Controllability of Hahn Difference Equations with Maxima

Abstract

In this paper, we investigate controllability for both linear and nonlinear Hahn difference equations, whose nonlinear terms are given by involving maxima. Introducing a Grammian matrix, we derive a necessary and sufficient condition characterizing controllability for linear Hahn difference control systems. For the nonlinear case with maxima, we construct an appropriate control input that allows reformulation as a fixed-point problem, enabling the use of fixed-point techniques. In particular, Krasnoselskii’s fixed point theorem is employed to obtain a controllability result. Finally, two examples are given to demonstrate our theoretical findings, including computation of the required control functions and the inverse of the Grammian matrix.

1. Introduction

Hahn difference equations, as a generalization of classical difference equations, have attracted great interest from researchers in recent years [1,2,3,4,5,6,7,8,9,10,11,12]. The Hahn difference operator was introduced in 1949 by the German mathematician Wolfgang Hahn [13]; it is a more general q-difference operator. As q approaches 1, the Hahn difference operator degenerates into an ordinary difference operator. In addition, $q=1$, $\omega \to 0$ (see Definition 1), the Hahn difference operator degenerates into an differential operator [14]. Therefore, Hahn difference equations can be viewed as a unification of differential equations and difference equations. Hahn difference equations have important applications in fields such as hypergeometric series, quantum calculus, and combinatorics [15,16,17,18]. In recent years, the applications of Hahn difference equations in control theory have also gained increasing attentions.

In modern control theory, controllability is a core concept in system design and analysis. Simply put, controllability is the property that, with suitable control inputs, a system can be steered from any initial state to a desired state in finite time. With the rapid advancement of science and technology, the application of control theory has expanded to encompass a wide range of fields, including aerospace, robotics, biomedical engineering, and economic management [19,20,21,22,23,24]. Therefore, in-depth research on the controllability of various control systems has both theoretical and practical significance.

Furthermore, it is worth noting that nonlinear phenomena are ubiquitous in practical control systems. Among them, nonlinear systems governed by maximum terms constitute a central class in nonlinear dynamics [25,26,27,28]. These systems are widely found in various fields. For example, in neural networks, neuron activation functions often include maximum operations; in economic models, investment decisions may depend on the maximum values of multiple factors; and in engineering control, maximum limits on certain variables are often required to avoid system overload. Therefore, studying nonlinear systems with maximum terms is of great practical significance.

So far, some results have been obtained for difference equations with maximum terms. However, research on Hahn difference equations with maximum terms is still relatively limited. In particular, studies on the controllability of Hahn difference equations with maximum terms are especially scarce.

Based on the above background, this paper aims to study the controllability of Hahn difference equations with maximum terms. Specifically, we will consider the following form of Hahn difference systems:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}& D_{q,\omega }y\left(t\right)=Ay\left(t\right)+Bu\left(t\right),t\in J:=[{\omega }_0,b],\hfill \\ & y\left({\omega }_0\right)=y_0,\hfill \end{array}\right.\end{array}\end{array}$$

(1)

where $D_{q,\omega }$ is the Hahn difference operator, $0<q<1,\omega >0$, $b-{\omega }_0<{\displaystyle \frac{1}{\parallel A\parallel (1-q)}}$, ${\omega }_0:={\displaystyle \frac{\omega }{1-q}}$, $y_0\in {\mathbb{R}}^n$ is the initial value, $b\in \mathbb{R}$ and $b>{\omega }_0$, $A,B\in {\mathbb{R}}^{n\times n}$, $y:J\to {\mathbb{R}}^n$, and the control function $u(·)$ takes value from $L_{q,\omega }^2(J,{\mathbb{R}}^n)$, which will be defined below.

For the nonlinear case

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}& D_{q,\omega }y\left(t\right)=Ay\left(t\right)+g(t,y\left(t\right),\underset{{\omega }_0\le \xi \le t}{\max }y\left(\xi \right))+Bu\left(t\right),t\in J^{★}:={[{\omega }_0,b]}^{*},\hfill \\ & y\left({\omega }_0\right)=y_0,\hfill \end{array}\right.\end{array}\end{array}$$

(2)

Due to the absolute value integral inequality on Hahn difference equations (see Lemma 2), we need to study the controlability of (2) on $J^{*}={[{\omega }_0,b]}^{★}:=\{q^{k}b+\omega {\left[k\right]}_q,k\in \mathbb{N}\}\bigcup \left\{{\omega }_0\right\}$ (see (3)). Here, $g:J^{*}\times {\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ is continuous at $({\omega }_0,y\left({\omega }_0\right),y\left({\omega }_0\right))$.

2. Preliminaries

We denote $C_{q,\omega }(J,{\mathbb{R}}^n)$ as all the vector functions, which are $q,\omega$-differentiable (see Definition 1 and Lemma 1) in ${\mathbb{R}}^n$. Also $L_{q,\omega }(J,{\mathbb{R}}^n)$ and $L_{q,\omega }^2(J,{\mathbb{R}}^n)$ are all the vector functions, which are respectively $q,\omega$-integral (see Definition 2) and square $q,\omega$-integral in ${\mathbb{R}}^n$.

Definition 1

([12,13]). Let $\beta :J\to \mathbb{R}$. The Hahn difference operator is then defined as follows.

$$\begin{array}{c}\hfill D_{q,\omega }\beta \left(t\right)=\left\{\begin{array}{c}{\displaystyle \frac{\beta (qt+\omega )-\beta \left(t\right)}{t(q-1)+\omega },t\ne {\omega }_0,}\hfill \\ {\beta }^{\prime }\left({\omega }_0\right),t={\omega }_0,\hfill \end{array}\right.\end{array}$$

where $q\in (0,1)$ and $\omega >0$ are fixed real constants, and ${\omega }_0$ is given by ${\omega }_0={\displaystyle \frac{\omega }{1-q}}$.

Lemma 1

([12,29]). Let $\beta ,f:J\to \mathbb{R}$ be $q,\omega$-differentiable at $t\in J$. Then

(i)   

$D_{q,\omega }(\beta +f)\left(t\right)=D_{q,\omega }\beta \left(t\right)+D_{q,\omega }f\left(t\right);$

(ii)  

$D_{q,\omega }\left(c\beta \right)\left(t\right)=c{D}_{q,\omega }\left(\beta \left(t\right)\right),\mathit{for}\mathit{every}\mathit{constant}c\in \mathbb{R};$

(iii) 

$D_{q,\omega }\left(\beta f\right)\left(t\right)=D_{q,\omega }\left(\beta \left(t\right)\right)f\left(t\right)+\beta (qt+\omega )D_{q,\omega }f\left(t\right);$

(iv) 

$D_{q,\omega }(\beta /f)\left(t\right)={\displaystyle \frac{D_{q,\omega }\left(\beta \left(t\right)\right)f\left(t\right)-\beta \left(s\right)D_{q,\omega }f\left(t\right)}{f\left(t\right)f(qt+\omega )}},forf\left(t\right)f(qt+\omega )\ne 0.$

Defining $\mu \left(t\right)=qt+\omega$, the k-th iterate of $\mu$ is given by

$$\begin{array}{c}\hfill {\mu }^k\left(t\right)=q^{k}t+\omega {\left[k\right]}_q,{\left[k\right]}_q={\displaystyle \frac{1-q^k}{1-q}},k\in \mathbb{N},t\in J.\end{array}$$

(3)

This allows us to introduce the notion of a $q,\omega$-integral, known as the Jackson–Nörlund integral.

Definition 2

([12,29]). Let $\beta :J\to \mathbb{R}$ and let $a,b\in J$. The $q,\omega$-integral of β over $[a,b]$ is defined by

$$\begin{array}{c}\hfill {\int }_a^b\beta \left(t\right)d_{q,\omega }t={\int }_{{\omega }_0}^b\beta \left(t\right)d_{q,\omega }t-{\int }_{{\omega }_0}^a\beta \left(t\right)d_{q,\omega }t,\end{array}$$

where

$$\begin{array}{c}\hfill {\int }_{{\omega }_0}^x\beta \left(t\right)d_{q,\omega }t=(x(1-q)-\omega )\sum _{k=0}^{\infty }{q}^k\beta \left({\mu }^k\left(x\right)\right),x\in J,\end{array}$$

provided that the series converges at $x=a$ and $x=b$.

Lemma 2

([29,30]). Suppose that $\beta ,f:J\to \mathbb{R}$ are $q,\omega$-integrable on J. If $\beta \left(t\right)\le f\left(t\right)$ for all $t\in {\{q^{k}s+{\left[k\right]}_q\omega \}}_{k=0}^{\infty },s\in J$, then for $a,b\in {\{q^{k}s+{\left[k\right]}_q\omega \}}_{k=0}^{\infty }$, it holds that

$$\begin{array}{c}\hfill \left|{\int }_{{\omega }_0}^b\beta \left(t\right)d_{q,\omega }t\right|\le {\int }_{{\omega }_0}^{b}f\left(t\right)d_{q,\omega }tand\left|{\int }_a^b\beta \left(t\right)d_{q,\omega }t\right|\le {\int }_a^{b}f\left(t\right)d_{q,\omega }t.\end{array}$$

For the basic properties and conclusions about being $q,\omega$-differentiable and $q,\omega$-integral, readers can refer to references [12,29,31,32,33,34].

Definition 3

([29]). Let $\varphi \left(t\right)$ be continuous at ${\omega }_0$ and satisfy $1-\varphi \left({\mu }^k\left(t\right)\right)q^k(t(1-q)-\omega )\ne 0$ for all $t\in J,k\in \mathbb{N}$. The $q,\omega$-exponential functions $e_{\varphi }\left(t\right)$ and $E_{\varphi }\left(t\right)$ are then defined by

$$\begin{array}{ccc}\hfill e_{\varphi }\left(t\right)& =& {\displaystyle \frac{1}{{\displaystyle \prod _{k=0}^{\infty }}(1-\varphi \left({\mu }^k\left(t\right)\right)q^k(t(1-q)-\omega ))}}\hfill \end{array}$$

(4)

and

$$\begin{array}{ccc}\hfill E_{\varphi }\left(t\right)& =& \prod _{k=0}^{\infty }(1+\varphi \left({\mu }^k\left(t\right)\right)q^k(t(1-q)-\omega )).\hfill \end{array}$$

(5)

We deduce that $e_{\varphi }\left(t\right)$ converges to a nonzero value for all $t\in J$. In particular, the convergence of products (4) and (5) follows from the convergence ${\displaystyle \sum _{k=0}^{\infty }}\left|\varphi \left({\mu }^k\left(t\right)\right)\right|q^k(t(1-q)-\omega )$ (see [3] for details). In the special case $\varphi \left(t\right)=z\in \mathbb{R}$, we obtain

$$\begin{array}{ccc}\hfill e_z\left(t\right)& =& {\displaystyle \frac{1}{{\displaystyle \prod _{k=0}^{\infty }}(1-z{q}^k(t(1-q)-\omega ))}}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill E_{-z}\left(t\right)& =& \prod _{k=0}^{\infty }(1+z{q}^k(t(1-q)-\omega )).\hfill \end{array}$$

Moreover, to guarantee the convergence of the function $e_z\left(s\right)$, we additionally assume that

$$|t-{\omega }_0|<{\displaystyle \frac{1}{\left|z\right(1-q\left)\right|}}.$$

According to Theorem 3.2 [12], suppose $u\in L_{q,\omega }^2(J,{\mathbb{R}}^n)$, and $I-A{q}^k(t(1-q)-\omega )$ is invertible for all $t\in J,k\in \mathbb{N}$. Then system (1)/(2) admits a unique solution $y\left(t\right)$, which can be expressed as follows.

$$\begin{array}{c}\hfill y\left(t\right)=e_A\left(t\right)\left(y_0+{\int }_{{\omega }_0}^t{E}_{-A}\left(\mu \left(\eta \right)\right)Bu\left(\eta \right)d_{q,\omega }\eta \right),t\in J,\end{array}$$

(6)

and

$$\begin{array}{c}\hfill y\left(t\right)=e_A\left(t\right)\left(y_0+{\int }_{{\omega }_0}^t{E}_{-A}\left(\mu \left(\eta \right)\right)(g(\eta ,y\left(\eta \right),\underset{{\omega }_0\le \xi \le \eta }{\max }y\left(\xi \right))+Bu\left(\eta \right))d_{q,\omega }\eta \right),t\in J,\end{array}$$

(7)

where

$$\begin{array}{c}\hfill e_A\left(t\right)={\left[\prod _{k=0}^{\infty }(I-A{q}^k(t(1-q)-\omega ))\right]}^{-1}=\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(A(t(1-q)-\omega )\right)}^k}{{(q;q)}_k}},\end{array}$$

(8)

and

$$\begin{array}{c}\hfill E_{-A}\left(t\right)=\prod _{k=0}^{\infty }(I+A{q}^k(t(1-q)-\omega ))=\sum _{k=0}^{\infty }{\displaystyle \frac{q^{\frac{1}{2}k(k-1)}{\left(A(t(1-q)-\omega )\right)}^k}{{(q;q)}_k}},\end{array}$$

(9)

$$\begin{array}{c}\hfill {(q:q)}_k=\left\{\begin{array}{cc}{\displaystyle \prod _{j=1}^k(1-q^j),}\hfill & \mathrm{if}k=1,2,···,\hfill \\ 1,\hfill & k=0.\hfill \end{array}\right.\end{array}$$

We introduce the following lemma and definition.

Lemma 3

(Krasnoselskii’s fixed point theorem, see [35,36]). Let $\mathbb{B}$ be a bounded closed and convex subset of Banach space X, and let ${\mathbb{P}}_1$, ${\mathbb{P}}_2$ be maps of $\mathbb{B}$ into X such that ${\mathbb{P}}_{1}x+{\mathbb{P}}_{2}y\in \mathbb{B}$ for every pair $x,y\in \mathbb{B}$. If ${\mathbb{P}}_1$ is a contraction and ${\mathbb{P}}_2$ is compact and continuous, then the equation ${\mathbb{P}}_{1}x+{\mathbb{P}}_{2}x=x$ has a solution on $\mathbb{B}$.

Definition 4

(see [37]). System (1)/(2) is said to be controllable, if for any initial function $y_0\in C_{q,\omega }(J,{\mathbb{R}}^n)$, any desired terminal state $y_1\in {\mathbb{R}}^n$, and any terminal time $t_1$, there exists a control $u\in L_{q,\omega }^2(J,{\mathbb{R}}^n)$ such that the corresponding solution $y\in C_{q,\omega }(J,{\mathbb{R}}^n)$ of (1)/(2) satisfies $y\left({\omega }_0\right)=y_0$ and $y\left(t_1\right)=y_1$.

3. Main Results

3.1. Linear Case

We examine the controllability of system (1) and define the following new Grammian matrix:

$$\begin{array}{c}\hfill W[{\omega }_0,t_1]=e_A\left(t_1\right){\int }_{{\omega }_0}^{t_1}{E}_{-A}\left(\mu \left(\eta \right)\right)B{B}^T{E}_{-A}^T\left(\mu \left(\eta \right)\right)e_A^T\left(t_1\right)d_{q,\omega }\eta ,t_1\in J.\end{array}$$

(10)

We are now prepared to present a necessary and sufficient condition that ensures system (1) is controllable.

Theorem 1.

System (1) is controllable on J if and only if the matrix $W[{\omega }_0,t_1]$ defined in (10) is nonsingular.

Proof. 

Sufficiency. As $W[{\omega }_0,t_1]$ is non-singular, the inverse $W^{-1}[{\omega }_0,t_1]$ exists. Accordingly, we can define the control as:

$$\begin{array}{c}\hfill u\left(t\right)=B^T{E}_{-A}^T\left(\mu \left(t\right)\right)e_A^T\left(t_1\right)W^{-1}[{\omega }_0,t_1]\zeta ,\end{array}$$

(11)

where

$$\begin{array}{c}\hfill \zeta =y_1-e_A\left(t_1\right)y_0,\end{array}$$

(12)

and the vector $y_1\in {\mathbb{R}}^n$ is arbitrarily prior to being specified.

Substituting (11) into (6) yields

$$\begin{array}{c}\hfill y\left(t_1\right)=e_A\left(t_1\right)y_0+e_A\left(t_1\right){\int }_{{\omega }_0}^{t_1}{E}_{-A}\left(\mu \left(\eta \right)\right)B{B}^T{E}_{-A}^T\left(\mu \left(\eta \right)\right)e_A^T\left(t_1\right)W^{-1}[{\omega }_0,t_1]\zeta d_{q,\omega }\eta .\end{array}$$

(13)

Linking (10) and (13) via (12), it follows readily that

$$\begin{array}{c}\hfill y\left(t_1\right)=e_A\left(t_1\right)y_0+\zeta =y_1.\end{array}$$

(14)

The initial condition $y\left({\omega }_0\right)=y_0$ is satisfied by (6). Using (14) together with Definition 4, the system (1) is controllable on J.

Necessity. We argue by contradiction. Suppose $W[{\omega }_0,t_1]$ is singular, then there exists a nonzero vector $\tilde{y}\in {\mathbb{R}}^n$ such that

$$\begin{array}{c}\hfill {\tilde{y}}^{\top }W[{\omega }_0,t_1]\tilde{y}=0.\end{array}$$

Further, one can obtain

$$\begin{array}{ccc}\hfill 0& =& {\tilde{y}}^{\top }W[{\omega }_0,t_1]\tilde{y}\hfill \\ & =& {\int }_{{\omega }_0}^{t_1}\left[{\tilde{y}}^{\top }{e}_A\left(t_1\right)E_{-A}\left(\mu \left(\eta \right)\right)B\right]{\left[{\tilde{y}}^{\top }{e}_A\left(t_1\right)E_{-A}\left(\mu \left(\eta \right)\right)B\right]}^{\top }{d}_{q,\omega }\eta \hfill \\ & =& {\int }_{{\omega }_0}^{t_1}{\parallel {\tilde{y}}^{\top }{e}_A\left(t_1\right)E_{-A}\left(\mu \left(\eta \right)\right)B\parallel }^2{d}_{q,\omega }\eta ,\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill {\tilde{y}}^{\top }{e}_A\left(t_1\right)E_{-A}\left(\mu \left(\eta \right)\right)B=(\underset{n}{\underbrace{0,\dots ,0}}),\forall \eta \in J.\end{array}$$

(15)

Because system (1) is controllable, by Definition 4 there exists a control $u_1\left(t\right)$ that steers the initial state to zero at $t_1$, i.e.,

$$\begin{array}{c}\hfill y\left(t_1\right)=e_A\left(t_1\right)y_0+e_A\left(t_1\right){\int }_{{\omega }_0}^{t_1}{E}_{-A}\left(\mu \left(\eta \right)\right)B{u}_1\left(\eta \right)d_{q,\omega }\eta =\mathbf{0},\end{array}$$

(16)

where $\mathbf{0}$ represents the n-dimensional zero vector.

Likewise, there exists a control $\overline{u}(·)$ that steers the initial state to $\overline{y}$ at time $t_1$, i.e.,

$$\begin{array}{c}\hfill y\left(t_1\right)=e_A\left(t_1\right)y_0+e_A\left(t_1\right){\int }_{{\omega }_0}^{t_1}{E}_{-A}\left(\mu \left(\eta \right)\right)B\overline{u}\left(\eta \right)d_{q,\omega }\eta =\overline{y}.\end{array}$$

(17)

Then by (16) and (17), we have

$$\begin{array}{c}\hfill \overline{y}=e_A\left(t_1\right){\int }_{{\omega }_0}^{t_1}{E}_{-A}\left(\mu \left(\eta \right)\right)B[\overline{u}\left(\eta \right)-u_1\left(\eta \right)]d_{q,\omega }\eta .\end{array}$$

(18)

Premultiplying (18) by ${\overline{y}}^{\top }$ yields

$$\begin{array}{c}\hfill {\overline{y}}^{\top }\overline{y}={\int }_{{\omega }_0}^{t_1}{\overline{y}}^{\top }{e}_A\left(t_1\right)E_{-A}\left(\mu \left(\eta \right)\right)B[\overline{u}\left(\eta \right)-u_1\left(\eta \right)]d_{q,\omega }\eta .\end{array}$$

From (15) we deduce ${\overline{y}}^{\top }\overline{y}=0$, hence $\overline{y}$ = $\mathbf{0}$, contradicting the assumption that $\overline{y}$ is not equal to zero. Therefore, $W[{\omega }_0,t_1]$ is non-singular. This completes the proof. □

3.2. Nonlinear Case

We impose the following assumption:

$\left[H_1\right]:$ The operator $W:L_{q,\omega }^2(J^{*},{\mathbb{R}}^n)\to {\mathbb{R}}^n$ given by

$$Wu=e_A\left(t_1\right){\int }_{{\omega }_0}^{t_1}{E}_{-A}\left(\mu \left(\eta \right)\right)Bu\left(\eta \right)d_{q,\omega }\eta ,$$

admits an inverse $W^{-1}:{\mathbb{R}}^n\to L_{q,\omega }^2(J^{*},{\mathbb{R}}^n)$ and a constant $M>0$ such that

$$\parallel W^{-1}{\parallel }_{L_b({\mathbb{R}}^n,L_{q,\omega }^2(J^{★},{\mathbb{R}}^n)/kerW)}\le M.$$

Remark 1.

From [36] [Remark 2], it follows that

$$\begin{array}{c}\hfill M=\sqrt{\parallel W{[{\omega }_0,t_1]}^{-1}\parallel }.\end{array}$$

(19)

$\left[H_2\right]:$ There exists a constant $L_g>0$ such that the following Lipschitz condition

$$\parallel g(t,y_1,y_2)-g(t,y_1^{\prime },y_2^{\prime })\parallel \le L_g\sum _{i=1}^2\parallel y_i-y_i^{\prime }\parallel$$

holds for all $t\in J^{*}$, $y_i,y_i^{\prime }\in {\mathbb{R}}^n,i=1,2$.

Given $\left[H_1\right]$, let us consider an arbitrary function $y(·)\in C_{q,\omega }(J^{*},{\mathbb{R}}^n)$ and define a control function $u_y(·)$ as follows:

$$\begin{array}{cc}\hfill u_y(t)& =W^{-1}[y_1-e_A(t_1)y_0\hfill \\ \hfill & -e_A(t_1){\int }_{{\omega }_0}^{t_1}{E}_{-A}(\mu (\eta ))(g(\eta ,y(\eta ),\underset{{\omega }_0\le \xi \le \eta }{\max }y(\xi ))d_{q,\omega }\eta ](t),t\in J^{*}.\hfill \end{array}$$

(20)

We now present the central idea used to establish our main result via the fixed point method. We begin by showing that, under control (20), the operator $\mathcal{P}:C_{q,\omega }(J^{*},{\mathbb{R}}^n)\to C_{q,\omega }(J^{*},{\mathbb{R}}^n)$ given by

$$\begin{array}{ccc}\hfill \left(\mathcal{P}y\right)\left(t\right)& =& e_A\left(t\right)y_0+e_A\left(t\right){\int }_{{\omega }_0}^t{E}_{-A}\left(\mu \left(\eta \right)\right)g(\eta ,y\left(\eta \right),\underset{{\omega }_0\le \xi \le \eta }{\max }y\left(\xi \right))d_{q,\omega }\eta \hfill \\ & & +e_A\left(t\right){\int }_{{\omega }_0}^t{E}_{-A}\left(\mu \left(\eta \right)\right)B{u}_y\left(\eta \right)d_{q,\omega }\eta \hfill \end{array}$$

admits a fixed point y, which corresponds to a solution of system (2). Next, we verify that $\left(\mathcal{P}y\right)\left(t_1\right)=y_1$ and $\left(\mathcal{P}y\right)\left({\omega }_0\right)=y_0$, showing that the control $u_y$ transfers system (2) from $y_0$ to $y_1$ within the finite time $t_1$. Consequently, system (2) is controllable on $J^{*}$.

For any $r>0$, let ${\mathcal{B}}_r=\{y\in C_{q,\omega }(J^{*},{\mathbb{R}}^n){:\parallel y\parallel }_C\le r\}$. It is clear that each ${\mathcal{B}}_r$ is a bounded, closed, and convex subset of $C_{q,\omega }(J^{*},{\mathbb{R}}^n)$. For convenience, denote $R_g=\underset{t\in J^{*}}{\sup }\parallel g(t,0,0)\parallel$.

Next, Krasnoselskii’s fixed point theorem (see Lemma 3) is employed to establish a controllability result for system (2).

Theorem 2.

Assume that conditions $\left[H_1\right]$ and $\left[H_2\right]$ hold. Then system (2) is controllable on $J^{*}$ if

$$\begin{array}{c}\hfill M_1\left(1+\delta M\parallel B\parallel \right)<1.\end{array}$$

(21)

where $M_1={\displaystyle \frac{2L_g{e}_{\parallel A\parallel }\left(b\right)(E_{\parallel A\parallel }\left(b\right)-1)}{\parallel A\parallel }}$, and

$$\begin{array}{ccc}\hfill \delta & =& e_{\parallel A\parallel }\left(b\right)\parallel y_0\parallel +{\displaystyle \frac{R_g{e}_{\parallel A\parallel }\left(b\right)(E_{\parallel A\parallel }\left(b\right)-1)}{\parallel A\parallel }}\hfill \end{array}$$

(22)

and M is given by (19).

Proof. 

The proof is organized into several steps.

Step 1. We show that there exists some $r>0$ for which $\mathcal{P}\left({\mathcal{B}}_r\right)\subseteq {\mathcal{B}}_r$.

Taking into account (20), using $\left[H_1\right]$, $\left[H_2\right]$, we have

$$\begin{array}{ccc}\hfill \parallel u_y(t)\parallel & \le & \parallel W^{-1}{\parallel }_{L_b({\mathbb{R}}^n,L_{q,\omega }^2(J^{*},{\mathbb{R}}^n)/kerW)}\parallel y_1-e_A(t_1)y_0\hfill \\ & & -e_A(t_1){\int }_{{\omega }_0}^{t_1}{E}_{-A}(\mu (\eta ))g(\eta ,y(\eta ),\underset{{\omega }_0\le \xi \le \eta }{\max }y(\xi ))d_{q,\omega }\eta \parallel \hfill \\ & \le & \parallel W^{-1}{\parallel }_{L_b({\mathbb{R}}^n,L_{q,\omega }^2(J^{*},{\mathbb{R}}^n)/kerW)}(\parallel y_1\parallel +\parallel e_A(t_1)\parallel \parallel y_0\parallel \hfill \\ & & +\parallel e_A(t_1)\parallel {\int }_{{\omega }_0}^{t_1}{E}_{\parallel -A\parallel }\parallel g(\eta ,y(\eta ),\underset{{\omega }_0\le \xi \le \eta }{\max }y(\xi ))\parallel d_{q,\omega }\eta )\hfill \\ & \le & \parallel W^{-1}{\parallel }_{L_b({\mathbb{R}}^n,L_{q,\omega }^2(J^{*},{\mathbb{R}}^n)/kerW)}(\parallel y_1\parallel +\parallel e_A(t_1)\parallel \parallel y_0\parallel \hfill \\ & & +\parallel e_A(t_1)\parallel {\int }_{{\omega }_0}^{t_1}{E}_{\parallel -A\parallel }\parallel g(\eta ,y(\eta ),\underset{{\omega }_0\le \xi \le \eta }{\max }y(\xi ))-g(\eta ,0,0)+g(\eta ,0,0)\parallel d_{q,\omega }\eta )\hfill \\ & \le & \parallel W^{-1}{\parallel }_{L_b({\mathbb{R}}^n,L_{q,\omega }^2(J^{*},{\mathbb{R}}^n)/kerW)}(\parallel y_1\parallel +\parallel e_A(t_1)\parallel \parallel y_0\parallel \hfill \\ & & +\parallel e_A(t_1)\parallel {\int }_{{\omega }_0}^{t_1}{E}_{\parallel -A\parallel }{L}_g(\parallel y(\eta )\parallel +\parallel \underset{{\omega }_0\le \xi \le \eta }{\max }y(\xi )\parallel )d_{q,\omega }\eta \hfill \\ & & +\parallel e_A(t_1)\parallel {\int }_{{\omega }_0}^{t_1}{E}_{\parallel -A\parallel }\parallel g(\eta ,0,0)\parallel d_{q,\omega }\eta )\hfill \\ & \le & \parallel W^{-1}{\parallel }_{L_b({\mathbb{R}}^n,L_{q,\omega }^2(J^{*},{\mathbb{R}}^n)/kerW)}(\parallel y_1\parallel +\parallel e_A(t_1)\parallel \parallel y_0\parallel \hfill \\ & & +2L_g\parallel e_A(t_1){\parallel \parallel y\parallel }_C{\int }_{{\omega }_0}^{t_1}{\displaystyle \frac{\parallel -A\parallel E_{\parallel -A\parallel }(\mu (\eta ))}{\parallel -A\parallel }}{d}_{q,\omega }\eta \hfill \\ & & +R_g\parallel e_A(t_1)\parallel {\int }_{{\omega }_0}^{t_1}{\displaystyle \frac{\parallel -A\parallel E_{\parallel -A\parallel }(\mu (\eta ))}{\parallel -A\parallel }}{d}_{q,\omega }\eta )\hfill \\ & \le & M\parallel y_1\parallel +M\delta +M{M}_{1}r,\hfill \end{array}$$

where $\delta$ and $M_1$ are as defined earlier.

From assumptions $\left[H_1\right]$ and $\left[H_2\right]$, it follows that

$$\begin{array}{ccc}\hfill \parallel (\mathcal{P}y)(t)\parallel & \le & \parallel e_A(t)\parallel \parallel y_0\parallel \hfill \\ & & +\parallel e_A(t)\parallel {\int }_{{\omega }_0}^t{E}_{\parallel -A\parallel }\parallel g(\eta ,y(\eta ),\underset{{\omega }_0\le \xi \le \eta }{\max }y(\xi ))-g(\eta ,0,0)+g(\eta ,0,0)\parallel d_{q,\omega }\eta \hfill \\ & & +\parallel e_A(t)\parallel {\int }_{{\omega }_0}^t{E}_{\parallel -A\parallel }\parallel B\parallel \parallel u_y(\eta )\parallel d_{q,\omega }\eta \hfill \\ & \le & \parallel e_A(t)\parallel \parallel y_0\parallel \hfill \\ & & +\parallel e_A(t)\parallel {\int }_{{\omega }_0}^t{\displaystyle \frac{\parallel -A\parallel E_{\parallel -A\parallel }(\mu (\eta ))}{\parallel -A\parallel }}{L}_g(\parallel y(\eta )\parallel +\parallel \underset{{\omega }_0\le \xi \le \eta }{\max }y(\xi )\parallel )d_{q,\omega }\eta \hfill \\ & & +\parallel e_A(t)\parallel {\int }_{{\omega }_0}^t{\displaystyle \frac{\parallel -A\parallel E_{\parallel -A\parallel }(\mu (\eta ))}{\parallel -A\parallel }}\parallel g(\eta ,0,0)\parallel d_{q,\omega }\eta \hfill \\ & & +\parallel e_A(t)\parallel {\int }_{{\omega }_0}^t{\displaystyle \frac{\parallel -A\parallel E_{\parallel -A\parallel }(\mu (\eta ))}{\parallel -A\parallel }}\parallel B\parallel \parallel u_y(\eta )\parallel d_{q,\omega }\eta \hfill \\ & \le & M_1{\parallel y\parallel }_C+\delta +{\displaystyle \frac{M_1}{2L_g}}\parallel B\parallel (M\parallel y_1\parallel +M\delta +M{M}_1{\parallel y\parallel }_C)\hfill \\ & \le & M_1{\parallel y\parallel }_C+\delta +{\displaystyle \frac{M_{1}M\parallel B\parallel (\parallel y_1\parallel +\delta )}{2L_g}}+{\displaystyle \frac{M_1^{2}M}{2L_g}}{\parallel y\parallel }_C\hfill \\ & \le & {\displaystyle \frac{2L_g\delta +M_{1}M\parallel B\parallel (\parallel y_1\parallel +\delta )}{2L_g}}+M_1[1+{\displaystyle \frac{M_{1}M}{2L_g}}]r=r\hfill \end{array}$$

for

$$r={\displaystyle \frac{M_{1}M\parallel B\parallel \parallel y_1\parallel +\delta (2L_g+M_{1}M\parallel B\parallel )}{2L_g[1-M_1(1+\frac{M_{1}M}{2L_g})]}}.$$

Therefore, for this choice of r, we have $\mathcal{P}\left({\mathcal{B}}_r\right)\subseteq {\mathcal{B}}_r$.

Next, we decompose $\mathcal{P}$ into two operators ${\mathcal{P}}_1$ and ${\mathcal{P}}_2$ on ${\mathcal{B}}_r$ as follows:

$$\begin{array}{ccc}\hfill \left({\mathcal{P}}_{1}y\right)\left(t\right)& =& e_A\left(t\right)y_0+e_A\left(t\right){\int }_{{\omega }_0}^t{E}_{-A}\left(\mu \left(\eta \right)\right)B{u}_y\left(\eta \right)d_{q,\omega }\eta ,\hfill \\ \hfill \left({\mathcal{P}}_{2}y\right)\left(t\right)& =& e_A\left(t\right){\int }_{{\omega }_0}^t{E}_{-A}\left(\mu \left(\eta \right)\right)g(\eta ,y\left(\eta \right),\underset{{\omega }_0\le \xi \le \eta }{\max }y\left(\xi \right))d_{q,\omega }\eta ,\hfill \end{array}$$

for $t\in J^{*}$, respectively.

Step 2. We demonstrate that ${\mathcal{P}}_1$ is a contraction operator.

Let $y,z\in {\mathcal{B}}_r$. In light of assumptions $\left[H_1\right]$ and $\left[H_2\right]$, for every $t\in J^{★}$, we obtain

$$\begin{array}{ccc}\hfill \parallel u_y(t)-u_z(t)\parallel & \le & M\parallel e_A(t_1)\parallel \parallel {\int }_{{\omega }_0}^{t_1}{E}_{-A}(\mu (\eta ))[g(\eta ,y(\eta ),\underset{{\omega }_0\le \xi \le \eta }{\max }y(\xi ))\hfill \\ & & -g(\eta ,z(\eta ),\underset{{\omega }_0\le \xi \le \eta }{\max }z(\xi ))]d_{q,\omega }\eta \parallel \hfill \\ & \le & M\parallel e_A(t_1)\parallel \hfill \\ & & \times {\int }_{{\omega }_0}^{t_1}{E}_{\parallel -A\parallel }\parallel g(\eta ,y(\eta ),\underset{{\omega }_0\le \xi \le \eta }{\max }y(\xi ))-g(\eta ,z(\eta ),\underset{{\omega }_0\le \xi \le \eta }{\max }z(\xi ))\parallel d_{q,\omega }\eta \hfill \\ & \le & M\parallel e_A(t_1)\parallel \hfill \\ & & \times L_g{\int }_{{\omega }_0}^{t_1}{E}_{\parallel -A\parallel }(\parallel y(\eta )-z(\eta )\parallel +\parallel \underset{{\omega }_0\le \xi \le \eta }{\max }y(\xi ))-\underset{{\omega }_0\le \xi \le \eta }{\max }z(\xi ))\parallel )d_{q,\omega }\eta \hfill \\ & \le & 2L_{g}M{e}_{\parallel A\parallel }(t_1){\parallel y-z\parallel }_C{\int }_{{\omega }_0}^{t_1}{\displaystyle \frac{\parallel -A\parallel E_{\parallel -A\parallel }(\mu (\eta ))}{\parallel -A\parallel }}{d}_{q,\omega }\eta \hfill \\ & \le & {\displaystyle \frac{2L_{g}M{e}_{\parallel A\parallel }(t_1)(E_{\parallel A\parallel }(t_1)-1)}{\parallel A\parallel }}{\parallel y-z\parallel }_C\hfill \\ & \le & M{M}_1{\parallel y-z\parallel }_C.\hfill \end{array}$$

Using this observation, we deduce that

$$\begin{array}{ccc}\hfill \parallel \left({\mathcal{P}}_{1}y\right)\left(t\right)-\left({\mathcal{P}}_{1}z\right)\left(t\right)\parallel & \le & \parallel e_A\left(t\right)\parallel \parallel {\int }_{{\omega }_0}^t{E}_{-A}\left(\mu \left(\eta \right)\right)B(u_y\left(\eta \right)-u_z\left(\eta \right))d_{q,\omega }\eta \parallel \hfill \\ & \le & \parallel e_A\left(t\right)\parallel \parallel B\parallel {\int }_{{\omega }_0}^t{\displaystyle \frac{\parallel -A\parallel E_{\parallel -A\parallel }\left(\mu \left(\eta \right)\right)}{\parallel -A\parallel }}\parallel u_y\left(\eta \right)-u_z\left(\eta \right)\parallel d_{q,\omega }\eta \hfill \\ & \le & {\displaystyle \frac{\parallel B\parallel M{M}_1{e}_{\parallel A\parallel }\left(t\right)(E_{\parallel A\parallel }\left(t\right)-1)}{\parallel A\parallel }}{\parallel y-z\parallel }_C\hfill \\ & \le & \delta M{M}_1{\parallel B\parallel \parallel y-z\parallel }_C,\hfill \end{array}$$

which gives that

$$\parallel {\mathcal{P}}_{1}y-{\mathcal{P}}_1{z\parallel }_C\le {V\parallel y-z\parallel }_C,V:=\delta M{M}_1\parallel B\parallel .$$

By inequality (21), we obtain $V<1$, and thus ${\mathcal{P}}_1$ is a contraction.

Step 3. We prove that ${\mathcal{P}}_2$ is a continuous and compact operator.

Let $y_n\in {\mathcal{B}}_r$ with $y_n\to y$ in ${\mathcal{B}}_r$. Set ${\mathcal{G}}_n(·)=g(·,y_n(·),\max y_n(·))$ and $\mathcal{G}(·)=g(·,y(·),\max y(·\left)\right)$. By assumption $\left[H_2\right]$, we have ${\mathcal{G}}_n\to \mathcal{G}$ in $C_{q,\omega }(J^{*},{\mathbb{R}}^n)$, and thus

$$\begin{array}{ccc}\hfill \parallel \left({\mathcal{P}}_2{y}_n\right)\left(t\right)-\left({\mathcal{P}}_{2}y\right)\left(t\right)\parallel & \le & \parallel e_A\left(t\right)\parallel {\int }_{{\omega }_0}^t\parallel E_{-A}\left(\mu \left(\eta \right)\right)\parallel \parallel {\mathcal{G}}_n\left(\eta \right)-\mathcal{G}\left(\eta \right)\parallel d_{q,\omega }\eta \to 0asn\to \infty \hfill \end{array}$$

uniformly for $t\in J^{*}$, which shows that ${\mathcal{P}}_2$ is continuous on ${\mathcal{B}}_r$.

To verify the compactness of ${\mathcal{P}}_2$, we show that ${\mathcal{P}}_2\left({\mathcal{B}}_r\right)\subset C_{q,\omega }(J^{*},{\mathbb{R}}^n)$ is both equicontinuous and bounded. Indeed, for any $y\in {\mathcal{B}}_r$ and $t_1\ge t+h\ge t>0$, we have

$$\begin{array}{ccc}\hfill ({\mathcal{P}}_{2}y)(t+h)-({\mathcal{P}}_{2}y)(t)& =& e_A(t+h){\int }_{{\omega }_0}^{t+h}{E}_{-A}(\mu (\eta ))g(\eta ,y(\eta ),\underset{{\omega }_0\le \xi \le \eta }{\max }y(\xi ))d_{q,\omega }\eta \hfill \\ & & -e_A(t){\int }_{{\omega }_0}^t{E}_{-A}(\mu (\eta ))g(\eta ,y(\eta ),\underset{{\omega }_0\le \xi \le \eta }{\max }y(\xi ))d_{q,\omega }\eta \hfill \\ & =& [e_A(t+h)-e_A(t)]{\int }_{{\omega }_0}^t{E}_{-A}(\mu (\eta ))g(\eta ,y(\eta ),\underset{{\omega }_0\le \xi \le \eta }{\max }y(\xi ))d_{q,\omega }\eta \hfill \\ & & +e_A(t+h){\int }_t^{t+h}{E}_{-A}(\mu (\eta ))g(\eta ,y(\eta ),\underset{{\omega }_0\le \xi \le \eta }{\max }y(\xi ))d_{q,\omega }\eta \hfill \\ & :=& I_1+I_2,\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill I_1& =& [e_A(t+h)-e_A(t)]{\int }_{{\omega }_0}^t{E}_{-A}(\mu (\eta ))g(\eta ,y(\eta ),\underset{{\omega }_0\le \xi \le \eta }{\max }y(\xi ))d_{q,\omega }\eta ,\hfill \\ \hfill I_2& =& e_A(t+h){\int }_t^{t+h}{E}_{-A}(\mu (\eta ))g(\eta ,y(\eta ),\underset{{\omega }_0\le \xi \le \eta }{\max }y(\xi ))d_{q,\omega }\eta .\hfill \end{array}$$

From the above assumptions, we conclude that

$$\begin{array}{c}\hfill \parallel \left({\mathcal{P}}_{2}y\right)(t+h)-\left({\mathcal{P}}_{2}y\right)\left(t\right)\parallel \le \parallel I_1\parallel +\parallel I_2\parallel .\end{array}$$

Next, we verify that $\parallel I_i\parallel \to 0$ as $h\to 0$, $i=1,2$ uniformly in t.

For $I_1$, by the definition of $e_A\left(t\right)$ and $e_A(t+h)$, we obtain $\parallel e_A(t+h)-e_A\left(t\right)\parallel \to 0$ as $h\to 0$; therefore

$$\begin{array}{ccc}\hfill \parallel I_1\parallel & \le & \parallel e_A(t+h)-e_A\left(t\right)\parallel ∥{\int }_{{\omega }_0}^t{E}_{-A}\left(\mu \left(\eta \right)\right)g(\eta ,y\left(\eta \right),\underset{{\omega }_0\le \xi \le \eta }{\max }y\left(\xi \right))d_{q,\omega }\eta ∥\to 0,\mathrm{as}h\to 0.\hfill \end{array}$$

In effect, note that for operators in ${\mathbb{R}}^{n\times n}$, the norm satisfies:

$$\parallel A^k-B^k\parallel \le \parallel A-B\parallel \sum _{j=0}^{k-1}{\parallel A\parallel }^j{\parallel B\parallel }^{k-1-j}.$$

Given that the sequence:

$$A_k\left(t\right):=A(t(1-q)-\omega ),$$

is continuous in t, the difference:

$$A_k(t+h)-A_k\left(t\right)=A((t+h)(1-q)-\omega )-A(t(1-q)-\omega )=Ah(1-q)$$

has norm:

$$\parallel A\left(h\right(1-q\left)\right)\parallel \le \parallel A\parallel ·\left|h\right(1-q\left)\right|,$$

which tends to zero as h→ 0.

Based on the above, we have:

$${\parallel (A((t+h)(1-q)-\omega ))}^k-(A{(t(1-q)-\omega )}^k\parallel \to 0,$$

for each fixed k, as $h\to 0$.

Furthermore, because the series converges uniformly, dominated by terms involving $\parallel A\parallel$ and the boundedness of t, series convergence and the boundedness imply that the entire sum:

$$\parallel e_A(t+h)-e_A\left(t\right)\parallel \to 0,$$

as $h\to 0$.

Assumption $\left[H_2\right]$ can then be applied to obtain

$$\begin{array}{ccc}\hfill \parallel I_2\parallel & \le & \parallel e_A(t+h)\parallel \parallel {\int }_t^{t+h}{E}_{-A}(\mu (\eta ))g(\eta ,y(\eta ),\underset{{\omega }_0\le \xi \le \eta }{\max }y(\xi ))d_{q,\omega }\eta \parallel \hfill \\ & \le & \parallel e_A(t+h)\parallel {\int }_t^{t+h}{E}_{\parallel -A\parallel }(\mu (\eta ))\parallel g(\eta ,y(\eta ),\underset{{\omega }_0\le \xi \le \eta }{\max }y(\xi ))-g(\eta ,0,0)+g(\eta ,0,0)\parallel d_{q,\omega }\eta \hfill \\ & \le & \parallel e_A(t+h)\parallel {\int }_t^{t+h}{\displaystyle \frac{\parallel -A\parallel E_{\parallel -A\parallel }(\mu (\eta ))}{\parallel -A\parallel }}{L}_g(\parallel y(\eta )\parallel +\parallel \underset{{\omega }_0\le \xi \le \eta }{\max }y(\xi )\parallel )d_{q,\omega }\eta \hfill \\ & & +\parallel e_A(t+h)\parallel {\int }_t^{t+h}{\displaystyle \frac{\parallel -A\parallel E_{\parallel -A\parallel }(\mu (\eta ))}{\parallel -A\parallel }}\parallel g(\eta ,0,0)\parallel d_{q,\omega }\eta \hfill \\ & \le & {\displaystyle \frac{\parallel e_A(t+h)\parallel (2L_g{\parallel y\parallel }_C+R_g)[E_{\parallel A\parallel }(t+h)-E_{\parallel A\parallel }(t)]}{\parallel A\parallel }}.\hfill \end{array}$$

Since $E_{\parallel A\parallel }(t+h)-E_{\parallel A\parallel }\left(t\right)\to 0$ as $h\to 0$. Thus, $\parallel I_2\parallel \to 0$ as $h\to 0$.

In effect, since

$$E_{\parallel A\parallel }\left(t\right)=\sum _{k=0}^{\infty }{c}_k\left(t\right),\mathrm{where}{c}_k\left(t\right)={\displaystyle \frac{q^{\frac{1}{2}k(k-1)}{\left(\parallel A\parallel \left(t\right(1-q)-\omega )\right)}^k}{{(q;q)}_k}},$$

and

$$E_{\parallel A\parallel }(t+h)=\sum _{k=0}^{\infty }{c}_k(t+h),$$

where

$$c_k(t+h)={\displaystyle \frac{q^{\frac{1}{2}k(k-1)}{\left(\parallel A\parallel \left(\right(t+h\left)\right(1-q)-\omega )\right)}^k}{{(q;q)}_k}}.$$

Thus, by the uniform convergence of the series, the limit as $h\to 0$ of

$$E_{\parallel A\parallel }(t+h)-E_{\parallel A\parallel }\left(t\right)=\sum _{k=0}^{\infty }[c_k(t+h)-c_k\left(t\right)],$$

goes to zero because each term:

$$c_k(t+h)-c_k\left(t\right)\to 0,$$

and the sum converges uniformly.

Based on the preceding discussion, we can directly conclude that

$$\parallel \left({\mathcal{P}}_{2}y\right)(t+h)-\left({\mathcal{P}}_{2}y\right)\left(t\right)\parallel \to 0,h\to 0,$$

uniformly for every t and every $y\in {\mathcal{B}}_r$. Hence, the family ${\mathcal{P}}_2\left({\mathcal{B}}_r\right)\subset C_{q,\omega }(J^{*},{\mathbb{R}}^n)$ is equicontinuous.

Next, by carrying out the same calculations as before, we obtain

$$\begin{array}{ccc}\hfill \parallel \left({\mathcal{P}}_{2}y\right)\left(t\right)\parallel & \le & \parallel e_A\left(t\right)\parallel \parallel {\int }_{{\omega }_0}^t{E}_{-A}\left(\mu \left(\eta \right)\right)g(\eta ,y\left(\eta \right),\underset{{\omega }_0\le \xi \le \eta }{\max }y\left(\xi \right))d_{q,\omega }\eta \parallel \hfill \\ & \le & \parallel e_A\left(t\right)\parallel {\int }_{{\omega }_0}^t{\displaystyle \frac{\parallel -A\parallel E_{\parallel -A\parallel }\left(\mu \left(\eta \right)\right)}{\parallel -A\parallel }}\hfill \\ & & \times \parallel g(\eta ,y\left(\eta \right),\underset{{\omega }_0\le \xi \le \eta }{\max }y\left(\xi \right))-g(\eta ,0,0)+g(\eta ,0,0)\parallel d_{q,\omega }\eta \hfill \\ & \le & 2\parallel e_A\left(t\right)\parallel L_g{\parallel y\parallel }_C{\displaystyle \frac{(E_{\parallel A\parallel }\left(t\right)-1)}{\parallel A\parallel }}+\parallel e_A\left(t\right)\parallel R_g{\displaystyle \frac{(E_{\parallel A\parallel }\left(t\right)-1)}{\parallel A\parallel }}\hfill \\ & \le & {\displaystyle \frac{e_{\parallel A\parallel }\left(b\right)(2L_g{\parallel y\parallel }_C+R_g)(E_{\parallel A\parallel }\left(b\right)-1)}{\parallel A\parallel }}.\hfill \end{array}$$

Thus, ${\mathcal{P}}_2\left({\mathcal{B}}_r\right)$ is bounded. By Arzela–Ascoli theorem, the set ${\mathcal{P}}_2\left({\mathcal{B}}_r\right)\subset C_{q,\omega }(J^{*},{\mathbb{R}}^n)$ is relatively compact in $C_{q,\omega }(J^{*},{\mathbb{R}}^n)$.

Consequently, ${\mathcal{P}}_2$ is a compact and continuous operator. Therefore, by Krasnoselskii’s fixed point theorem, $\mathcal{P}$ admits a fixed point y in ${\mathcal{B}}_r$. Clearly, this fixed point y is a solution of system (2) with $y\left(t_1\right)=y_1$, and it also satisfies the initial condition $y\left({\omega }_0\right)=y_0$. This completes the proof. □

4. Examples

In this section, we primarily present an example to validate our theoretical findings. In addition, we introduce the following norms defined by a compact and continuous operator.

$$\begin{array}{c}\hfill \parallel A\parallel =(\sum _{i=1}^n\sum _{j=1}^n|a_{ij}{|}^2{)}^{\frac{1}{2}},\parallel y\parallel =(\sum _{i=1}^n|y_i{{|}^2)}^{\frac{1}{2}}.\end{array}$$

Example 1.

Consider the linear Hahn difference control system

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}& D_{\frac{1}{2},\frac{1}{2}}y\left(t\right)=Ay\left(t\right)+Bu\left(t\right),t\in J_1:=\left[1,{\displaystyle \frac{14}{5}}\right],\hfill \\ & y\left(1\right)=\left(\begin{array}{c}{\displaystyle \frac{3}{2}}\\ 1\end{array}\right),\hfill \end{array}\right.\end{array}\end{array}$$

(23)

where

$$q=\frac{1}{2},\omega =\frac{1}{2},{\omega }_0=1,t_1={\displaystyle \frac{14}{5}}<{\omega }_0+{\displaystyle \frac{1}{\parallel A\parallel (1-q)}}=1+{\displaystyle \frac{1}{1-\frac{1}{2}}}=3,$$

$$A={\displaystyle \frac{1}{2}}\left(\begin{array}{cc}1& \sqrt{2}\\ 0& -1\end{array}\right),B=\left(\begin{array}{cc}{\displaystyle \frac{1}{2}}& 0\\ 0& 1\end{array}\right).$$

For this purely linear system, the Grammian matrix on $\left[1,\frac{14}{5}\right]$ is defined exactly as in (10):

$$\begin{array}{c}\hfill W\left[1,\frac{14}{5}\right]=e_A\left(\frac{14}{5}\right){\int }_1^{\frac{14}{5}}{E}_{-A}\left(\mu \left(\eta \right)\right)B{B}^T{E}_{-A}^T\left(\mu \left(\eta \right)\right)e_A^T\left(\frac{14}{5}\right)d_{q,\omega }\eta .\end{array}$$

(24)

Using the explicit form of A and B, together with the definitions of the Hahn matrix exponentials $e_A$ and $E_{-A}$, we evaluate the integral in (24) numerically. This yields the $2\times 2$ Grammian matrix

$$W\left[1,\frac{14}{5}\right]\approx \left(\begin{array}{cc}62.0019& 8.4385\\ 8.4385& 1.9227\end{array}\right),$$

with

$$\det W\left[1,\frac{14}{5}\right]=48.0011\ne 0.$$

Hence $W\left[1,\frac{14}{5}\right]$ is nonsingular, and by Theorem 1, system (23) is controllable on $J_1$.

Example 2.

We examine the following system

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}& D_{\frac{1}{4},\frac{1}{4}}y(t)=Ay(t)+g(t,y(t),\underset{\frac{1}{3}\le \xi \le \frac{14}{9}}{\max }y(\xi ))+Bu(t),t\in J_1^{*}:={\left[{\displaystyle \frac{1}{3}},{\displaystyle \frac{14}{9}}\right]}^{*},\hfill \\ & y\left({\displaystyle \frac{1}{3}}\right)={\left({\displaystyle \frac{1}{5}}{\displaystyle \frac{2}{5}}\right)}^T,\hfill \end{array}\right.\end{array}\end{array}$$

(25)

where we set $A=\left(\begin{array}{cc}\frac{1}{3\sqrt{3}}& \frac{1}{3\sqrt{3}}\\ 0& -\frac{1}{3\sqrt{3}}\end{array}\right)$ and $g(t,y\left(t\right),\underset{\frac{1}{3}\le \xi \le \frac{14}{9}}{\max }y\left(\xi \right))=\left(\begin{array}{c}\frac{1}{1000}(t+0.1)y_1\left(t\right)+1\\ \frac{1}{1000}(t+0.1)y_2\left(t\right)+1\end{array}\right),$

$B=\left(\begin{array}{cc}\frac{1}{6}& \frac{1}{4}\\ 0& -\frac{1}{3\sqrt{2}}\end{array}\right)$, $u\left(t\right)=\left(\begin{array}{c}t\\ \sin t\end{array}\right).$

Here, we have

$$∥\left(\begin{array}{cc}\frac{1}{3\sqrt{3}}& \frac{1}{3\sqrt{3}}\\ 0& -\frac{1}{3\sqrt{3}}\end{array}\right)∥={\left({\displaystyle \sum _{i=1}^2}{\displaystyle \sum _{j=1}^2}{\left|a_{ij}\right|}^2\right)}^{\frac{1}{2}}=\sqrt{{(\frac{1}{3\sqrt{3}})}^2+{(\frac{1}{3\sqrt{3}})}^2+0^2+{|-\frac{1}{3\sqrt{3}}|}^2}=\frac{1}{3},$$

$$∥\left(\begin{array}{c}\frac{1}{5}\\ \frac{2}{5}\end{array}\right)∥={\left(\sum _{i=1}^2{\left|y_i\right|}^2\right)}^{\frac{1}{2}}=\sqrt{{\left(\frac{1}{5}\right)}^2+{\left(\frac{2}{5}\right)}^2}=\frac{\sqrt{5}}{5},$$

$$∥\left(\begin{array}{cc}\frac{1}{6}& \frac{1}{4}\\ 0& -\frac{1}{3\sqrt{2}}\end{array}\right)∥={\left(\sum _{i=1}^2\sum _{j=1}^2{\left|q_{ij}\right|}^2\right)}^{\frac{1}{2}}=\sqrt{{\left(\frac{1}{6}\right)}^2+{\left(\frac{1}{4}\right)}^2+0^2+{|-\frac{1}{3\sqrt{2}}|}^2}=\frac{\sqrt{3}}{3},$$

$$R_g=\underset{t\in J_1^{*}}{\sup }\parallel g(t,0,0)\parallel =\frac{1}{2}.$$

Further, it is easy to see that for any $y\left(t\right),z\left(t\right)\in {\mathbb{R}}^2$, and $t\in J_1^{*}$,

$$\begin{array}{ccc}& & \parallel g(t,y,\max y)-g(t,z,\max z)\parallel \hfill \\ & =& {\displaystyle \frac{1}{1000}}(t+0.1)\sqrt{{(y_1\left(t\right)-z_1\left(t\right))}^2+{(y_2\left(t\right)-z_2\left(t\right))}^2)}\hfill \\ & \le & {\displaystyle \frac{1}{1000}}(t+0.1)\parallel y-z\parallel .\hfill \end{array}$$

Therefore, g meets condition $\left[H_2\right]$, with $L_g$ chosen as $0.0017$.

From (8) and (9), since $\parallel A\parallel =\frac{1}{3}$, $\parallel y_0\parallel =\frac{\sqrt{5}}{5}$, then

$$\begin{array}{ccc}\hfill e_{\parallel A\parallel }\left({\displaystyle \frac{14}{9}}\right)& =& \sum _{k=0}^{\infty }{\displaystyle \frac{{(\frac{1}{3}\times (\frac{14}{9}(1-\frac{1}{4})-\frac{1}{4}))}^k}{{(\frac{1}{4};\frac{1}{4})}_k}}\hfill \\ & =& \sum _{k=0}^{\infty }{\displaystyle \frac{{\left(\frac{11}{36}\right)}^k}{(1-\frac{1}{4})(1-\frac{1}{4^2})\dots (1-\frac{1}{4^k})}}\hfill \\ & =& 1.5996,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill E_{\parallel A\parallel }\left({\displaystyle \frac{14}{9}}\right)& =& \sum _{k=0}^{\infty }{\displaystyle \frac{q^{\frac{1}{2}k(k-1)}{\left(\frac{1}{3}(\frac{14}{9}(1-\frac{1}{4})-\frac{1}{4})\right)}^k}{{(\frac{1}{4};\frac{1}{4})}_k}}\hfill \\ & =& \sum _{k=0}^{\infty }{\displaystyle \frac{{\left(\frac{1}{2}\right)}^{k(k-1)}\times {\left(\frac{11}{36}\right)}^k}{{(\frac{1}{4};\frac{1}{4})}_k}}\hfill \\ & =& \sum _{k=0}^{\infty }{\displaystyle \frac{{\left(\frac{1}{2}\right)}^{k(k-1)}\times {\left(\frac{11}{36}\right)}^k}{(1-\frac{1}{4})(1-\frac{1}{4^2})\dots (1-\frac{1}{4^k})}}\hfill \\ & =& 2.5386.\hfill \end{array}$$

Consequently, we obtain

$$M=\sqrt{∥W^{-1}[{\displaystyle \frac{1}{3}},{\displaystyle \frac{14}{9}}]∥}\approx 8.4549,$$

$$\begin{array}{ccc}\hfill M_1& =& {\displaystyle \frac{2L_g{e}_{\parallel A\parallel }\left(\frac{14}{9}\right)(E_{\parallel A\parallel }\left(\frac{14}{9}\right)-1)}{\parallel A\parallel }}\hfill \\ & =& 0.0251,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \delta & =& e_{\parallel A\parallel }\left({\displaystyle \frac{14}{9}}\right)\parallel y_0\parallel +{\displaystyle \frac{R_g{e}_{\parallel A\parallel }\left(\frac{14}{9}\right)(E_{{\parallel A\parallel }_{{\mathbb{R}}^{2\times 2}}}\left(\frac{14}{9}\right)-1)}{\parallel A\parallel }}\hfill \\ & =& 4.4072.\hfill \end{array}$$

Therefore,

$$\begin{array}{c}\hfill M_1(1+\delta M\parallel B\parallel )=0.5651<1,\end{array}$$

which shows that condition (21) is satisfied.

Therefore, all the hypotheses of Theorem 2 are fulfilled, and consequently system (2) is controllable on ${\left[\frac{1}{3},\frac{14}{9}\right]}^{*}$.

5. Conclusions

This paper aims to establish controllability results for linear and nonlinear Hahn difference control systems whose nonlinear terms are given by involving maxima. We derive new criteria for controllability: for the linear case via a constructed Grammian matrix, and for the nonlinear case by reformulating the problem and applying fixed-point techniques.