized Hilfer control system described by (9) and (10) is globally controllable on the closed interval $[0,T]$.

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 $[0,T]$ into halves, $[0,{\displaystyle \frac{T}{2}}]$ and $[{\displaystyle \frac{T}{2}},T]$, 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:

$${}_{ }{}^{s}D_{ }^{{\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{2}}}y\left(t\right)=Ay\left(t\right)+B\left(t\right)u\left(t\right)+f(t,y\left(t\right)),$$

(23)

$$I^{(1-\alpha )(1-\beta )}y(0+)=0,$$

(24)

for $A=\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right]$, $B\left(t\right)=\left[\begin{array}{cc}0& t\\ 1& 0\end{array}\right]$, $f(t,y\left(t\right))=\left[\begin{array}{c}cosy\left(t\right)\\ -siny\left(t\right)\end{array}\right],$ and $t\in [0,2]$.

We examine its controllability using Theorem 3.

It can be easily calculated that for $n=2$ and $\alpha ={\displaystyle \frac{1}{3}}$, $\Phi \left(t\right)$, defined by (17), has the following form:

$$\Phi \left(t\right)=\sum _{k=0}^1{\displaystyle \frac{A^k{t}^{(k+1)\frac{1}{3}-1}}{\frac{1}{3}\Gamma \left((k+1)\right)}}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]{\displaystyle \frac{t^{-\frac{2}{3}}}{\Gamma \left(\frac{1}{3}\right)}}+\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right]{\displaystyle \frac{t^{-\frac{1}{3}}}{\Gamma \left(\frac{2}{3}\right)}}$$

$${\displaystyle =\left[\begin{array}{cc}\frac{t^{-\frac{2}{3}}}{\Gamma \left(\frac{1}{3}\right)}+\frac{t^{-\frac{1}{3}}}{\Gamma \left(\frac{2}{3}\right)}& 0\\ 0& \frac{t^{-\frac{2}{3}}}{\Gamma \left(\frac{1}{3}\right)}+\frac{2t^{-\frac{1}{3}}}{\Gamma \left(\frac{2}{3}\right)}\end{array}\right].}$$

Therefore, the controllability matrix $W\left(T\right)$, given by (18), for the Hilfer system (23) on the interval $[0,2]$ is as follows:

$$W\left(2\right)=\underset{0}{\overset{1}{\int }}\Phi (2-\sigma )B\left(\sigma \right)B{\left(\sigma \right)}^T{\Phi }^T(2-\sigma )d\sigma +\underset{1}{\overset{2}{\int }}\Phi (\sigma -2)B\left(\sigma \right)B{\left(\sigma \right)}^T{\Phi }^T(2-\sigma )d\sigma$$

$$=\left[\begin{array}{cc}{\displaystyle \frac{6\sqrt[3]{4}}{5{\left(\Gamma \left(\frac{1}{3}\right)\right)}^2}}+{\displaystyle \frac{2}{\Gamma \left(\frac{1}{3}\right)\Gamma \left(\frac{2}{3}\right)}}+{\displaystyle \frac{12\sqrt[3]{2}}{7{\left(\Gamma \left(\frac{2}{3}\right)\right)}^2}}& 0\\ 0& {\displaystyle \frac{6\sqrt[3]{4}}{5{\left(\Gamma \left(\frac{1}{3}\right)\right)}^2}}+{\displaystyle \frac{4}{\Gamma \left(\frac{1}{3}\right)\Gamma \left(\frac{2}{3}\right)}}+{\displaystyle \frac{48\sqrt[3]{2}}{7{\left(\Gamma \left(\frac{2}{3}\right)\right)}^2}}\end{array}\right].$$

Thus, $W\left(2\right)$ 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 $[0,2].$

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 $n-1<\alpha <n$ for $n\in \mathbb{N}$, then

$$D^{\alpha ,\beta }f\left(t\right)=D^{\alpha -(n-1),\beta }{D}^{n-1}f\left(t\right).$$

This property enables the application of the Hilfer fractional derivative of order $\alpha \in (0,1)$ to fractional problems of any order.

Let us adopt the simpler notation $Df\left(t\right)=f^{\prime }\left(t\right)$. We consider the symmetrized Hilfer fractional dynamical system described by the Euler–Lagrange equation of motion as follows:

$${\displaystyle \frac{\partial L^{\prime }}{\partial y\left(t\right)}}{+}^s{D}^{\alpha ,\beta }\left({\displaystyle \frac{\partial L^{\prime }}{{\partial }^s{D}^{\alpha ,\beta }y\left(t\right)}}\right)=0,$$

(25)

where $t\in [0,T]$, y is a differentiable real-valued function, and the action functional S is defined as follows:

$$S={\int }_0^T{\displaystyle \frac{1}{2}}{\left({}_{ }{}^{s}D_{ }^{\alpha ,\beta }y\left(\sigma \right)\right)}^{2}d\sigma ,$$

(26)

indicating that it depends only on the Hilfer derivatives and is independent of the Lagrangian L.

Assume that $y_{\left[\nu \right]}^{\prime }$ for $\left[\nu \right]=[{\nu }_1,{\nu }_2]$ and ${\nu }_i\in \{0,1\}$ is the projection of $y^{\prime }$ defined by the formula (14), $0<\alpha <1,$ and $0⩽\beta ⩽1,$ for $n\in \mathbb{N}$. Then, considering (26), the Euler–Lagrange equation on $[0,T]$ takes the following form:

$${}_{ }{}^{s}D_{[0,T]}^{\alpha ,\beta }{\left({}_{ }{}^{s}D_{ }^{\alpha ,\beta }{y}^{\prime }\left(t\right)\right)}_{\left[\nu \right]}=0.$$

(27)

Calculating the ${\nu }_i$-th components of the projection, we have the following:

$${\left({}_{ }{}^{s}D_{ }^{\alpha ,\beta }{y}^{\prime }\left(t\right)\right)}_{\left[0\right]}={}_{ }{}^{s}D_{[0,T]}^{\alpha ,\beta }{y}_{\left[0\right]}^{\prime }:={\gamma }_{\left[0\right]}$$

$${\left({}_{ }{}^{s}D_{ }^{\alpha ,\beta }{y}^{\prime }\left(t\right)\right)}_{\left[1\right]}={}_{ }{}^{s}D_{[0,T]}^{\alpha ,\beta }{y}_{\left[1\right]}^{\prime }:={\gamma }_{\left[1\right]}.$$

Hence, the system as follows:

$$\left\{\begin{array}{c}{}_{ }{}^{s}D_{[0,T]}^{\alpha ,\beta }{\gamma }_{\left[\nu \right]}=0\hfill \\ {}_{ }{}^{s}D_{[0,T]}^{\alpha ,\beta }{y}_{\left[\nu \right]}^{\prime }={\gamma }_{\left[\nu \right]},\hfill \end{array}\right.$$

(28)

for $\nu \in \{0,1\}$.

Applying Theorem 2, we obtain the following system of Hilfer fractional Equations (28) in the subinterval $[{\displaystyle \frac{T}{2}},T]$:

$$\left\{\begin{array}{c}{(1+{(-1)}^{\nu }{\mathcal{R}}_{[0,T]})}^2{}_{ }{}^{s}D_{[{\displaystyle \frac{T}{2}},T]}^{\alpha ,\beta }{\gamma }_{\left[\nu \right]}=0\hfill \\ {(1+{(-1)}^{\nu }{\mathcal{R}}_{[0,T]})}^2{}_{ }{}^{s}D_{[{\displaystyle \frac{T}{2}},T]}^{\alpha ,\beta }{y}_{\left[\nu \right]}^{\prime }={\gamma }_{\left[\nu \right]}.\hfill \end{array}\right.$$

(29)

Finally, we transform the system (29) into an equivalent system of fractional integral equations:

$$\left\{\begin{array}{c}{(1+{(-1)}^{\nu }{\mathcal{R}}_{[0,T]})}^2{}_{ }{}^{s}I_{[{\displaystyle \frac{T}{2}},T]}^{2-\alpha }{\gamma }_{\left[\nu \right]}=P\left(t\right)\hfill \\ {(1+{(-1)}^{\nu }{\mathcal{R}}_{[0,T]})}^2{}_{ }{}^{s}I_{[{\displaystyle \frac{T}{2}},T]}^{2-\alpha }{y}_{\left[\nu \right]}^{″}={\gamma }_{\left[\nu \right]},\hfill \end{array}\right.$$

(30)

where P is a polynomial of the form $P\left(t\right)=C_{1}t+C_2$ for arbitrary $C_1,C_2\in \mathbb{R}$.

In this way, we obtain the general solution of Equation (25) in the subinterval $[{\displaystyle \frac{T}{2}},T]$.

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 $f$, 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.