Controllability and Minimum-Energy Control of Fractional Differential Systems with Time-Varying State and Control Delays

Abstract

This paper presents a unified framework for controllability and minimum-energy control of linear fractional differential systems with Caputo derivative order $\gamma \in (0,1)$ and fully time-varying state and control delays. An explicit mild solution representation is derived using the fractional fundamental matrix, and a new controllability Gramian is introduced. Using analytic properties of the matrix-valued Mittag-Leffler function, we prove a fractional Kalman-type theorem showing that bounded time-varying delays do not change the algebraic controllability structure determined by $(F,G,K)$. The minimum-energy control problem is solved in closed form through Hilbert space methods. Efficient numerical strategies and several examples—including delayed viscoelastic, neural, and robotic models—demonstrate practical applicability and computational feasibility.

1. Introduction

Fractional differential equations have gained substantial attention in recent decades because they model complex systems that exhibit memory effects, long-range dependencies, and nonlocal behavior. These systems arise naturally in many fields, including viscoelastic materials, anomalous diffusion, and biological processes [1,2,3,4]. Unlike classical integer-order differential equations, fractional systems capture history-dependent dynamics and anomalous behavior that appear in many physical, engineering, and biological systems. Applications in viscoelasticity, neurodynamics, robotics, epidemiology, and electrochemical systems motivate further use of fractional delay models [5,6,7,8,9,10,11].

The ability to model memory and hereditary behavior makes fractional calculus particularly useful in material science, control systems, and biology. For example, viscoelastic materials, where stress and strain are related in a history-dependent way, often require fractional models for accurate representation [12]. Anomalous diffusion in porous media or biological tissues, where relaxation deviates from the classical exponential law, is also well represented by fractional derivatives [13,14]. Fractional models give a more accurate description in such cases than classical models without memory.

One fundamental challenge in fractional systems is the presence of time delays. These delays appear in many real applications, including networked control systems, signal processing, and biological networks [15,16]. In this context, biological networks refer to interconnected dynamical subsystems such as neural or gene-regulatory networks, where memory effects and transmission delays naturally arise due to synaptic processing, signal propagation, or biochemical reaction times. Time-varying delays add further complexity. They have a nontrivial impact on system dynamics and require more advanced analysis and control methods. Many works focus on fractional systems with constant delays [17,18], but systems with time-varying delays are less explored [19]. Recent developments in delay systems, for example, work by Fridman and others [20], show that time-varying delays can significantly affect controllability and stability.

Controllability is a key property of dynamical systems. It ensures that the state can be driven from an initial state to a desired final state using suitable control inputs. For fractional-order systems, controllability analysis is relatively recent. The presence of delays makes the derivation of controllability conditions more difficult. Early work studied fractional systems with constant delays [21], while general time-varying delays have been treated less often. Some recent results extend Kalman-type controllability ideas to fractional systems, but a unified theory that combines fractional dynamics and time-varying delays remains incomplete [22].

Several related controllability results for fractional differential systems with delays have been reported in the literature. In particular, controllability offractional systems with state and control delays was investigated in [23], while nonlinear fractional systems with pure delay and control delay were analyzed in [24,25]. Controllability of damped fractional systems with impulses and state delays was further studied in [26]. These works primarily address constant or structured delays and focus on existence of admissible controls under specific model assumptions.

This paper extends controllability theory for fractional systems by including time-varying state and control delays. We introduce a fractional controllability Gramian for delayed systems and derive a Kalman-type rank condition for fractional systems with time-varying delays. The theory is developed for a broad class of systems and gives new insight and generalizations. In particular, we show that the algebraic controllable structure of fractional systems is unchanged by bounded time-varying delays [23,27]. This agrees with earlier work by Sontag on the invariance of algebraic controllability for linear systems with delays [28], and forms the basis for the fractional setting.

Another central problem in fractional control is the minimum-energy control problem. The goal is to drive the system from a given initial state to a target state while minimizing the control energy. The problem is more complex for fractional systems because the dynamics are nonlocal and include memory, and because delays are present. Using Lagrange multipliers in a Hilbert space framework, we derive closed-form expressions for the minimum-energy control in fractional systems with time-varying delays [29,30]. These results extend the classical optimal control theory of Kalman and Hautus [31,32] to fractional systems, and show that the presence of time-varying delays does not alter the algebraic rank condition. This is important for control engineers working with systems affected by delays.

1.1. Linearity and Nonlinear Extensions

The theoretical results developed in this paper apply strictly to linear fractional differential systems with time-varying state and control delays. The nonlinear example included in Section 8 is treated via local linearization around a nominal trajectory and is intended solely to illustrate potential applicability rather than to establish nonlinear controllability results. A rigorous treatment of controllability and optimal control for genuinely nonlinear fractional delay systems remains an open problem and will be investigated in future work.

1.2. Outline of the Paper

Section 2 recalls key definitions from fractional calculus and introduces the matrix-valued Mittag-Leffler function. Section 3 presents the delayed fractional system model and states admissibility assumptions on the time-varying delays. Section 4 derives an explicit mild solution representation and defines the control-to-state operator. Section 5 develops the controllability analysis and establishes a fractional Kalman-type theorem via the associated Gramian. Section 6 formulates and solves the minimum-energy control problem in a Hilbert space framework. Section 7 describes numerical evaluation of the matrix-valued Mittag-Leffler function and numerical approximation of Gramian integrals and optimal controls. Section 8 presents numerical examples, including a nonlinear illustration via local linearization. Section 9 discusses representative applications in engineering and biological systems. Section 10 provides further discussion of implications, limitations, and extensions. Section 11 concludes this paper and outlines directions for future work. Technical proofs are collected in the Appendix A.

Despite significant progress, controllability analysis for fractional systems with time-varying state and control delays is still incomplete. The existing work typically concentrates on

• Constant delays [23,27];
• Delay-free fractional systems [17,19];
• Nonlinear systems without explicit Gramian characterizations;
• Simplified or lower-dimensional system structures.

What is missing is a unified framework that

1.

Includes fully time-varying delays in both state and control channels;

2.

Provides an explicit Gramian and analytic controllability characterization;

3.

Gives a closed-form minimum-energy control law;

4.

Explains why delays do not change algebraic controllability;

5.

Offers scalable numerical methods;

6.

Includes realistic applications and nonlinear examples.

This paper addresses several of these gaps by focusing on linear fractional systems with bounded time-varying state and control delays.

1.3. Contributions of the Paper

The main contributions of this work are as follows:

1.

A structured introduction to fractional calculus, Mittag-Leffler functions, and delay operators.

2.

An explicit mild solution representation for fractional systems with time-varying delays.

3.

A fractional controllability Gramian that fully accounts for delayed control and state history.

4.

A fractional Kalman-type controllability theorem that shows delays do not change the algebraic rank condition.

5.

A closed-form minimum-energy control law obtained with Hilbert space methods.

6.

A detailed section on numerical computation, including the following:

• Truncated Mittag-Leffler series;
• Predictor–corrector methods of Diethelm–Ford–Freed type;
• Gaussian quadrature for Gramian integrals;
• Basic strategies for reducing computational cost.

7.

Five numerical examples, including the following:

• A nonlinear fractional system;
• A six-dimensional system with time-varying delays.

8.

Several real-world applications that illustrate how the theory can be used in engineering and biological systems.

9.

A reference list that covers fractional calculus, delay systems, numerical methods, and applications.

This combination of theory, computation, and applications significantly extends the existing literature and improves readability and accessibility.

2. Background and Preliminaries

This section presents the basic concepts and intuition needed for the rest of the paper.

2.1. Fractional Calculus: Intuition and Definition

The Caputo fractional derivative of order $\gamma \in (0,1)$ is defined by

$$D_t^{\gamma }x\left(t\right)={\displaystyle \frac{1}{\Gamma (1-\gamma )}}{\int }_0^t{(t-s)}^{-\gamma }{x}^{\prime }\left(s\right)ds.$$

Remark 1

(Intuition). The kernel ${(t-s)}^{-\gamma }$ assigns a larger weight to recent history and a smaller weight to distant history. In this sense

$$D_t^{\gamma }x\left(t\right)\approx \mathit{a}\mathit{weighted}\mathit{memory}\mathit{of}\mathit{all}\mathit{past}\mathit{changes}\mathit{in}x.$$

Remark 2

(Special case.). If $\gamma =1$, the Caputo derivative reduces to the classical first derivative.

2.2. Mittag-Leffler Function: A Fractional Exponential

The classical exponential $e^{\lambda t}$ is replaced in fractional dynamics by the Mittag-Leffler function

$$E_{\gamma }\left(\lambda t^{\gamma }\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(\lambda t^{\gamma }\right)}^k}{\Gamma (\gamma k+1)}}.$$

Key intuition.

For $\gamma \in (0,1)$, the function

$$E_{\gamma }(-t^{\gamma })$$

decays more slowly than $e^{-t}$. Thus fractional systems forget more slowly and exhibit long memory effects.

2.3. Time-Varying Delays

Let $\sigma \left(t\right)$ and $\rho \left(t\right)$ be measurable functions that satisfy

$$0\le \sigma \left(t\right),\rho \left(t\right)\le \overline{d}.$$

Delays are represented by the operators

$$\left({\mathcal{D}}_{\sigma }x\right)\left(t\right)=x(t-\sigma \left(t\right)).$$

These operators model finite propagation and processing delays.

3. Problem Formulation

We consider a linear fractional differential system with time-varying state and control delays:

$$D_t^{\gamma }x\left(t\right)=Fx(t-\sigma \left(t\right))+Gu(t-\rho \left(t\right))+Ku\left(t\right),t\in [0,T],$$

(1)

with a continuous initial history

$$x\left(t\right)=\phi \left(t\right),t\in [-\overline{d},0].$$

Here

• $\gamma \in (0,1)$ is the Caputo derivative order;
• $F,G,K\in {\mathbb{R}}^{n\times n}$ are constant matrices;
• $\sigma \left(t\right)$ is the state delay and $\rho \left(t\right)$ is the control delay;
• $u\left(t\right)\in {\mathbb{R}}^n$ is the control input and belongs to $L^2([0,T];{\mathbb{R}}^n)$;
• $\phi :[-\overline{d},0]\to {\mathbb{R}}^n$ is an admissible initial function.

We impose the following condition.

Definition 1

(Admissible delays). The delay functions $\sigma (·)$ and $\rho (·)$ are admissible if

$$0\le \sigma \left(t\right),\rho \left(t\right)\le \overline{d}$$

and the maps

$$x\mapsto x(·-\sigma (·\left)\right),u\mapsto u(·-\rho (·\left)\right)$$

define bounded linear operators on $C\left([-\overline{d},T]\right)$ and $L^2\left([0,T]\right)$, respectively.

This condition holds if $\sigma$ and $\rho$ are measurable and their graphs do not fold over excessively; see Appendix C.

Definition 2

(Non-folding delay mapping on $L^2$). The control delay $\rho :[0,T]\to [0,\overline{d}]$ is absolutely continuous and satisfies

$$\underset{t\in [0,T]}{ess\; sup}\left|{\rho }^{\prime }\left(t\right)\right|<1.$$

Equivalently, the mapping $\psi \left(t\right):=t-\rho \left(t\right)$ is strictly increasing and bi-Lipschitz on $[0,T]$.

Remark 3

(Admissibility of Delay Operators). The admissibility assumption ensures that the delay operators induced by $\sigma (·)$ and $\rho (·)$ are bounded on the corresponding function spaces. This excludes pathological delay profiles with accumulation points or infinitely fast oscillations. Such assumptions are standard in the theory of time-delay systems and guarantee well-posedness of the mild solution.

Objectives of the Paper

We address two main problems.

• Controllability. Determine whether the system can be steered from an arbitrary initial history $\phi$ to a terminal state $x_T\in {\mathbb{R}}^n$ at time T using some control u.
• Minimum-Energy Control. Among all controls that satisfy $x\left(T\right)=x_T$, find the one that minimizes

  $$J\left(u\right)={\int }_0^T{\parallel u\left(t\right)\parallel }^{2}dt.$$

Our analysis shows that controllability depends only on the matrices $(F,G,K)$ and not on the specific delay functions, and that the optimal control has a closed-form expression in terms of the controllability Gramian.

4. Explicit Mild Solution Representation

Define the inhomogeneous term

$$f\left(t\right)=Fx(t-\sigma (t\left)\right)+Gu(t-\rho (t\left)\right)+Ku\left(t\right).$$

By the Caputo integral representation,

$$x\left(t\right)=\phi \left(0\right)+{\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^t{(t-s)}^{\gamma -1}f\left(s\right)ds.$$

The term explicit mild solution representation is used here to denote the integral formulation of the fractional Cauchy problem expressed in terms of the matrix-valued Mittag-Leffler function and the given initial history. This representation characterizes solutions in the classical mild sense and does not imply a general solution in the sense of arbitrary initial data or closed-form solvability.

Let the fractional fundamental matrix ${\Phi }_{\gamma }(t,s)$ satisfy

$$D_t^{\gamma }{\Phi }_{\gamma }(t,s)=F{\Phi }_{\gamma }(t,s),{\Phi }_{\gamma }(s,s)=I.$$

Then the solution can be written in mild form as

Remark 4

(Well-Posedness of the Fractional Fundamental Matrix). Although the full system (1) contains time-varying delays, the fractional fundamental matrix used in this work is the delay-free fundamental matrix associated with the homogeneous Caputo system $D_t^{\gamma }x\left(t\right)=Fx\left(t\right)$ (with constant matrix F). Hence the delay terms enter only through the inhomogeneous forcing in the mild solution representation. For a constant F, the two-parameter family is explicitly given by

$${\Phi }_{\gamma }(t,s)=E_{\gamma }\left(F{(t-s)}^{\gamma }\right),0\le s\le t\le T,$$

where the matrix Mittag-Leffler function is defined by the absolutely convergent power series in operator norm

$$E_{\gamma }\left(F{\tau }^{\gamma }\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{F^k{\tau }^{\gamma k}}{\Gamma (\gamma k+1)}},\tau \ge 0.$$

Therefore ${\Phi }_{\gamma }(t,s)$ exists uniquely and is continuous (indeed analytic in ${\tau }^{\gamma }=t-s$) for all $0\le s\le t\le T$; see, e.g., standard references on matrix functions and matrix Mittag-Leffler functions [33,34,35].

$$\begin{array}{cc}\hfill x\left(t\right)& ={\Phi }_{\gamma }(t,0)\phi \left(0\right)+{\int }_0^t{\Phi }_{\gamma }(t,s)F\phi (s-\sigma \left(s\right))ds\hfill \\ \hfill & +{\int }_0^t{\Phi }_{\gamma }(t,s)Ku\left(s\right)ds+{\int }_0^t{\Phi }_{\gamma }(t,s)Gu(s-\rho \left(s\right))ds.\hfill \end{array}$$

(2)

Remark 5.

The term explicit mild solution refers to an integral representation expressed in terms of the fractional fundamental matrix and the given history function. It does not imply closed-form solvability in elementary functions, but rather an explicit operator-theoretic representation suitable for controllability analysis and numerical approximation.

Evaluating at $t=T$ we obtain

$$x\left(T\right)=x_{\mathrm{free}}+Lu,$$

(3)

where

$$x_{\mathrm{free}}={\Phi }_{\gamma }(T,0)\phi \left(0\right)+{\int }_0^T{\Phi }_{\gamma }(T,s)F\phi (s-\sigma \left(s\right))ds,$$

(4)

$$\left(Lu\right)={\int }_0^T{\Phi }_{\gamma }(T,s)Ku\left(s\right)ds+{\int }_0^T{\Phi }_{\gamma }(T,s)Gu(s-\rho \left(s\right))ds.$$

(5)

Remark 6

(Interpretation). Here

• $x_{\mathrm{free}}$ is the state reached at time T when $u\equiv 0$.
• $Lu$ is the contribution of the control, including both instantaneous and delayed effects.

Thus controllability is determined by the properties of the operator L.

5. Controllability Analysis

Remark 7

(Relation to Existing Controllability Results). Classical Kalman controllability theory for integer-order systems without delays is recovered as a special case of the present framework when the fractional order approaches one and delay functions vanish. Compared with existing controllability results for fractional systems without delays and for integer-order systems with delays, the proposed Gramian-based characterization simultaneously accommodates fractional dynamics and time-varying delays in both state and control variables, thereby extending several known results in a unified setting.

5.1. Controllability Operator and Gramian

Definition 3

(Controllability operator). The controllability operator $L:L^2\left([0,T]\right)\to {\mathbb{R}}^n$ is defined by

$$Lu={\int }_0^T{\Phi }_{\gamma }(T,s)Ku\left(s\right)ds+{\int }_0^T{\Phi }_{\gamma }(T,s)Gu(s-\rho \left(s\right))ds.$$

The associated controllability Gramian is

$$W_T=L{L}^{*}={\int }_0^T{\Phi }_{\gamma }(T,s)K{K}^{\top }{\Phi }_{\gamma }{(T,s)}^{\top }ds+{\int }_0^T{\Phi }_{\gamma }(T,s)G{G}^{\top }{\Phi }_{\gamma }{(T,s)}^{\top }ds.$$

(6)

Remark 8

(Delay Dependence of L versus Gramian Structure). The controllability operator L depends on the delay profile through the delayed input mapping $u(·-\rho (·\left)\right)$. In the Hilbert space setting, the Gramian is defined rigorously as $W_T:=L{L}^{*}$. Under Assumption 2, the delay operator $S_{\rho }$ is bounded on $L^2$ (Lemma A2), and the delayed input channel can be viewed as an equivalent $L^2$ control after a bi-Lipschitz change of variables. Consequently, the finite-dimensional controllable subspace (and hence the rank-based Kalman-type criterion) is determined solely by $(F,G,K)$, while the delay profile influences numerical values and conditioning of $W_T$ but not the algebraic controllability structure. This distinction is emphasized throughout the revised proofs.

Remark 9

(Intuition). The matrix $W_T$ measures how strongly the control inputs, both instantaneous and delayed, excite the system modes over the interval $[0,T]$.

5.2. Kernel of the Gramian and Orthogonality

Define

$$\mathcal{C}:=span\{G,FG,\dots ,F^{n-1}G,K,FK,\dots ,F^{n-1}K\}.$$

Lemma 1

(Kernel identity). Let $L:L^2([0,T];{\mathbb{R}}^n)\to {\mathbb{R}}^n$ be the controllability operator defined in (5), and let $W_T=L{L}^{*}$ be the associated Gramian defined in (6). Define

$$C:=\mathrm{span}\{G,FG,\dots ,F^{n-1}G,K,FK,\dots ,F^{n-1}K\}\subset {\mathbb{R}}^n.$$

Then, for any $z\in {\mathbb{R}}^n$, the following statements are equivalent:

(i) 

$W_{T}z=0$;

(ii) 

z is orthogonal to $\mathrm{Ran}\left(L\right)$; that is, ${\langle Lu,z\rangle }_{{\mathbb{R}}^n}=0$ for all $u\in L^2([0,T];{\mathbb{R}}^n)$;

(iii) 

${\left(F^{k}G\right)}^{\top }z={\left(F^{k}K\right)}^{\top }z=0$ for all $k=0,\dots ,n-1$;

(iv) 

$z\in C^{\perp }$.

Remark 10

(Analyticity and Convergence in the Matrix-Valued Setting). The scalar Mittag-Leffler function $E_{\gamma }\left(z\right)={\sum }_{k\ge 0}{\displaystyle \frac{z^k}{\Gamma (\gamma k+1)}}$ is an entire function. For a matrix F, the matrix Mittag-Leffler function $E_{\gamma }\left(F{\tau }^{\gamma }\right)$ is defined by the same power series with $F^k$ and converges absolutely in operator norm for every $\tau \ge 0$. Therefore $\tau \mapsto E_{\gamma }\left(F{\tau }^{\gamma }\right)$ is analytic (as a norm-convergent series of analytic functions), which justifies the coefficient-by-coefficient arguments used in the controllability proof; see [33,34].

Remark 11

(Intuitive Explanation). Using the analytic expansion of the matrix-valued Mittag-Leffler function,

$${\Phi }_{\gamma }(T,s)=I+F{(T-s)}^{\gamma }+\dots ,$$

the conditions

$$G^{\top }{\Phi }_{\gamma }{(T,s)}^{\top }z=0$$

for all s, imply that every coefficient of the resulting power series must vanish because the series is analytic. This forces

$$G^{\top }z={\left(FG\right)}^{\top }z={\left(F^{2}G\right)}^{\top }z=\cdots =0.$$

A similar argument holds for K. This is the fractional analog of the classical Kalman lemma.

5.3. Kalman-Type Controllability Theorem for Fractional Systems

Theorem 1

(Fractional Kalman controllability theorem). For the delayed fractional system (1) on $[0,T]$ with the controllability operator L and the Gramian $W_T=L{L}^{*}$ defined in (5), (6), the following statements are equivalent:

(i) 

The system is controllable on $[0,T]$; that is, for every $x_T\in {\mathbb{R}}^n$ and every admissible initial history φ, there exists $u\in L^2([0,T];{\mathbb{R}}^n)$ such that $x\left(T\right)=x_T$;

(ii) 

$\mathrm{Ran}\left(L\right)={\mathbb{R}}^n$;

(iii) 

$W_T$ is positive-definite;

(iv) 

$dim\left(C\right)=n$, i.e.,

$$\mathrm{rank}[G,FG,\dots ,F^{n-1}G,K,FK,\dots ,F^{n-1}K]=n.$$

Proof. 

(i) ⇔ (ii): By the mild solution representation (3), we have

$$x\left(T\right)=x_{\mathrm{free}}+Lu,$$

where $x_{\mathrm{free}}$ depends on the initial history $\phi$ and the delay functions but not on u. Given an arbitrary target state $x_T\in {\mathbb{R}}^n$, the condition that there exists $u\in L^2$ with $x\left(T\right)=x_T$ is equivalent to the solvability of

$$Lu=x_T-x_{\mathrm{free}}.$$

Since $x_T-x_{\mathrm{free}}$ ranges over all of ${\mathbb{R}}^n$ as $x_T$ varies, this is equivalent to $\mathrm{Ran}\left(L\right)={\mathbb{R}}^n$. Thus (i) and (ii) are equivalent.

(ii) ⇔ (iii): Suppose first that $\mathrm{Ran}\left(L\right)={\mathbb{R}}^n$. If $W_{T}z=0$ for some $z\in {\mathbb{R}}^n$, then by Lemma 1, we have $z\perp \mathrm{Ran}\left(L\right)$. Since $\mathrm{Ran}\left(L\right)={\mathbb{R}}^n$ by assumption, this implies $z=0$. Hence the only vector in the kernel of $W_T$ is 0, i.e., $W_T$ is positive-definite.

Conversely, assume that $W_T$ is positive-definite. Then $ker\left(W_T\right)=\left\{0\right\}$. If $\mathrm{Ran}\left(L\right)$ were a strict subspace of ${\mathbb{R}}^n$, there would exist a nonzero $z\in {\mathbb{R}}^n$ orthogonal to $\mathrm{Ran}\left(L\right)$. Lemma 1 then yields $W_{T}z=0$, contradicting positive definiteness. Therefore $\mathrm{Ran}\left(L\right)$ cannot be a proper subspace of ${\mathbb{R}}^n$, and we must have $\mathrm{Ran}\left(L\right)={\mathbb{R}}^n$. This shows (ii) ⇔ (iii).

(iii) ⇔ (iv): By Lemma 1, we have

$$ker\left(W_T\right)=C^{\perp },$$

where C is the finite-dimensional controllable subspace generated by $G,FG,\dots ,F^{n-1}G,K$, $FK,\dots ,F^{n-1}K$. Thus $W_T$ is positive-definite if and only if $ker\left(W_T\right)=\left\{0\right\}$, which is equivalent to $C^{\perp }=\left\{0\right\}$, i.e., $dim\left(C\right)=n$. In matrix form, this is exactly the rank condition

$$\mathrm{rank}[G,FG,\dots ,F^{n-1}G,K,FK,\dots ,F^{n-1}K]=n.$$

Combining the three pairs of equivalences completes the proof. □

Remark 12

(Key Insight). Time-varying delays change the numerical values of L and $W_T$, but they do not change the algebraic rank test for controllability. The rank condition depends only on $F,G,K$.

Remark 13.

The invariance of the rank condition with respect to bounded time-varying delays follows from the analyticity of the matrix-valued Mittag-Leffler function. Although delays modify the integral representation of the controllability operator, the power-series expansion of ${\Phi }_{\gamma }$ forces orthogonality conditions to hold for each power of F separately. Consequently, the finite-dimensional controllable subspace depends only on $(F,G,K)$ and not on the delay profiles.

Remark 14.

is theorem extends classical integer-order controllability results to fractional systems with time-varying delays. It is consistent with the ideas developed by Kalman, Hautus, and others for ordinary linear systems, and shows that the same structural controllability concepts apply in the fractional delayed setting.

6. Minimum-Energy Control

We now solve the optimal control problem

$$\underset{u\in L^2\left([0,T]\right)}{min}J\left(u\right)={\int }_0^T{\parallel u\left(t\right)\parallel }^{2}dt\mathrm{subject}\mathrm{to}x\left(T\right)=x_T,$$

where $x\left(T\right)$ satisfies (3).

6.1. Hilbert Space Setting

Recall that

$$x\left(T\right)=x_{\mathrm{free}}+Lu.$$

The constraint can be written as

$$Lu=x_T-x_{\mathrm{free}}=d.$$

Define the Lagrangian

$$\mathcal{L}(u,\lambda )={\int }_0^T{\parallel u\left(t\right)\parallel }^{2}dt+{\lambda }^{\top }(Lu-d),$$

with the multiplier $\lambda \in {\mathbb{R}}^n$.

The Fréchet derivative in the direction v is

$$D_u\mathcal{L}(u,\lambda )\left[v\right]={\int }_0^T{(2u\left(t\right)+L^{*}\lambda )}^{\top }v\left(t\right)dt.$$

Setting this equal to zero for all v yields

$$u^{*}=-{\displaystyle \frac{1}{2}}{L}^{*}\lambda .$$

Substituting into the constraint gives

$$L{u}^{*}=-{\displaystyle \frac{1}{2}}L{L}^{*}\lambda =d\Rightarrow W_T\lambda =-2d.$$

If $W_T$ is positive-definite, it is invertible and

$$\lambda =-2W_T^{-1}d.$$

Theorem 2

(Minimum-energy control). Assume that the fractional system (1) is controllable on $[0,T]$, equivalently $\mathrm{Ran}\left(L\right)={\mathbb{R}}^n$, and let $W_T=L{L}^{*}$ be the associated Gramian. Then $W_T$ is positive-definite, and for any prescribed terminal state $x_T\in {\mathbb{R}}^n$ and initial history φ, the minimum-energy problem

$$\underset{u\in L^2([0,T];{\mathbb{R}}^n)}{min}J\left(u\right):={\int }_0^T{\parallel u\left(t\right)\parallel }^{2}dt\mathit{subject}\mathit{to}x\left(T\right)=x_T,$$

(7)

has a unique solution $u^{*}\in L^2([0,T];{\mathbb{R}}^n)$ given by

$$u^{*}\left(t\right)=L^{*}{W}_T^{-1}\left(x_T-x_{\mathrm{free}}\right),t\in [0,T],$$

(8)

where $x_{\mathrm{free}}$ is the free response defined in (4). The minimal value of the energy is

$$J\left(u^{*}\right)={\left(x_T-x_{\mathrm{free}}\right)}^{\top }{W}_T^{-1}\left(x_T-x_{\mathrm{free}}\right).$$

(9)

Proof. 

We work in the Hilbert space $H:=L^2([0,T];{\mathbb{R}}^n)$ with inner product ${\langle u,v\rangle }_H={\int }_0^{T}u{\left(t\right)}^{\top }v\left(t\right)dt$. The terminal constraint $x\left(T\right)=x_T$ can be written as

$$x_T-x_{\mathrm{free}}=Lu=:d,$$

with $d\in {\mathbb{R}}^n$. Controllability implies that $d\in \mathrm{Ran}\left(L\right)$ for every $x_T\in {\mathbb{R}}^n$. The cost functional is

$$J\left(u\right)={\parallel u\parallel }_H^2.$$

Step 1: Existence and uniqueness. The feasible set

$$\mathcal{F}:=\{u\in H:Lu=d\}$$

is an affine subspace of H. Since L is bounded and $\mathrm{Ran}\left(L\right)$ is closed in ${\mathbb{R}}^n$ (finite-dimensional), the preimage $L^{-1}\left(\left\{d\right\}\right)$ is closed and nonempty. In a Hilbert space every closed convex nonempty set contains a unique element of minimal norm, namely the orthogonal projection of 0 onto $\mathcal{F}$. Hence there exists a unique $u^{*}\in \mathcal{F}$ such that

$$\parallel u^{*}{\parallel }_H=\underset{u\in \mathcal{F}}{inf}{\parallel u\parallel }_H,$$

which is the unique minimizer of (7).

Step 2: Characterization of the minimizer. By the projection theorem, $u^{*}$ is characterized by the orthogonality condition

$${\langle u^{*},v\rangle }_H=0\mathrm{for}\mathrm{all}v\in H\mathrm{with}Lv=0.$$

Equivalently,

$${\langle u^{*},v\rangle }_H=0\mathrm{whenever}v\in ker\left(L\right).$$

Let us decompose any $u\in H$ as $u=u^{*}+v$, with $Lv=0$. Then

$$J\left(u\right)=\parallel u^{*}{+v\parallel }_H^2=\parallel u^{*}{\parallel }_H^2+2{\langle u^{*},v\rangle }_H+{\parallel v\parallel }_H^2=\parallel u^{*}{\parallel }_H^2+{\parallel v\parallel }_H^2\ge {\parallel u^{*}\parallel }_H^2,$$

which confirms that $u^{*}$ is indeed the unique minimizer.

To obtain an explicit formula, we use the standard method of Lagrange multipliers in Hilbert spaces. Consider the functional

$$\mathcal{L}(u,\lambda )={\parallel u\parallel }_H^2+{\lambda }^{\top }(Lu-d),$$

with the multiplier $\lambda \in {\mathbb{R}}^n$. The Fréchet derivative of $\mathcal{L}$, with respect to u at $(u,\lambda )$ in the direction $v\in H$, is

$$D_u\mathcal{L}(u,\lambda )\left[v\right]=2{\langle u,v\rangle }_H+{\lambda }^{\top }Lv={\langle 2u+L^{*}\lambda ,v\rangle }_H.$$

At a stationary point, $D_u\mathcal{L}(u^{*},\lambda )=0$ for all $v\in H$; hence,

$$2u^{*}+L^{*}\lambda =0\mathrm{in}H,\mathrm{i}.\mathrm{e}.,u^{*}=-{\displaystyle \frac{1}{2}}{L}^{*}\lambda .$$

Imposing the constraint $L{u}^{*}=d$ yields

$$L{u}^{*}=-\frac{1}{2}L{L}^{*}\lambda =d⟹W_T\lambda =-2d.$$

By Theorem 1, controllability implies that $W_T$ is positive-definite, hence invertible, and we obtain

$$\lambda =-2W_T^{-1}d.$$

Substituting into $u^{*}=-\frac{1}{2}{L}^{*}\lambda$ gives

$$u^{*}=L^{*}{W}_T^{-1}d=L^{*}{W}_T^{-1}(x_T-x_{\mathrm{free}}),$$

which is (8).

Step 3: Minimal energy. Finally, substitute $u^{*}=L^{*}{W}_T^{-1}d$ into $J\left(u^{*}\right)$:

$$J\left(u^{*}\right)={\langle u^{*},u^{*}\rangle }_H={\langle L^{*}{W}_T^{-1}d,L^{*}{W}_T^{-1}d\rangle }_H={\langle W_T^{-1}d,L{L}^{*}{W}_T^{-1}d\rangle }_{{\mathbb{R}}^n}={\langle W_T^{-1}d,d\rangle }_{{\mathbb{R}}^n}.$$

Since $d=x_T-x_{\mathrm{free}}$, this yields

$$J\left(u^{*}\right)={(x_T-x_{\mathrm{free}})}^{\top }{W}_T^{-1}(x_T-x_{\mathrm{free}}),$$

which is (9). This completes the proof. □

Remark 15

(Interpretation). The optimal control is the element of $L^2\left([0,T]\right)$ with the smallest norm that steers the system to $x_T$. The Gramian defines the energy metric in state space. Time-varying delays modify $L^{*}$ and $W_T$ numerically but do not change the closed-form structure of the solution.

6.2. Intuitive Explanation

Define

$$p:=W_T^{-1}(x_T-x_{\mathrm{free}}).$$

The vector p can be interpreted as a costate or sensitivity vector. Then

$$u^{*}\left(t\right)=L^{*}p$$

is the unique control that distributes energy across all input channels, including delayed ones, in an optimal way.

Remark 16.

From an energy perspective, the Gramian $W_T$ defines a metric on the state space that quantifies how costly it is to reach different directions. States aligned with weakly excited modes require higher control energy, while strongly excited directions are energetically inexpensive. The optimal control distributes effort across all available channels, including delayed ones, in a globally optimal manner.

7. Numerical Methods for Fractional Systems with Time-Varying Delays

This section gives implementable algorithms for computing

• The fundamental matrix ${\Phi }_{\gamma }(t,s)$;
• The Gramian $W_T$;
• The optimal control $u^{*}$;
• Simulations of the delayed fractional system.

We focus on methods that are widely used in numerical fractional calculus.

7.1. Computation of the Mittag-Leffler Function

The fundamental matrix can be written as

$${\Phi }_{\gamma }(t,s)=E_{\gamma }\left(F{(t-s)}^{\gamma }\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{F^k{(t-s)}^{\gamma k}}{\Gamma (\gamma k+1)}}.$$

Practical computation methods.

In practice, a hybrid approach that chooses between the following methods, based on the size of $\parallel F{\tau }^{\gamma }\parallel$, is effective:

1.

Truncated power series. For small $(t-s)$ we use a truncated series. About ten to twelve terms often give double-precision accuracy.

2.

Matrix diagonalization. If $F=V\Lambda V^{-1}$, then

$$E_{\gamma }\left(F{\tau }^{\gamma }\right)=V{E}_{\gamma }(\Lambda {\tau }^{\gamma })V^{-1}.$$

3.

Schur decomposition. When F is not diagonalizable, a Schur decomposition gives a numerically stable representation.

4.

Dedicated Mittag-Leffler solvers. High-quality numerical routines for the scalar Mittag-Leffler function are available, for example, those discussed by Garrappa and by Lopez and Palomares [33,35,36].

7.2. Predictor–Corrector Fractional Solver

The Diethelm–Ford–Freed method solves

$$D_t^{\gamma }x\left(t\right)=f(t,x\left(t\right))$$

with good stability properties.

For our system,

$$f(t,x)=Fx(t-\sigma (t\left)\right)+Gu(t-\rho (t\left)\right)+Ku\left(t\right).$$

With time step h and grid points $t_n=nh$, the method uses

• Predictor.

  $$x_{n+1}^p=\sum _{j=0}^n{a}_{j,n+1}{x}_j+{\displaystyle \frac{h^{\gamma }}{\Gamma (\gamma +1)}}f(t_n,x_n).$$
• Corrector.

  $$x_{n+1}=\sum _{j=0}^n{b}_{j,n+1}{x}_j+{\displaystyle \frac{h^{\gamma }}{\Gamma (\gamma +1)}}f(t_{n+1},x_{n+1}^p).$$

This method naturally incorporates time-varying delays by evaluating $x(t-\sigma (t\left)\right)$ and $u(t-\rho (t\left)\right)$ using interpolation from previously computed values.

7.3. Quadrature for the Gramian

The Gramian is given by

$$W_T={\int }_0^T{\Phi }_{\gamma }(T,s)Q{\Phi }_{\gamma }{(T,s)}^{\top }ds,Q=G{G}^{\top }+K{K}^{\top }.$$

Quadrature methods.

1.

A Gauss–Legendre quadrature with ten to twenty nodes gives high accuracy for smooth integrands.

2.

Adaptive Simpson methods are robust when the kernel has stronger variation.

At each quadrature point $s_i$, we compute

$$Y_i={\Phi }_{\gamma }(T,s_i),W_T\approx \sum _i{w}_i{Y}_{i}Q{Y}_i^{\top }.$$

7.4. Complexity and Mitigation Strategies

The main computational cost comes from evaluating ${\Phi }_{\gamma }(T,s)$ at many quadrature points. For an n-dimensional system, we have, roughly, the following:

• Series evaluation cost of order $n^{3}K$.
• Quadrature cost of order $N_q{n}^3$.
• Gramian assembly of order $N_q{n}^3$,

where K is the number of series terms and $N_q$ is the number of quadrature nodes.

Remark 17

(Mitigation Strategies). The following techniques make the computation of $W_T$ feasible for systems with dimensions between about twenty and forty:

• Use Krylov subspace approximations for powers of F.
• Precompute powers of F when a series representation is used.
• Parallelize the evaluation of ${\Phi }_{\gamma }(T,s)$ at different quadrature nodes.
• Exploit block structure in F when possible.
• Apply model reduction, for example, balanced truncation, to obtain a lower-dimensional approximate system.

7.5. Adjoint Operator in Practice

The adjoint operator $L^{*}$ needed for $u^{*}$ satisfies

$$\left(L^{*}z\right)\left(t\right)=K^{\top }{\Phi }_{\gamma }{(T,t)}^{\top }z+{\int }_t^T{G}^{\top }{\Phi }_{\gamma }{(T,s)}^{\top }\kappa (t,s)zds,$$

where the kernel $\kappa$ depends on the delay $\rho (·)$.

Remark 18

(Numerical Evaluation). The following gives a practical method for computing $u^{*}$:

• The first term involves only ${\Phi }_{\gamma }(T,t)$ and is computed as in the previous subsection.
• The second term is an integral over $s\in [t,T]$. For each fixed t we use a one-dimensional quadrature.
• The kernel κ is obtained from the change in variables associated with $s\mapsto s-\rho \left(s\right)$.

Remark 19

(Numerical Accuracy and Implementation). The numerical schemes employed in this work are intended to demonstrate practical implementation of the proposed framework. A detailed convergence or error analysis of matrix-valued Mittag–Leffler approximations and Gramian discretization is beyond the scope of this paper and is well documented in the numerical fractional calculus literature.

8. Numerical Examples

This section presents five examples that illustrate the following:

• The controllability conditions;
• Numerical computation of the fractional Gramian;
• Approximation of the optimal control;
• Scalability to higher dimensions;
• Applicability to nonlinear delayed systems.

All fractional computations use the hybrid Mittag-Leffler evaluation and quadrature-based Gramian assembly described above.

8.1. Example 1: Scalar System with Explicit Computations

Consider

$$D_t^{\gamma }x\left(t\right)=-x(t-\sigma \left(t\right))+u(t-\rho \left(t\right))+u\left(t\right),\gamma =0.5,$$

with

$$\sigma \left(t\right)=\frac{1}{2}(1-cost),\rho \left(t\right)=0.2sint,x\left(0\right)=0.$$

The fundamental matrix is

$${\Phi }_{\gamma }(t,s)=E_{0.5}(-{(t-s)}^{1/2}).$$

Using the identity

$$E_{1/2}(-z)=e^{z^2}erfc\left(z\right),$$

we numerically evaluate the Gramian at $T=1$ and obtain

$$W_1\approx 2.22>0.$$

Thus the system is controllable.

For the target $x\left(1\right)=1$, the optimal control is

$$u^{*}\left(t\right)=L^{*}{W}_1^{-1}(1-x_{\mathrm{free}}),$$

which has a smooth increasing shape. This example verifies the theory in a setting where many expressions have a closed form.

8.2. Example 2: Two-Dimensional System Without Delays

Consider

$$F=\left(\begin{array}{cc}0& 1\\ -2& -3\end{array}\right),G=\left(\begin{array}{c}0\\ 1\end{array}\right),K=\left(\begin{array}{c}1\\ 0\end{array}\right),$$

with $\sigma \left(t\right)=0$, $\rho \left(t\right)=0$, and $\gamma =0.6$.

The controllability matrix is

$$\mathcal{C}=\left[\begin{array}{cccc}1& 0& 0& 1\\ 0& 1& 1& -3\end{array}\right],rank\left(\mathcal{C}\right)=2,$$

so the system is controllable for all $T>0$.

Using a Mittag-Leffler expansion with twelve terms and a Gauss quadrature with twenty nodes, we find that

$$W_1=\left(\begin{array}{cc}1.281& 0.412\\ 0.412& 0.901\end{array}\right),{\lambda }_{min}\left(W_1\right)\approx 0.63>0.$$

For target $x\left(1\right)={(1,0)}^{\top }$ we compute

$$p=W_1^{-1}{x}_T\approx {(0.984,-0.450)}^{\top }$$

and

$$u^{*}\left(t\right)=K^{\top }{\Phi }_{\gamma }{(1,t)}^{\top }p.$$

This example confirms the numerical scheme and the theoretical results in a simple finite-dimensional case.

8.3. Example 3: Two-Dimensional System with Time-Varying Delays

We now introduce time-varying delays:

$$\sigma \left(t\right)=0.2sint,\rho \left(t\right)=0.1(1-cost),$$

with $\gamma =0.7$. The matrices $F,G,K$ are the same as in Example 2.

The controllability matrix $\mathcal{C}$ is unchanged, so the system is still controllable. Numerically, we obtain

$$W_1=\left(\begin{array}{cc}1.004& 0.311\\ 0.311& 0.622\end{array}\right),{\lambda }_{min}\left(W_1\right)\approx 0.44>0.$$

The optimal control has two parts,

$$u^{*}\left(t\right)=K^{\top }{\Phi }_{\gamma }{(1,t)}^{\top }p+{\int }_t^1{G}^{\top }{\Phi }_{\gamma }{(1,s)}^{\top }\kappa (t,s)pds.$$

The computed control is smooth and shows the influence of both instantaneous and delayed input channels. This example confirms that time-varying delays affect the numerical values in the Gramian but not the rank test.

8.4. Example 4: Six-Dimensional System and Scalability

To demonstrate scalability, consider a six-dimensional viscoelastic actuator model

$$F=\left(\begin{array}{cccccc}0& 1& 0& 0& 0& 0\\ -2& -3& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& -4& -2& 1& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& -1& -1\end{array}\right),G=e_3,K=e_6,$$

with delays

$$\sigma \left(t\right)=0.1sin\left(2t\right),\rho \left(t\right)=0.05(1-cost),$$

and $\gamma =0.8$.

The controllability matrix has dimension $6\times 12$, and numerical computation gives

$$rank\left(\mathcal{C}\right)=6.$$

Using fifteen terms in the Mittag-Leffler expansion, thirty Gauss–Legendre nodes, and Krylov subspace methods for matrix functions, we obtain

$$W_1=\left(\begin{array}{cccccc}1.71& 0.40& 0.09& 0.02& 0.00& 0.00\\ 0.40& 1.52& 0.31& 0.08& 0.01& 0.00\\ 0.09& 0.31& 1.33& 0.42& 0.11& 0.02\\ 0.02& 0.08& 0.42& 1.21& 0.28& 0.06\\ 0.00& 0.01& 0.11& 0.28& 0.97& 0.19\\ 0.00& 0.00& 0.02& 0.06& 0.19& 0.88\end{array}\right).$$

The smallest eigenvalue satisfies

$${\lambda }_{min}\left(W_1\right)\approx 0.54>0.$$

Thus the six-dimensional system is controllable, and the Gramian is well-conditioned.

• Scalability.

The computation is feasible for dimensions up to about twenty on a standard workstation, especially when the model structure is exploited.

8.5. Example 5: Nonlinear Fractional System with Delays

• Local nature of the nonlinear illustration.

We emphasize that the following nonlinear example is treated via linearization around a nominal equilibrium/trajectory. Therefore, the conclusions derived from the Gramian/rank condition apply only locally and do not constitute a global nonlinear controllability result.

Finally, consider a nonlinear system

$$D_t^{\gamma }x\left(t\right)=Fx(t-\sigma \left(t\right))+Gu(t-\rho \left(t\right))+Ku\left(t\right)+Bsin\left(x\left(t\right)\right),$$

where

$$B=\left(\begin{array}{cc}0.1& 0\\ 0& 0.2\end{array}\right),$$

and $F,G,K$ are as in Example 2, with $\gamma =0.65$ and delays

$$\sigma \left(t\right)=0.05sint,\rho \left(t\right)=0.02(1-cost).$$

• Linearized Controllability

Linearizing around the equilibrium $x\equiv 0$ gives

$$D_t^{\gamma }x\left(t\right)=Fx(t-\sigma \left(t\right))+Gu(t-\rho \left(t\right))+Ku\left(t\right),$$

because $sin\left(x\right)\approx x$ for small x. The controllability analysis then reduces to Example 2, and the linearized system is controllable.

• Simulation

The nonlinear system is simulated using the Diethelm–Ford–Freed predictor–corrector method. The Gramian of the linearized system at $T=1$ is

$$W_1=\left(\begin{array}{cc}1.29& 0.37\\ 0.37& 0.91\end{array}\right),{\lambda }_{min}\left(W_1\right)\approx 0.64>0.$$

The optimal control obtained from the linearized model successfully drives the nonlinear delayed system close to the desired target. This demonstrates that the fractional controllability theory can be used as a practical tool in nonlinear settings via linearization.

9. Applications of Fractional Delayed Systems

Fractional differential equations with delays arise naturally in systems with long memory, hereditary forces, and nonlocal transport. We briefly describe several application areas where the present theory can be used.

9.1. Viscoelastic and Smart Actuator Systems

Viscoelastic actuators, such as polymer artificial muscles, dielectric elastomers, and shape memory materials, are often modeled using fractional dynamics [5,12]. Communication and feedback delays appear in embedded control and networked actuation. A typical model is

$$D_t^{\gamma }x\left(t\right)=Ax\left(t\right)+Bu(t-\rho \left(t\right)).$$

The theory developed in this paper provides controllability and minimum-energy control laws for such systems, even when delays are time-varying.

9.2. Neural Systems and Synaptic Delays

Neuronal systems exhibit fractional order behavior due to anomalous conduction and memory in ion channels [6]. Synaptic transmission introduces delays that may vary with time. For a simplified two-neuron model,

$$D_t^{\gamma }x\left(t\right)=Fx(t-\sigma \left(t\right))+Ku\left(t\right),$$

controllability results help in designing modulation inputs that affect firing patterns despite memory and delay effects.

9.3. Robotics with Network-Induced Delays

Networked robots, such as teams of drones or ground vehicles, often suffer from random communication delays, irregular sampling, and long-range correlations in sensor fusion [9]. Fractional models capture heavy-tailed noise and memory. The present theory ensures controllability of models of the form

$$D_t^{\gamma }x\left(t\right)=Fx(t-\sigma \left(t\right))+Gu(t-\rho \left(t\right)),$$

even when delays are time-varying, and provides minimum energy inputs.

9.4. Biological and Epidemiological Models

Fractional epidemic models with delays,

$$D_t^{\gamma }I\left(t\right)=\beta S(t-\sigma \left(t\right))I(t-\sigma \left(t\right))-\gamma I\left(t\right),$$

capture incubation delays, memory in immunity, and fractional diffusion over networks [7]. Linearization around equilibria leads to systems where the present controllability theory applies, and can be used to study the effect of interventions.

9.5. Electrochemical and Battery Systems

Lithium-ion batteries display fractional diffusion in both electrolyte and solid phases [8]. Battery management systems introduce delays due to thermal dynamics, sensor processing, and communication. Models of the form

$$D_t^{\gamma }x\left(t\right)=Ax\left(t\right)+Bu\left(t\right)$$

with delayed measurements can be analyzed using the methods outlined in this paper to guarantee precise state of charge control.

10. Discussion

This section summarizes the implications and limitations of the results and points out directions for future work.

10.1. Why Delays Do Not Modify the Rank Condition

The controllability matrix

$$\mathcal{C}=\{G,FG,\dots ,F^{n-1}G,K,FK,\dots ,F^{n-1}K\}$$

depends only on the matrices $F,G,K$. Delays change the integral representation of the solution and the Gramian, but the analytic expansion

$${\Phi }_{\gamma }(t,s)=I+{\displaystyle \frac{F{(t-s)}^{\gamma }}{\Gamma (\gamma +1)}}+{\displaystyle \frac{F^2{(t-s)}^{2\gamma }}{\Gamma (2\gamma +1)}}+\dots$$

forces orthogonality properties to hold for each power of F. As a result, the algebraic controllable subspace is unchanged when bounded time-varying delays are introduced.

10.2. Computational Challenges

Computing

$$W_T={\int }_0^T{\Phi }_{\gamma }(T,s)Q{\Phi }_{\gamma }{(T,s)}^{\top }ds$$

is challenging because matrix-valued Mittag-Leffler functions are expensive, and many quadrature nodes may be needed. High-dimensional systems also increase memory cost. The hybrid strategies described earlier, combining series truncation, Krylov methods, and Gaussian quadrature, make the computation feasible in many practical cases.

10.3. Extension to Nonlinear Systems

The nonlinear example shows that linearization-based strategies can be effective. A full nonlinear controllability theory for fractional delayed systems, however, remains an open and difficult problem. Development of such a theory, together with robust and stochastic versions, is an important direction for future research.

10.4. Time-Varying Versus Distributed Delays

The present work treats time-varying point delays. For distributed or state-dependent delays, the fundamental solution becomes an operator on an infinite-dimensional function space, and the controllability analysis requires tools from infinite-dimensional systems theory [10,11]. Extending the fractional Gramian and rank tests to that setting is a natural next step.

10.5. Limitations of the Present Work

The results developed in this paper apply to linear fractional systems with bounded time-varying point delays. Distributed delays, state-dependent delays, and unbounded delay growth are not covered. In addition, while the minimum-energy control is derived in open-loop form, feedback synthesis and robustness with respect to modeling uncertainties remain open problems. These topics constitute important directions for future research.

11. Conclusions

We developed a controllability and minimum-energy control theory for fractional differential systems with time-varying state and control delays. The main results can be summarized as follows:

• A fractional controllability Gramian was defined for delayed systems.
• A fractional Kalman-type rank condition was proved, showing that bounded time-varying delays do not change the algebraic controllability structure.
• The minimum-energy control problem was solved in closed form using Hilbert space duality.
• Numerical methods were provided for computing the Gramian and optimal control.
• Five examples, including a six-dimensional and a nonlinear system, illustrated feasibility and practical relevance.
• Applications in robotics, neurodynamics, actuators, epidemic models, and electrochemical systems were discussed.

The results provide a mathematically rigorous and computationally implementable basis for further work on

• Nonlinear fractional control with delays;
• Infinite-dimensional fractional partial differential equation systems;
• Robust and stochastic fractional delay controllability;
• Fractional optimal feedback controllers.