Investigating the controllability of differential systems with nonlinear fractional delays, characterized by the order 0 < η ≤ 1 < ζ ≤ 2

Rajveer Singh; Sachin Kumar; Ahmed H. Arnous; Hassen Aydi; Manuel De La Sen

doi:10.1515/dema-2025-0105

Article Open Access

Investigating the controllability of differential systems with nonlinear fractional delays, characterized by the order 0 < η ≤ 1 < ζ ≤ 2

Rajveer Singh , Sachin Kumar , Ahmed H. Arnous , Hassen Aydi and Manuel De La Sen

Published/Copyright: March 14, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

In this study, we investigate systems known as nonlinear fractional delay differential (nLFDD) systems, characterized by finite state delays and fractional orders within the range of 0 < η ≤ 1 < ζ ≤ 2 , situated infinite-dimensional settings. We utilize the controllability Gramian matrix to establish both necessary and sufficient conditions for the controllability of linear fractional delay differential systems that fall within the order range of 0 < η ≤ 1 < ζ ≤ 2 . Moreover, the Schauder fixed point theorem is employed to delineate the sufficient conditions required for the controllability of nLFDD systems, which are defined by finite state delays and fractional orders in the specified range. To substantiate the theoretical constructs put forth, we provide two illustrative examples.

Keywords: controllability; non-linear fractional delay differential system; Caputo fractional derivative

MSC 2010: 34K05; 26A33

1 Introduction

Fractional calculus [1–4] is an area of mathematics that extends the scope of classical calculus. It generalizes the integer-valued derivative and integral to fractional-valued. This unique mathematical framework provides the means to describe phenomena that cannot be adequately captured by traditional calculus. As a result, it has gained immense popularity in diverse scientific disciplines. Fractional calculus has proven invaluable in modeling and understanding systems characterized by non-local interactions, memory effects, and state-dependent delay. The fractional derivative has played a crucial role in various fields, including describing the dynamics of biological populations [5], simulating cell behavior [6,7], examining the development of illnesses [8–10], modeling anomalous diffusion [11,12], analyzing viscoelastic materials [13,14], understanding fractal phenomena [15], controlling systems [16], studying heat conduction in non-homogeneous materials [17], and analyzing electrical circuits [18].

Control theory [19–22] is a branch of mathematics and engineering that deals with manipulating or controlling the state of dynamical systems to achieve desired objectives. It has various applications in the fields of engineering, economics, biology, etc. Fractional differential equations (FDEs) have drawn more and more attention in control theory due to their ability to model systems with memory and long-range dependencies. In control theory, controllability is a key idea that determines whether control inputs can be used to bring a system’s state under desired control.

Kalman [23] presented the ideas of controllability in 1960. He formed the basis of the control theorem and was soon extended to an infinite-dimensional space. Recent research has shown that FDEs with a delay in the state variable are a useful tool for describing a wide range of phenomena in numerous domains. Numerous studies [24–30] have recently explored the concept of controllability within the realm of fractional control systems. The work of Kumar et al. [25] delves into the controllability aspects of fractional Langevin delay dynamical systems, while Selvam et al. [31] investigated the controllability of fractional dynamical systems employing the ψ -Caputo fractional derivative. Additionally, Zhou et al. [32] have articulated the controllability of fractional linear time-invariant neutral dynamical systems. Motivated by these foundational studies, this research further probes the controllability characteristics of the subsequent nonlinear fractional delay differential (nLFDD) system:

(1) D η C ( D ζ C z ( t ) + M z ′ ( t ) ) = N D η C z ( t − γ ) + g ( t , z ( t ) , z ( t − γ ) , u ( t ) ) + P u ( t ) , 0 < η ≤ 1 < ζ ≤ 2 , t ∈ I , D ζ C z ( t ) ∣ t = 0 = y 0 , z ( t ) = ψ ( t ) , t ∈ [ − γ , 0 ] , z ′ ( 0 ) = z 1 ,

where the vector y ∈ R m , the control vector u ∈ R k . M , N , and P are the real matrices with order m × m , m × m , and m × k , respectively, and g : I × R m × R m × R k → R m is continuous, and m > k . For ζ = 2 and η = 0 equation (1) reduces to the second-order equation of the Kelvin-Voigt model, and for ζ = 0 and η = 0 equation (1) reduces to the first-order equation of the Kelvin-Voigt model [33].

In this study, we will begin by introducing fundamental concepts and terminology that will be essential for establishing our controllability results. Moving on to Section 3, we will unveil the sufficient and necessary criteria for determining the controllability of a linear fractional delay differential (LFDD) system. The controllability results for a state-delayed nLFDD system will be examined in detail in Section 4. Finally, the last section will discuss the two examples to illustrate the theoretical concepts.

2 Preliminaries

For simplicity, throughout this article, we are taking Riemann-Liouville fractional integral as I ζ and Caputo fractional derivative as D ζ C .

Definition 1

[1] For a function j ( w ) , the Riemann-Liouville fractional integral of order ζ > 0 is defined by

I ζ j ( w ) = 1 Γ ( ζ ) ∫ 0 w ( w − t ) ζ − 1 j ( t ) d t ,

where Γ ( ⋅ ) is the gamma function.

Definition 2

[1] For a suitable function j ( w ) , the Caputo fractional derivative of order m − 1 < ζ ≤ m ( m ∈ N ) is given as

( D ζ C j ) ( w ) = 1 Γ ( m − ζ ) ∫ 0 w ( w − t ) m − ζ − 1 j ( m ) ( t ) d t ,

where j ( m ) ( t ) = d m j d t m . For ζ ∈ ( 0 , 1 ] ,

( D ζ C j ) ( w ) = 1 Γ ( 1 − ζ ) ∫ 0 w ( w − t ) − ζ j ′ ( t ) d t ,

and for ζ ∈ ( 1 , 2 ] ,

( D ζ C j ) ( w ) = 1 Γ ( 2 − ζ ) ∫ 0 w ( w − t ) 1 − ζ j ” ( t ) d t .

Definition 3

[1] The Mittag-Leffler function with one parameter ( ζ ) for a matrix M is defined by

E ζ ( M ) = ∑ i = 0 ∞ M i Γ ( ζ i + 1 ) .

Definition 4

[1] The Mittag-Leffler function with two parameters ( ζ , η ) for a matrix M is defined by

E ζ , η ( M ) = ∑ i = 0 ∞ M i Γ ( ζ i + η ) .

Definition 5

[34] The definition of the Laplace transformation of a function j ( w ) for w ∈ ( 0 , ∞ ) is

ℒ [ j ( w ) ] = ∫ 0 ∞ e − s w j ( w ) d w .

The convolution of two functions j ( w ) and h ( w ) is given by

(2) h ( w ) ⁎ j ( w ) = ∫ 0 w h ( w − v ) j ( w ) d v ,

and the Laplace transformation of (2) is expressed as

ℒ [ h ( w ) ⁎ j ( w ) ] = ℒ [ h ( w ) ] ℒ [ j ( w ) ] .

The Laplace transformation of the Mittag-Leffler function with one parameter is given by

ℒ − 1 [ E ζ ( ± M w ) ] ( s ) = s ζ − 1 s ζ ∓ M , Re ( ζ ) > 0 .

The Laplace transformation of the Mittag-Leffler function with two parameters is given by

ℒ − 1 [ w η − 1 E ζ , η ( ± M w ) ] ( s ) = s ζ − η s ζ ∓ M , Re ( ζ ) , Re ( η ) > 0 .

If J ( s ) = ℒ [ j ( w ) ] ( s ) for Re ( s ) > 0 , then

J ( s − a ) = ℒ [ e a w j ( w ) ] ( s ) , ℒ [ u a ( w ) j ( w − a ) ] ( s ) = e − a s J ( s ) ,

and

ℒ − 1 [ e − a s J ( s ) ] ( w ) = u a ( w ) j ( w − a ) ,

where u a ( w ) is a step function.

3 Linear systems with finite state delay

Consider the following LFDD system:

(3) D η C ( D ζ C z ( t ) + M z ′ ( t ) ) = N D η C z ( t − γ ) + P u ( t ) , 0 < η ≤ 1 < ζ ≤ 2 , t ∈ I , z ′ ( 0 ) = z 1 , D ζ C z ( t ) ∣ t = 0 = y 0 , z ( t ) = ψ ( t ) , t ∈ [ − γ , 0 ] ,

where z ( t ) ∈ R m , z ( 0 ) = z o and the control vector u ( t ) ∈ R k . M , N , and P are the real matrices with order m × m , m × m , and m × k , respectively, with m > k . ψ ( t ) is a continuous function on [ − γ ,0]. Using the Laplace transformation on (3), we have

(4) s η ( s ζ z ˜ ( s ) − s ζ − 1 ψ ( 0 ) ) − s η − 1 y 0 + M ( s η ( s z ˜ ( s ) − ψ ( 0 ) ) − s η − 1 z 1 ) = N s η ∫ 0 ∞ e − s t z ( t − γ ) d t − s η − 1 ψ ( − γ ) + P u ˜ ( s ) ,

which can be simplified to

(5) ( s ζ I + M s − N s − 1 e − s γ ) z ˜ ( s ) = s ζ − 2 ψ ( 0 ) + s ζ − 3 z 1 + s − 2 y 0 + M s − 1 ψ ( 0 ) + M s − 2 z 1 + N s − 1 e − s γ ∫ − γ 0 e − s t ψ ( t ) d t − N s − 2 ψ ( − γ ) + P s − η − 1 u ˜ ( s ) .

After simplifying, system (5) can be expressed as

(6) z ˜ ( s ) = s ζ − 2 s ζ I + M s − N s − 1 e − s γ ψ ( 0 ) + s ζ − 3 s ζ I + M s − N s − 1 e − s γ z 1 + s ζ − 2 s ζ I + M s − N s − 1 e − s γ s − ζ y 0 + s ζ − 2 s ζ I + M s − N s − 1 e − s γ s 1 − ζ M ψ ( 0 ) + s ζ − 2 s ζ I + M s − N s − 1 e − s γ s − ζ M z 1 + s ζ − 2 s ζ I + M s − N s − 1 e − s γ s 1 − ζ N e − s γ ∫ − γ 0 e − s t ψ ( t ) d t − s ζ − 2 s ζ I + M s − N s − 1 e − s γ s − ζ N ψ ( − γ ) + s ζ − 2 s ζ I + M s − N s − 1 e − s γ s 1 − ζ − η P u ˜ ( s ) .

Using inverse Laplace transformation on (6), we obtain

(7) z ( t ) = ℒ − 1 s ζ − 2 s ζ I + M s − N s − 1 e − s γ ( t ) ψ ( 0 ) + ℒ − 1 s ζ − 3 s ζ I + M s − N s − 1 e − s γ ( t ) z 1 + ℒ − 1 s ζ − 2 s ζ I + M s − N s − 1 e − s γ s − ζ ( t ) y 0 + ℒ − 1 s ζ − 2 s ζ I + M s − N s − 1 e − s γ s 1 − ζ ( t ) M ψ ( 0 ) + ℒ − 1 s ζ − 2 s ζ I + M s − N s − 1 e − s γ s − ζ ( t ) M z 1 + ℒ − 1 s ζ − 2 s ζ I + M s − N s − 1 e − s γ s 1 − ζ ( t ) ⁎ N ℒ − 1 e − s γ ∫ − γ 0 e − s t ψ ( t ) d t ( t ) − ℒ − 1 s ζ − 2 s ζ I + M s − N s − 1 e − s γ s − ζ ( t ) N ψ ( − γ ) + ℒ − 1 s ζ − 2 s ζ I + M s − N s − 1 e − s γ s 1 − ζ − η ( t ) ⁎ P u ( t ) .

For simplicity of notation, let us take

(8) Ξ ζ ( t ) = ℒ − 1 s ζ − 2 s ζ I + M s − N s − 1 e − s γ ( t )

Ξ ζ , 2 ( t ) = t − 1 ℒ − 1 s ζ − 3 s ζ I + M s − N s − 1 e − s γ ( t ) Ξ ζ , ζ + 1 ( t ) = t − ζ ℒ − 1 s ζ − 2 s ζ I + M s − N s − 1 e − s γ s − ζ ( t ) Ξ ζ , ζ ( t ) = t 1 − ζ ℒ − 1 s ζ − 2 s ζ I + M s − N s − 1 e − s γ s 1 − ζ ( t ) Ξ ζ , ζ + ζ ( t ) = t 1 − ζ − η ℒ − 1 s ζ − 2 s ζ I + M s − N s − 1 e − s γ s 1 − ζ − η ( t ) .

Let us define ω (t): [ − γ , ∞ ) → [ 0 , 1 ] by ω ( t ) = 1 for t < 0 and ω ( t ) = 0 for t ≥ 0 . Extend the function ψ ( t ) from [ − γ , 0 ] to [ − γ , ∞ ) by taking ψ ( t ) = ψ ( 0 ) for t ≥ 0 , then

(9) e − s γ ∫ − γ 0 e − s t ψ ( t ) d t = e − s γ ∫ 0 γ e − s ( − γ + μ ) ψ ( − γ + μ ) d μ = ∫ 0 ∞ e − s μ ω ( − γ + μ ) ψ ( − γ + μ ) d μ = ℒ [ ω ( − γ + . ) ψ ( − γ + . ) ] ( s ) .

Thus, the solution for (3) is as follows:

(10) z ( t ) = z ( t ; ψ , y 0 , z 1 ) + ∫ 0 t ( t − s ) ζ + η − 1 Ξ ζ , ζ + η ( t − s ) P u ( s ) d s ,

where

z ( t ; ψ , y 0 , z 1 ) = Ξ ζ ( t ) ψ ( 0 ) + t ζ Ξ ζ , ζ + 1 ( t ) y 0 + t Ξ ζ , 2 ( t ) z 1 + t ζ − 1 Ξ ζ , ζ ( t ) M ψ ( 0 ) + t ζ Ξ ζ , ζ + 1 ( t ) M z 1 − t ζ Ξ ζ , ζ + 1 ( t ) N ψ ( − γ ) + ∫ 0 t ( t − s ) ζ − 1 Ξ ζ , ζ ( t − s ) N ψ ( s − γ ) d s .

Definition 6

If there is a continuous control function u ( t ) such that the solution to (3) satisfies z ( T ) = z T for each y 0 , z 1 , z T ∈ R m and initial function ψ (t), then system (3) is completely controllable on I .

The controllability Gramian matrix is defined by

(11) ℋ = ∫ 0 T ( T − s ) 2 ( ζ + η − 1 ) [ Ξ ζ + η ( T − s ) P ] [ Ξ ζ + η ( T − s ) P ] ⊺ d s ,

where the symbol “ ⊺ ” represents the matrix transpose.

Theorem 1

The linear system (3)is completely controllable on [ 0 , T ] if and only if the Gramian matrix

(12) ℋ = ∫ 0 T ( T − s ) 2 ( ζ + η − 1 ) [ Ξ ζ + η ( T − s ) P ] [ Ξ ζ + η ( T − s ) P ] ⊺ d s

is non-singular.

Proof

Let ℋ be non-singular. Therefore, its inverse is well defined. Let the initial function ψ (t) be continuous on [ − γ ,0]. Define the control function as

(13) u ( t ) = ( T − t ) ζ + η − 1 [ Ξ ζ , ζ + η ( T − t ) P ] ⊺ ℋ − 1 ( z T − z ( T ; ψ , y 0 , z 1 ) ) .

Substituting u ( t ) into (10) at t = T yields

(14) z ( T ) = z ( T ; ψ , y 0 , z 1 ) + ∫ 0 T ( T − s ) 2 ( ζ + η − 1 ) Ξ ζ , ζ + η − 1 ( T − s ) P [ Ξ ζ , ζ + η × ( T − s ) P ] ⊺ d s × ℋ − 1 [ z T − z ( T ; ψ , y 0 ) ] ,

which simplified to

z ( T ) = z T .

Therefore, the linear system (1) exhibits the controllability.

Conversely, let ℋ be singular. It implies there exists a vector v ≠ 0 such that v ⊺ ℋ v = 0 , i.e.,

v ⊺ ∫ 0 T ( T − s ) 2 ( ζ + η − 1 ) [ Ξ ζ , ζ + η ( T − s ) P ] [ Ξ ζ , ζ + η ( T − s ) P ] ⊺ v d s = 0 ,

which implies

(15) v ⊺ ( T − s ) ζ + η − 1 ( Ξ ζ , ζ + η ( T − s ) P ) = 0 , for all s in [ 0 , T ] .

Consider the initial function ψ = 0 , y 0 = 0 , z 1 = 0 and the final point z T = v , which gives z ( T ; ψ , y 0 , z 1 ) = 0 . Since the given system is controllable, there exists a control u ( t ) on I that steers the z ( 0 ) = z o to z ( T ) = v ; thus,

(16) v = ∫ 0 T ( T − s ) ζ + η − 1 Ξ ζ , ζ + η ( T − s ) P u ( s ) d s .

From (15) and (16), we obtain

v ⊺ v = 0 .

This contradicts that v ≠ 0 . Thus, ℋ is non-singular.□

4 Nonlinear systems with finite state delay

Consider the nLFDD systems of the form:

(17) D η C ( D ζ C z ( t ) + M z ′ ( t ) ) = N D η C z ( t − γ ) + g ( t , z ( t ) , z ( t − γ ) , u ( t ) ) + P u ( t ) , t ∈ I , 0 < η ≤ 1 < ζ ≤ 2 z ′ ( 0 ) = z 1 , D ζ C z ( t ) ∣ t = 0 = y 0 , z ( t ) = ψ ( t ) , t ∈ [ − γ , 0 ] ,

where z ( t ) is a state variable of system in R m , z ( 0 ) = z o , and u ( t ) is a control input of system in R k . g : I × R m × R m × R k → R m is continuous. M , N , and P are the real matrices with order m × m , m × m , and m × k , respectively, with m > k . Then, for t ∈ I , the solution z ( t ) of system (17) may be expressed as

z ( t ) = ψ ( t ) for t ∈ [ − γ , 0 ] ,

(18) z ( t ) = z ( t ; ψ , y 0 , z 1 ) + ∫ 0 t ( t − s ) ζ + η − 1 Ξ ζ , ζ + η ( t − s ) g ( s , z ( s ) , z ( s − γ ) , u ( s ) ) d s + ∫ 0 t ( t − s ) ζ + η − 1 Ξ ζ , ζ + η ( t − s ) P u ( s ) d s ,

for t ∈ I .

Theorem 2

The nLFDD system (17) is completely controllable on I if the linear system (3) is completely controllable on I and the continuous function g fulfils the criterion:

(19) lim ∣ ∣ q ∣ ∣ → ∞ ∣ ∣ g ( t , q ) ∣ ∣ ∣ ∣ q ∣ ∣ = 0 , u n i f o r m l y f o r a l l t ∈ I .

Proof

Consider C * to be the space of all continuous functions of

( z , u ) : [ − γ , T ] × [ 0 , T ] → R m × R k ,

with the norm

∣ ∣ ( z , u ) ∣ ∣ = ∣ ∣ z ∣ ∣ + ∣ ∣ u ∣ ∣ ,

where ∣ ∣ u ∣ ∣ = sup { ∣ u ( t ) for t ∈ [ 0 , T ] } and ∣ ∣ z ∣ ∣ = sup { ∣ z ( t ) ∣ for t ∈ [ − γ , T ] } .

Clearly, C * is a Banach space. Let ψ ( t ) be a continuous initial function on [ − γ ,0]. Consider the operator Q : C * → C * such that Q ( z , u ) = ( y , v ) , where y and v are defined as

y ( t ) = ψ ( t ) , for t ∈ [ − γ , 0 ] , y ( t ) = z ( t ; ψ , y 0 , z 1 ) + ∫ 0 t ( t − s ) ζ + η − 1 Ξ ζ , ζ + η ( t − s ) g ( s , z ( s ) , z ( s − γ ) , u ( s ) ) d s + ∫ 0 t ( t − s ) ζ + η − 1 Ξ ζ , ζ + η ( t − s ) P u ( s ) d s , for t ∈ I ,

and

v ( t ) = ( T − t ) ζ + η − 1 [ Ξ ζ , ζ + η ( T − t ) P ] ⊺ ℋ − 1 z T − z ( T ; ψ , y 0 , z 1 ) − ∫ 0 T ( T − s ) ζ + η − 1 Ξ ζ , ζ + η ( T − s ) g ( s , z ( s ) , z ( s − γ ) , u ( s ) ) d s ,

for t ∈ I . Let us assume

k 1 = sup { ∣ z ( t ; ψ , y 0 , z 1 ) ∣ + ∣ z T ∣ } , k 2 = ∣ ℋ − 1 ∣ , k 3 = sup ∣ ∣ Ξ ζ , ζ + η ( T − t ) P ∣ ∣ , k 4 = sup ∣ Ξ ζ , ζ + η ( T − t ) ∣ , l = max { k 3 T ζ + η ( ζ + η ) − 1 , 1 } , a 1 = 6 l k 3 k 2 k 4 T ζ + η ( ζ + η ) − 1 , a 2 = 6 k 4 T ζ + η ( ζ + η ) − 1 , b 1 = 6 l k 3 k 2 k 1 , b 2 = 6 k 1 , a = max { a 1 , a 2 } , b = max { b 1 , b 2 } .

Define

(20) C * ( r ) = ( y , u ) ∈ C * : ∣ ∣ y ∣ ∣ ≤ r 3 , ∣ ∣ u ∣ ∣ ≤ r 3 ,

since

(21) ∣ v ( t ) ∣ ≤ k 3 k 2 ( k 1 + k 4 T ζ + η ( ζ + η ) − 1 sup s ∈ I ∣ g ( s , z ( s ) , z ( s − γ ) , u ( s ) ) ∣ ) ≤ b 1 6 l + a 1 6 l sup s ∈ I ∣ g ( s , z ( s ) , z ( s − γ ) , u ( s ) ) ∣ ≤ 1 6 l [ b + a sup s ∈ I ∣ g ( s , z ( s ) , z ( s − γ ) , u ( s ) ) ∣ ]

and

(22) ∣ y ( t ) ∣ ≤ k 1 + k 3 ( T ) ζ + η ζ + η ∣ ∣ u ∣ ∣ + k 4 ( T ) ζ + η ζ + η sup s ∈ I ∣ g ( s , z ( s ) , z ( s − γ ) , u ( s ) ) ∣ ≤ b 6 + l ∣ ∣ u ∣ ∣ + a 6 sup s ∈ I ∣ g ( s , z ( s ) , z ( s − γ ) , u ( s ) ) ∣ .

By assumption, the function g fulfills the following criteria in [35]: there is a positive real number δ such that, for any a , b ∈ R + , if ∣ ∣ q ∣ ∣ ≤ δ , then for all t ∈ I ,

(23) a ∣ g ( t , q ) ∣ + b ≤ δ .

Choose r > δ such that sup − 1 ≤ t ≤ 0 ∣ ψ ( t ) ∣ ≤ r 3 . If ∣ ∣ y ∣ ∣ ≤ r 3 and ∣ ∣ u ∣ ∣ ≤ r 3 , implies ∣ z ( s ) ∣ + ∣ z ( s − γ ) ∣ + ∣ u ( s ) ∣ ≤ r , s ∈ I . By 23, we obtain ( b + a s u p ∣ g ( s , z ( s ) , z ( s − γ ) , u ( s ) ) ∣ ) ≤ r , for s ∈ I .

Thus, we obtain ∣ ∣ v ( t ) ∣ ∣ ≤ r 6 l ≤ r 3 and ∣ ∣ z ( t ) ∣ ∣ ≤ r 3 . Then, the operator Q maps C * ( r ) into itself. Also, Q is compact operator. Since C * ( r ) is bounded, closed, and convex, using Schauder’s fixed-point theorem [36], the operator Q has a fixed-point ( z , u ) in C * ( r ) . Thus,

(24) z ( t ) = z ( t ; ψ , y 0 , z 1 ) + ∫ 0 t ( t − s ) ζ + η − 1 Ξ ζ , ζ + η ( t − s ) g ( s , z ( s ) , z ( s − γ ) , u ( s ) ) d s + ∫ 0 t ( t − s ) ζ + η − 1 Ξ ζ , ζ + η ( t − s ) P u ( s ) d s ,

for t ∈ I , and z ( t ) = ψ ( t ) for t ∈ [ − γ , 0 ] . Hence, z ( t ) is a solution of system (17), and z ( T ) = z T . Consequently, system (17) is completely controllable on I .□

5 Illustrations

Example 1

Consider the following LFDD system:

(25) D η C ( D ζ C z ( t ) ) + M D η C z ′ ( t ) = N D η C z ( t − γ ) + P u ( t ) , z ( t ) = ψ ( t ) , t ∈ [ − γ , 0 ] , D ζ C z ( t ) ∣ t = 0 = y 0 , z ′ ( 0 ) = z 1 ,

where ζ = 3 ⁄ 2 , η = 1 ⁄ 2 , γ = 1 ,

M = 0 − 1 1 0 , N = 1 1 − 1 0 ,

z ( t ) = ( z 1 ( t ) , z 2 ( t ) ) ⊺ ∈ R 2 , P = ( 1 , 0 ) ⊺ , ψ ( t ) = ( t , 1 ) ⊺ , initial conditions are z ( 0 ) = ( 1 , 0 ) ⊺ , y 0 = ( 0 , 0 ) ⊺ and final condition is z ( 1 ) = ( 1 , 0 ) ⊺ . Here, the control input is denoted by u ( t ) and the system’s state is represented by z ( t ) . The solution of system (25) can be expressed as [37]

(26) z ( t ) = E 1 ⁄ 2 ( − M t 1 ⁄ 2 ) ψ ( 0 ) + t E 1 ⁄ 2 , 2 ( − M t 1 ⁄ 2 ) z 1 + t 3 ⁄ 2 E 1 ⁄ 2 , 5 ⁄ 2 ( − M t 1 ⁄ 2 ) y 0 + t 1 ⁄ 2 E 1 ⁄ 2 , 3 ⁄ 2 ( − M t 1 ⁄ 2 ) M ψ ( 0 ) + t 3 ⁄ 2 E 1 ⁄ 2 , 5 ⁄ 2 ( − M t 1 ⁄ 2 ) M z 1 − t 3 ⁄ 2 E 1 ⁄ 2 , 5 ⁄ 2 ( − M t 1 ⁄ 2 ) N ψ ( − 1 ) + ∫ 0 t ( t − s ) 1 ⁄ 2 E 1 ⁄ 2 ( − M ( t − s ) 1 ⁄ 2 ) N ψ ( s − γ ) d s + ∫ 0 t ( t − s ) E 1 ⁄ 2 , 2 ( − M ( t − s 1 ⁄ 2 ) ) P u ( s ) d s .

The controllability Gramian matrix can be calculated with a basic matrix calculation as

ℋ = 193.4344 278.6243 278.6243 414.0671 ,

implies ℋ is non-singular. Hence, system (25) is completely controllable on [0, 1]. Furthermore, the numerical simulation of both the system’s state and the control input is provided as follows:

(27) u ( t ) = ( 1 − t ) [ E 1 ⁄ 2 , 2 ( − M ( 1 − t ) 1 ⁄ 2 ) P ] ⊺ ℋ − 1 [ z ( 1 ) − E 1 ⁄ 2 ( − M ) ψ ( 0 ) − E 1 ⁄ 2 , 2 ( − M ) z 1

− E 1 ⁄ 2 , 5 ⁄ 2 ( − M ) y 0 − E 1 ⁄ 2 , 3 ⁄ 2 ( − M ) M ψ ( 0 ) − E 1 ⁄ 2 , 5 ⁄ 2 ( − M ) M z 1 + E 1 ⁄ 2 , 5 ⁄ 2 ( − M ) N ψ ( − 1 ) − ∫ 0 1 ( 1 − s ) 1 ⁄ 2 E 1 ⁄ 2 ( − M ( 1 − s ) 1 ⁄ 2 ) N ψ ( s − 1 ) d s ,

which steers z ( 0 ) = ( 0 , 1 ) ⊺ to z ( 1 ) = ( 1 , 0 ) ⊺ .

Figure 1 illustrates the state of system (25), showing that, in the absence of control, it begins at the point ( 0 , 1 ) ⊺ and does not reach the point ( 1 , 0 ) ⊺ . Figure 2 displays the state of the system (25) with the application of control (29), indicating that it starts from the point z ( 0 ) = ( 0 , 1 ) ⊺ and successfully reaches the point ( 1 , 0 ) ⊺ . Figure 3 illustrates the control input.

$Figure 1 z ( t ) z\left(t) without control.$

Figure 1

z ( t ) without control.

$Figure 2 z ( t ) z\left(t) with control.$

Figure 2

z ( t ) with control.

$Figure 3 Control u ( t ) u\left(t) .$

Figure 3

Control u ( t ) .

Example 2

Consider the nLFDD system with state delay

(28) D η C ( D ζ C z ( t ) ) + M D η C z ′ ( t ) = N D η C z ( t − γ ) + P u ( t ) + g ( t , z ( t ) , z ( t − γ ) , u ( t ) ) , z ′ ( 0 ) = z 1 , D ζ C z ( t ) ∣ t = 0 = y 0 , z ( t ) = ψ ( t ) , t ∈ [ − γ , 0 ] ,

where ζ = 3 ⁄ 2 , η = 1 ⁄ 2 , γ = 1 , ψ ( t ) = ( t , 1 ) ⊺ , P = ( 1 , 0 ) ⊺ ,

M = 0 − 1 1 0 , N = 1 1 − 1 0 ,

z ( t ) = ( z 1 ( t ) , z 2 ( t ) ) ⊺ ∈ R 2 , g ( t , z ( t ) , z ( t − 1 ) , u ( t ) ) = ( 0 , ( z 1 ( t ) + z 2 ( t ) ) ⁄ ( 1 + z 2 ( t − 1 ) + u ( t ) ) ) ⊺ , initial conditions are z ( 0 ) = ( 1 , 0 ) ⊺ , y 0 = ( 0 , 0 ) ⊺ , and final condition is z ( 1 ) = ( 1 , 0 ) ⊺ . Here, the control input is denoted by u ( t ) and the system’s state is represented by z(t). Since the linear system of (28) is the same as (25), the controllability of Example 1 implies the controllability of linear system of (28). It is easy to verify that g ( t , z ( t ) , z ( t − 1 ) , u ( t ) ) satisfies condition (19) and control u ( t ) given by

(29) u ( t ) = ( 1 − t ) [ E 1 ⁄ 2 , 2 ( − M ( 1 − t ) 1 ⁄ 2 ) P ] ⊺ ℋ − 1 [ z ( 1 ) − E 1 ⁄ 2 ( − M ) ψ ( 0 ) − E 1 ⁄ 2 , 2 ( − M ) z 1 − E 1 ⁄ 2 , 5 ⁄ 2 ( − M ) y 0 − E 1 ⁄ 2 , 3 ⁄ 2 ( − M ) M ψ ( 0 ) − E 1 ⁄ 2 , 5 ⁄ 2 ( − M ) M z 1 + E 1 ⁄ 2 , 5 ⁄ 2 ( − M ) N ψ ( − 1 ) − ∫ 0 1 ( 1 − s ) 1 ⁄ 2 E 1 ⁄ 2 ( − M ( 1 − s ) 1 ⁄ 2 ) N ψ ( s − 1 ) d s − ∫ 0 1 ( 1 − s ) E 1 ⁄ 2 , 2 ( − M ( 1 − s 1 ⁄ 2 ) ) g ( s , z ( s ) , z ( s − γ ) , u ( s ) ) d s ,

steers the point z ( 0 ) = ( 1 , 0 ) ⊺ to the final point z ( 1 ) = ( 1 , 0 ) ⊺ . Thus, the nonlinear system (28) satisfies the hypothesis of Theorem 2. Therefore, the nonlinear system (28) is controllable on interval [0, 1].

6 Conclusion

This study investigated systems known as nLFDD systems, characterized by a finite state delay and a fractional order, where 0 < η ≤ 1 < ζ ≤ 2 , within finite-dimensional spaces. By employing a controllability Gramian matrix, the research delineated both necessary and sufficient conditions for the controllability of LFDD systems within the same fractional order range. Additionally, leveraging Schauder’s fixed point theorem was crucial in establishing sufficient conditions for controlling nLFDD systems. To validate the theoretical principles discussed, this study provided two illustrative examples demonstrating the practical application of the methodologies and criteria introduced.

Acknowledgements

Rajveer Singh is very much thankful to UGC for providing financial support in the form of JRF fellowship via letter UGC ID: 211610023103. Sachin Kumar wants to acknowledge the financial support provided under the scheme “Fund for Improvement of S&T Infrastructure (FIST)” of the Department of Science & Technology (DST), Government of India, via letter no. SR/FST/MS-I/2021/104 to the Department of Mathematics and Statistics, Central University of Punjab. Manuel De la Sen would like to thank the Basque Government for funding his research work through Grant IT1555-22, and he also thanks MICIU/AEI/ 10.13039/501100011033 and ERDF/E for partially funding his research work through Grants PID2021-123543OB-C21 and PID2021-123543OB-C22.

Funding information: This research did not receive any external funding.
Author contributions: All authors contributed equally to this work.
Conflict of interest: The authors declare that they have no competing interests.
Data availability statement: No data were used for the research described in the article.

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Received: 2024-04-20

Revised: 2024-08-15

Accepted: 2025-02-10

Published Online: 2025-03-14

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2025-0105

Keywords for this article

controllability; non-linear fractional delay differential system; Caputo fractional derivative

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