Existence and controllability of Hilfer fractional neutral differential equations with time delay via sequence method

1.   Introduction

Fraction systems have been demonstrated to be important tools for providing many complex miracles in numerous sectors of science and engineering, and this pairing has received a lot of traction recently. Fractional differential equations (FDEs), calculations have become increasingly important in mathematics, see ^{[3,8,9,15,16,23,24,32,33]}. Hilfer ^[17] launched a separate sort of derivative, alongside Riemann-Liouville and Caputo fractional derivatives, that is, Hilfer fractional derivative (HFD). For additional information, see ^{[4,9,17,20,22,29,30,43]}.

Thermal science, chemical engineering, and mechanics all use the time-fractional advection-reaction-diffusion equation. An analytic solution to this equation is nearly impossible to find. Recently, numeral modalities are provided, including a finite differentiation optimization approach and a homotope perturbation method. The Taylor's formula, also known as the Delta function, was employed for three decades to build the replicating kernel space, which has proven to be an excellent technique for three decades, the Taylor's formula, also known as the Delta function, was used to construct the replicating kernel space and it has proven to be a useful method for resolving different forms. In ^[1], the authors proposed various new reproductive kernel spaces for numerical approaches to time-fractional advection-reaction-diffusion equations based on Legendre polynomials.

References ^[2,9] explored the approximate controllability of semilinear inclusions with respect to HFD. Furati, et al. ^[7] discussed the existence and uniqueness of a problem involving HFD.

Neutral systems have gotten increasing attention in the present generation because among their widespread applicability in various domains of pragmatic mathematics. Several neutral systems, including heat flow in materials, visco-elasticity, wave propagation, and several natural developments, benefit from neutral systems with or without delay. To know more details on neutral system and its application reader can refer ^[4,20,21,53].

The advancement of current mathematical control theory has been aided by approximate controllability. The difficulties of approximation controllability of differential systems are extensively used in theory connected to system analysis with control. The system with fractional order generated by the fractional evolution system has attracted attention in recent years, list of these distributions may be found in ^[21,52]. Li et al. ^[26] and He et al. ^[12] developed a fractal differential model as well as a fractal Duffing-Van der Pol oscillator (DVdP) with two-scale fractal derivatives.

An analytic approximate solution can be obtained using two-scale transforms and the He-Laplace method. He and Ji ^[1] focused on two-scale mathematics and fractional calculus for thermodynamics, and found it is required to show the information lost owing to the reduced dimensional method. In general, one scale is set by utlization, in which case regular calculus is used, and the other scale is determined by the need to reveal lost information, in which case the continuity assumption is allowed and fractional or fractal calculus must be used. For numerical results of space fractional variable coefficient kdv-modified kdv equation via Fourier spectral approach, see ^[49,50]. Many academics are now using the Sequence method to represent the approximate controllability outcomes using Riemann-Liouville fractional derivative, fractional evolution with damping, and an impulsive system. See articles ^{[4,5,6,18,19,26,28,29,31,34,35,36,37,38,39,40,41,42,44,45,46,47,48,54,55,57]} for further information.

Consider

The Hilfer fractional derivative is symbolized by $\mathcal {D}^{\alpha, \beta}_{0^+},$ whose order and type are $0 < \beta < 1$, $0\le\alpha\le 1$ on Hilbert space $\mathcal {H}$, $\mathcal{A}$ refers to a $C_0$ semigroup $\{S(\vartheta)\}_{\vartheta\geq 0}$'s infinitesimal generator.

On a Hilbert space $\mathcal {H}$, $\mathcal{A}_1$ denotes a bounded linear operator. We choose, $\mathcal {K}$ is a function space associate to $W,$ and $\mathcal {Y}$ is the space of values $\vartheta(\cdot),$ then the control function $w(\cdot)\in W,$ $B: W {\rightarrow} \mathcal {Y}$; Assume $L^b (\mathcal {H}, Y),$ $\| \mathcal {G}\| \in L^p (\mathcal {K}, R^+),$ for some $p$ with $0\le p \le\infty, \ \mathcal {G}$ mapping from $\mathcal {K}$ into $\mathcal {H}$ into $\mathcal {H},$ and $\hbar(\cdot)$ is from $\mathbb {C}([-\sigma, 0]; \mathcal {H})$.

We split this work into the sections below: The fundamentals of fractional differential systems, semigroup and control systems are addressed in Section 2. Existence outcomes for the system (1.1) is given in Section 3. The filter diagram is included in Section 4. Further we evaluated the results in Section 5 with respect to approximate controllability, 6 we establish the outcomes with time delay by utilizing the sequence method and nonlocal conditions. In 7, we provide an application to demonstrate our main arguments and some inference are established in the end.

2.   Preliminary results

$C(\mathcal {K}, \mathcal {H}): \mathcal {K} {\rightarrow} \mathcal {H}$ symbolizes the continuous function throughout this paper along with $\|x\|_{ \mathbb {C}}^{*} = \sup_{\vartheta \in \mathcal {K}} e^{- \mathfrak{r}\vartheta}\|x(\vartheta)\|$, where $\mathfrak{r}$ is a fixed positive constant. Now characterize $C_{1-\mathfrak{b}}(\mathcal {K}, \mathcal {H}) = \{x:\vartheta^{1-\mathfrak{b}}x(\vartheta)\in C(\mathcal {K}, \mathcal {H})\}$, $\|\cdot\|_\mathfrak{b}$ represented as $\|x\|_\mathfrak{b} = \sup\{\vartheta^{1-\mathfrak{b}}\|x(\vartheta)\|, \vartheta \in \mathcal {K}\},$ where $(1-\mathfrak{b}) = (1-\alpha)(1-\beta)$ since $\mathfrak{b} = \alpha+\beta-\alpha \beta.$

Following are the properties of $\mathcal {A}^{\kappa}$:

$\mathcal {A}^{\kappa}$ is a fractional power, $0 < \kappa\leq 1$, as a closed linear operator on $D(\mathcal {A}^{\kappa})$ along inverse $\mathcal {A}^{-\kappa}$.

$\rm(i)$ Let $D(\mathcal {A}^{\kappa})$ be a Banach space along $\|u\|_{\kappa} = \| \mathcal {A}^{\kappa}u\|$ for $x\in D(\mathcal {A}^{\kappa}).$

$\rm(ii)$ $S(\vartheta):U{\rightarrow} U_{\kappa}$ for $\vartheta\geq 0.$

$\rm(iii)$ $\mathcal {A}^{\kappa}S(\vartheta)x = S(\vartheta)A^{\kappa}x$ for all $u\in D(\mathcal {A}^{\kappa}),$ $\vartheta\geq 0.$

$\rm(iv)$ For each $\kappa\in(0, 1)$, $\mathcal {A}^{\kappa}S(\vartheta)$ is bounded, ${\mathbb N}_{\kappa} > 0$, such that

Definition 2.1. ^[33] Suppose $\vartheta:[d, +\infty){\rightarrow} \mathbb {R},$ then RLI is defined as

Definition 2.2. ^[33] Type $\beta\in [\jmath-1, \jmath)$, $\jmath\in {\mathbb Z}$ for $\mathcal {G}:[d, +\infty)\rightarrow {\mathbb R},$ the RLD is defined as

Definition 2.3. ^[33] Type $\beta\in [\jmath-1, \jmath)$, $\jmath\in {\mathbb Z}$ for $\mathcal {G}:[d, +\infty)\rightarrow {\mathbb R},$ we have the RLD in the form of

Definition 2.4. ^[33] $0\leq \beta \leq 1,$ $0 < \alpha < 1,$ for $\mathcal {G}(\vartheta),$ then the HFD is

Remark 2.5. ^[17] RLI and CFD's Hilfer fractional derivatives are characterized as follows:

Definition 2.6. ^[10,11] He's fractional derivative:

In fractal space, fractional evolution equations are established using He's fractional derivative. The fractional evolution equation is converted into its traditional form via He's fractional real transform, and the solutions are obtained using the homotopy perturbation method.

Definition 2.7. ^[14] Two-scale fractal derivative:

The standard differential derivatives and the two-scale fractal derivative are conformable. The two-scale transform is used to convert the nonlinear Zhiber-Shabat oscillator with the fractal derivatives to the traditional model.

where $x_0$ is the smallest scale beyond which there is no physical understanding and it is the porous size. Refer ^[10,11,14], for the variational iteration method refer^[13].

Definition 2.8. ^[33] $x(\cdot; w)\in C((0, d], \vartheta)$ is a mild solution of (1.1) only if for all $w\in L^2(\mathcal {K}, \mathcal {H}),$ the integral equation

where

(2.1) implies

Wright function: ($N_\beta(\vartheta)$):

where $N_\beta(\vartheta)$ belongs to $(0, \infty)$ satisfying

Lemma 2.9. ^[17,56]

● The $\mathcal {V}_\beta(\vartheta)$ is continuous.

● For $\vartheta > 0,$ $\{ \mathcal {R}_{\alpha, \beta}(\vartheta)\}$, $\{ \mathcal {Q}_\beta(\vartheta)\}$ are strongly continuous.

● For $\vartheta > 0$ and for all $x \in \mathcal {H}$, then

Lemma 2.10. ^[29]In any case $x\in \mathcal {B}$, $\kappa\in(0, 1]$, then

3.   Existence results

In order to obtain the existence of mild solution for the system (1.1), the following assumptions are made.

$\mathcal {F}_{\bf{1}}:$ There exists ${\mathbb N}\ge 0,$ such that the semigroup $S(\vartheta)$ is uniformly bounded on $\mathcal {H},$

$\mathcal {F}_{\bf{2}} :$ There exist $\vartheta_1\in (0, \beta)$ and $\eta \in L^{\frac{1}{\vartheta_1}}(\mathcal {K}, R^+)$, $x\in \mathcal {H},$ the function $\mathcal {G}(\vartheta, x)$ is continuous at $\vartheta$ then

with

$\mathcal {F}_{\bf{3}} :$

$\mathcal {F}_{\bf{4}}:$ $\mathfrak{g}: (0, \mathfrak{b}] \times \mathcal {J}{\rightarrow} \mathcal {B}$ is continuous and there is a $\kappa\in(0, 1)$ such that $\mathfrak{g}\in D(\mathcal {A}^\kappa),$ $x, \hat{x}\in \mathbb{C}$, $\vartheta\in \mathcal {J}$, $\mathcal {A}^\kappa \mathfrak{g}(\cdot, x)$ is strongly measurable, there exist $\mathcal {L}_1, \mathcal {L}_2 > 0$ and $\mathcal {A}^\kappa \mathfrak{g}(\vartheta, \cdot)$ satisfies

For convenience

Theorem 3.1. For every control function $\ w(\cdot)\in W$ and the assumptions $\bf{ \mathcal {F}_1}$ and $\bf{ \mathcal {F}_2}$ are true then $(1.1)$ has a mild solution on $C([-\sigma, d]; \mathcal {H}).$

Proof. $\Gamma$ has a fixed point in $\mathcal {H}$:

Define

Step 1: Fix $\mathfrak{q} > 0$ and $B_\mathfrak{q} = \{x\in C([-\sigma, d], \mathcal {H}):\|x\|^*_C\leq \mathfrak{q}\}.$

where

Combining $\mathcal {J}_1$ to $\mathcal {J}_7,$ we get

Where

A positive constant $\mathfrak{q}$ appearing from the norm $\|\cdot\|^*_C,$

and the radius of the sphere

From (3.2) and (3.3) we are getting a contradiction to $\mathcal {F}_3$. Therefore $\|\Gamma x\|^*\leq \mathfrak{q}.$

Step 2: Contraction: For every $\vartheta\in (0, d]$ using $\bf{(\mathcal {F}_2)}$ and there exists constants $x_1, x_2\in \mathbb {C}([-\sigma, d]; \mathcal {H}),$ we obtain

From the definition of $\mathfrak{r}$ from (3.2), we obtain

Therefore $\Gamma$ is contraction on $\mathbb {C}([-\sigma, d]; \mathcal {H}).$ Hence $x$ has a fixed point of $\Gamma$, i.e., it is a mild solution of (1.1).

4.   Filter system

By referring the articles ^[44,58], we have given a filter design for our system (1.1) shown in Figure 1 and it shows a rough diagram format, it contributes to the structure's practicality by reducing the number of input sources.

$\rm(a)$ Product modulators $1$ and $2$ accept the $\mathcal{A}$ and $\mathfrak{g}(\mathfrak{r}, x_{\mathfrak{r}})$, $u(\mathfrak{r})$ and $B$ gives the outputs as $\mathcal{A}\mathfrak{g}(\mathfrak{r}, x_{\mathfrak{r}})$ and $B u(\mathfrak{r})$.

$\rm (b)$ Product modulator $3$ accepts the input $[\hbar(0)-\mathfrak{g}(0, \hbar(0))]$ and $\mathcal {R}_{\alpha, \beta}(\vartheta)$ at time $\vartheta = 0$, gives the output as $\mathcal {R}_{\alpha, \beta}(\vartheta)[\hbar(0)-\mathfrak{g}(0, \hbar(0))]$.

$\rm(c)$ $\mathcal{A}_1$ and $x(\mathfrak{r}-\sigma)$, produced $\mathcal{A}_1 x(\mathfrak{r}-\sigma).$

$\rm(d)$ $\mathcal {Q}_\beta(\vartheta-\mathfrak{r})$, $\mathcal {G}(\mathfrak{r}, x(\mathfrak{r}-\sigma))$ are the inputs. Over $\vartheta,$ the inputs are joined and multiplied with an integrator output.

$\rm(e)$ $\mathcal {Q}_\beta(\vartheta-\mathfrak{r})$, $\mathcal{A}_1 x(\mathfrak{r}-\sigma)$ are the inputs. Over $\vartheta,$ the inputs are joined and multiplied with an integrator output.

$\rm(g)$ $\mathcal {Q} \beta(\vartheta-\mathfrak{r})$, $\mathcal {A}\mathfrak{g}(\mathfrak{r}, x_\mathfrak{r})$ are the inputs. Over $\vartheta,$ the inputs are joined and multiplied with an integrator output.

$\rm (h)$ $\mathcal {Q}_\beta(\vartheta-\mathfrak{r})$, $B u(\mathfrak{r})$ are the inputs. Over $\vartheta,$ the inputs are joined and multiplied with an integrator output.

$\rm(f)$ The following integrators sum up with the above mentioned modulators over the period $\vartheta,$

where $\mathcal {Q}_\beta(\vartheta-\mathfrak{r}) = (\vartheta-\mathfrak{r})^{\beta-1} \mathcal {V}_\beta(\vartheta-\mathfrak{r}).$

Finally, we move all integrator outputs to the network. As a result, we have our output result $x(\vartheta)$.

5.   Approximate controllability results

Nonlinear control systems with approximate controllability are operated by fractional-order with time delay.

Definition 5.1. Let $E(\mathcal {G}) = \{x(d; w):u(\cdot)\in U\}$ be the reachable set of (1.1) at time $d$. Suppose $\mathcal {G}$ is identically zero then (1.1) is said to be corresponding linear system and $E(0)$ is defined as the reachable set of (1.1).

Definition 5.2. Suppose $\overline{E}(\mathcal {G}) = \mathcal {K}$, then (1.1) is approximately controllable at time $d$ $(d > \sigma)$, where $\overline{E}(\mathcal {G})$ signifies the closure of $E(\mathcal {G})$. If $\overline{E}(0) = \mathcal {K}$ then (1.1) is also approximately controllable.

Following hypotheses are used to prove the main outcome.

$\mathcal {F}_{\bf{5}}:$ For every $\mu > 0$ and $\ell(\cdot)$ from $\mathcal {Y}$, then there exists $u(\cdot)\in U$ such that

$\mathcal {F}_{\bf{6}}:$ For $\nu > 0$ independent of $\ell(\cdot)\in \mathcal {Y}$ such that

Lemma 5.3. Assumptions $\bf{(\mathcal {F}_1)}, \bf{(\mathcal {F}_2)}$ are true then the mild solutions of (1.1) satisfies

where

Proof. Define $\mathcal {E}: \mathcal {Y}{\rightarrow} C((0, d], \mathcal {K})$ by

choosing a desired final state function $\Psi$ and $\mu > 0$ then we have

For any $\Psi\in D(\mathcal {A})$ and $x_0\in \mathcal {H},$ there exists $\ell > 0$ such that

In the above

and

is a mild solution (1.1) according to $w_{\mu}(\cdot)$ belongs to $\mathcal {K}$. Suppose $x(\cdot, w) = x(\cdot)$ is a mild solution of (1.1) with respect to $u(\cdot) \in W$ then

By using Gronwall's inequality, Mittag-Leffler function

(5.1) implies

where

Now

This completes the proof.

Theorem 5.4. If $\bf{(\mathcal {F}_1)}$–$\bf{(\mathcal {F}_3)}$ are true then (1.1) is approximately controllable.

Proof. To verify $\overline{D(\mathcal {A})} \subset \mathcal {H}$ for all $\Psi\in D(\mathcal {A})$, there is a control $w_{\mu}(\cdot)\in W$, such that

For any $x_0$ belongs to $\mathcal {H}$ and there exists a function $\ell(\cdot)$ belongs to $\mathcal {Y}$, then

Suppose $w_1(\cdot)$ belongs to $W$ and $\mu > 0$ then from $\bf{(\mathcal {F}_3)}$ we choose an arbitrary value $w_2(\cdot)$ belongs to $W$ such that

From the above we note $x_{1}(\vartheta)$ takes $x(\vartheta; w_{1})$ and $x_{2}(\vartheta)$ takes $x(\vartheta; w_{2})$ for $0 \leq \vartheta \leq d$.

Again from $\bf{(\mathcal {F}_3)}$ there exists $\varpi_{2}(\cdot)\in W$, such that

we now consider

Define $w_{3}(\vartheta) = w_{2}(\vartheta)-v_{2}(\vartheta)$, $w_{3}(\cdot)\in W$, then

Suppose there is a sequence $\{x_{k}(\cdot)\} \subset \mathcal {X},$ then

In the above $x_{k}(\cdot) = x(\cdot; w_{k})$ for $0\leq \vartheta \leq \mu$, and

By referring (5.3) and there exists a $\chi(\cdot)\in W,$ such that

As a result, for every $\mu > 0$, there exists a positive integer number $N$, such that

Hence, we get

Therefore $(1.1)$ is approximate controllability. Thus this ends the proof.

6.   Nonlocal condition

Byszewski ^[15,16] investigated the idea of "nonlocal conditions", proving the existence and uniqueness of mild, strong, and classical nonlocal Cauchy problem solutions for semilinear evolution equations. In ^[51] the authors considered the controllability with nonlocal conditions by utilizing fixed point methods and fractional calculus. A valuable conversation about the nonlocal conditions are given in ^[25,27,51].

Apparently, the controllability of neutral differential problems in particular of time delay with nonlocal conditions with respect to Hilfer fractional differential equations has not been explored at this point. Motivated by the articles ^[25,53,56], consider

Where $\mathcal {K}$ is a positive real, $0 < t_1 < t_2 < t_3 < \cdots < t_n \leq d$, $\mathfrak{p}:C([0, \mathcal {K}], \mathcal {H}){\rightarrow} \mathcal {H}$ and satisfying the following assumption:

$\mathcal {F}_{\bf{7}}:$ $\mathfrak{p}:C((0, \mathcal {K}], \mathcal {H}){\rightarrow} \mathbb {K}$ is continuous, there exists $\mathcal {N}_i(\mathfrak{p}) > 0$ such that

For every $\lambda_k, \gamma_k\in \mathbb {K}$ and consider $N_h = \sup\{\|\mathfrak{p}(\lambda_1, \lambda_2, \cdots, \lambda_n)\|:\lambda_k\in \mathbb {K}\}$.

Definition 6.1. If $x(\cdot; w)\in C((0, \mathcal {K}], \mathcal {H})$is a mild solution of (6.1) then $w\in L^2(\mathcal {K}, \mathcal {H})$ the integral equation

Theorem 6.2. If the assumptions $\bf{(\mathcal {F}_1)}$–$\bf{(\mathcal {F}_5)}$ are true then (6.2) is approximately controllable.

Proof. To verify $\overline{D(\mathcal {A})} \subset \mathcal {H}$ for every $\Psi\in D(\mathcal {A})$, suppose there is a control $u_{\mu}(\cdot)\in U$, then

For any $x_0\in \mathcal {H}$ and there exists a function $\ell(\cdot)\in \mathcal {Y}$, such that

Suppose $u_1(\cdot)\in U$ and $\mu > 0$ then from $\bf{(\mathcal {F}_3)}$ we choose an arbitrary value $w_2(\cdot)\in W$ such that

From the above we note $x_{1}(\vartheta)$ takes $x(\vartheta; u_{1})$ and $x_{2}(\vartheta)$ takes $x(\vartheta; u_{2})$ for $0 \leq \vartheta \leq d$.

Again from $\bf{(\mathcal {F}_3)}$ there exists $\varpi_{2}(\cdot)\in U$, such that

we now consider

Define $u_{3}(\vartheta) = u_{2}(\vartheta)-v_{2}(\vartheta)$, $u_{3}(\cdot)\in U$, then

Suppose there is a sequence $\{x_{k}(\cdot)\} \subset \mathcal {X},$ then

In the above $x_{k}(\cdot) = x(\cdot; u_{k})$ for $0\leq \vartheta \leq \mu$, and

By referring (6.4) and there exists a $\chi(\cdot)\in U$ such that

As a result, any $\mu > 0$, there is a positive integer number $N$, then

Hence, we get

As a consequence, the system $(6.1)$ is approximately controllable. This ends the proof.

7.   Application

Consider

The Hilfer fractional derivative is symbolized by $D^{\alpha, \frac{2}{3}}_{0+},$ whose order and type are $\frac{3}{2}$, $0\le\alpha\le 1$ and $I_{0^+}^{\frac{1}{3}(1-\alpha)}$ is the Reimann-Liouoville integral of order $\frac{1}{3}(1-\alpha).$ The function $\mathcal {G}(\cdot, \cdot)\in L^2([0, \pi]\times[0, \pi], {\mathbb R}^+)$, for $\mathfrak{m} > 0.$

Abstract form: Considering $\mathcal {A}: \mathcal {H}{\rightarrow} \mathcal {H}, \ \mathcal {H} = L^2([0, \pi], {\mathbb R})$ which is defined as $\mathcal {A}\nu = \nu''$, $\nu\in D(\mathcal {A})$, where

and

Also, $\mathcal {A}$ satisfies $\mathcal {C}_1, \mathcal {C}_2$ Then, we have

where

For all $x\in \mathcal {H}$,

The function $\mathfrak{g}:[0, d]\times{\rightarrow} \mathcal {H}$ is defined by $\mathfrak{g}(\vartheta, x_\vartheta) = \int_0^\pi c(\beta, \mathfrak{u})x(\vartheta, \beta)d\beta.$

Let $\frac{\partial}{\partial \beta}c(\beta, \mathfrak{u})$ be measurable, $c(0, u) = c(\pi, u) = 0,$ and

$c$ is measurable.

Hence, $X_1(\vartheta)\in D(\mathcal{A}^\frac{1}{2})$ and $\|\mathcal{A}^\frac{1}{2}\|^2\leq \mathcal {L}$.

Therefore,

Let's define the $B$ as

where $u_h(\vartheta) = \langle u(\vartheta, \varsigma_h) \rangle, h = 0, 1, 2, \cdots.$

the reader can understand by $\|Bu(\cdot)\|\le\|u(\cdot)\|,$ $B$ is bounded linear operator. The linear system of (7.1) is

Hence, for any given $f(\cdot)$

For any $f(\cdot)$ assume $\hat{u_h}$ as

where

as a consequence, ${\bf{ \mathcal {F}_6}}$ is fulfilled. Furthermore, we have

As a result, it can be observed that if the conditions ${\bf{ \mathcal {F}_6}},$ is satisfied then (7.1) is approximately controllable on $\mathcal {H}$.

8.   Conclusions

Our study investigates the existence and approximate controllability of HF neutral evolution equations with time delay. The Sequence method was used to derive the approximate controllability outcomes for HFD equations with time delay. An illustration is offered to support the analytical findings at the end and also given a filter diagram to represent the mild solution of the system with neutral term. Next, new research may use the sequence method with infinite delay to extend the Hilfer fractional stochastic differential evolution equations to approximate control results.

Acknowledgments

Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R157), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Conflict of interest

Conflicts of interest are not present in this study.