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Abstract

The primary goal of this work is to establish the approximate controllability of a semilinear neutral-type system with nonlocal conditions and impulses, demonstrating that the approximate controllability of a neutral linear equation is preserved when external forces, impulses and nonlocal conditions are introduced as perturbations to the system. The approach leverages the properties of sectorial operators, the compactness of the semigroup that governs the evolution of the linear part of the equation and Rothe’s fixed point theorem. To demonstrate the practical applicability of our method, we present a specific case that encompasses a broad family of examples, including a neutral-type heat equation.

approximate controllability, neutral semilinear evolution equations, impulses, nonlocal conditions, Rothe’s fixed point theorem

1. Introduction

The main goal of this paper is to establish the aproximate controllability of the following impulsive neutral semilinear evolution equation with nonlocal conditions on Hilbert spaces |$U$| and |$Z$|⁠,

(1.1)

where the operator |$A:D(A)\subset Z\to Z$| is sectorial and |$-A$| is the infinitesimal generator of a compact analytic semigroup of uniformly bounded linear operators |$\{T(t)\}_{t\geq 0}$|⁠, with |$0\in \rho (A)$|⁠, then fractional power operators |$A^\beta $|⁠, |$0<\beta \leq 1$|⁠, are well defined. Since |$A^\beta $| is a closed operator, its domain |$D(A^\beta )$| is a Banach space endowed with the graph norm

$$ \begin{align*} & \left\Vert z\right\Vert{}_\beta=\left\Vert A^\beta z\right\Vert, \quad z\in D(A^\beta). \end{align*} $$

This Banach space is denoted by |$Z^\beta =D(A^\beta )$| and it is dense in |$Z$|⁠. Moreover, for |$0<\delta <\beta \leq 1$|⁠, the embedding |$Z^\beta \hookrightarrow Z^\delta $| is compact whenever the resolvent operator of |$A$| is compact. Also, the following properties of the semigroup |$\{T(t)\}_{t\geq 0}$| and the operator |$A^\beta $| are well known (see, for instance, Pazy (1983)):

(1.2)
(1.3)
(1.4)

where |$\gamma>0$|⁠, |$M\geq 1$| and |$M_\beta \geq 0$| are real constants. For more properties of sectorial operators and strongly continuous semigroups, we referred to Henry (1981) and Jerome (1985).

Here, |$u\in L^{2}([0,\tau ],U)$| represents the control and |$B:U\to Z$| is a linear, bounded operator. |$0<t_{1}<\cdots <t_{p}<\tau $|⁠, |$0<\tau _{1}<\cdots <\tau _{q}<r<\tau $|⁠, |$I_{p}:=\{1,\dots ,p\}$|⁠, |$z_{t}$| is the time history function |$[-r,0]\ni \theta \mapsto z_{t}(\theta )=z(t+\theta )\in Z^\beta $| and |$J_{k}:Z^\beta \times U\to Z^\beta $|⁠, |$k\in I_{p}$|⁠.

If we denote by |$PW_{r\beta }$| to the Banach space

$$ \begin{align*} & PW_{r\beta}=PW([-r,0],Z^\beta)=\left\{\eta:[-r,0]\to Z^\beta| \ \eta \ \mbox{is piecewise continuous}\right\} \end{align*} $$

endowed with the supremum norm |$\left \Vert \cdot \right \Vert{}_{r\beta }$|⁠, then |$g:[0,\tau ]\times PW_{r\beta }\to Z^\beta $|⁠, |$f:[0,\tau ]\times PW_{r\beta }\times U\to Z$| and |$\eta \in PW_{r\beta }$|⁠.

On the other hand, considering the notation

$$ \begin{align*} & [0,\tau]^{\prime}=[0,\tau]\setminus\{t_k\}_{k\in I_P}, \quad \phi(t_k^-):=\displaystyle\lim_{t\to t_k^-}\phi(t), \quad \phi(t_k^+):=\displaystyle\lim_{t\to t_k^+}\phi(t), \end{align*} $$

a suitable Banach space to work impulsive differential systems is given by

$$ \begin{align*} PW_{p\beta}:=\Big\{ & z:[-r,\tau]\to Z^\beta\big| \ z\big|_{[-r,0]}\in PW_{r\beta}, \ z\big|_{[0,\tau]}\in C([0,\tau]^{\prime},Z^\beta) \ \mbox{and the} \\ & \ \mbox{one-side limits} \ z(t_{k}^{-}), \ z(t_{k}^{+}) \ \mbox{exist with} \ z(t_{k}^{-})=z(t_{k}) \ \mbox{for all} \ k\in I_{p}\Big\}, \end{align*} $$

equipped with the supremum norm |$\left \Vert \cdot \right \Vert{}_{p\beta }$|⁠. Also, the space

$$ \begin{align*} & \big(Z^\beta\big)^q=\underbrace{Z^\beta\times\cdots\times Z^\beta}_{q-times} \end{align*} $$

is a Banach space endowed with the norm

$$ \begin{align*} & \left\Vert z\right\Vert{}_{q\beta}=\displaystyle\sum_{i=1}^q\left\Vert z_i\right\Vert{}_\beta, \quad z=(z_1,\dots,z_q)\in \big(Z^\beta\big)^q. \end{align*} $$

Similarly, the space

$$ \begin{align*} & PW_{qp\beta}:=PW\big([-r,0],(Z^\beta)^q\big) \end{align*} $$

is a Banach space with the norm

$$ \begin{align*} & \left\Vert\eta\right\Vert{}_{qp\beta}=\displaystyle\sup_{[-r,0]}\left\Vert\eta(t)\right\Vert{}_{q\beta}=\displaystyle\sup_{[-r,0]} \left\{\displaystyle\sum_{i=1}^q\left\Vert\eta_i(t)\right\Vert{}_\beta\right\}, \quad \eta=(\eta_1,\dots,\eta_q)\in PW_{qp\beta}. \end{align*} $$

In this case |$h:PW_{qp\beta }\to PW_{r\beta }$|⁠.

In this work, we demonstrate that the approximate controllability of a neutral linear equation remains preserved when external forces, impulses and nonlocal conditions are introduced as perturbations to the system. This is a natural extension, as such perturbations, though often neglected, are inherent in real-life systems modeled by differential equations.

To establish the controllability of system (1.1), we employ Rothe’s fixed point theorem, which is particularly suited for this purpose because it only requires the operator in question to map the boundary of a ball into the ball’s interior. Sublinear perturbations further refine this by enabling the operator to map the boundary of a convex set, closed and containing zero in its interior, into the set’s interior. This provides a more effective approach to demonstrating that the controllability of a semilinear system is preserved under sublinear perturbations. While there are relatively few studies on neutral differential equations with impulses and nonlocal conditions simultaneously, notable contributions include Fu (2011); Jeet & Sukavanam (2020); Shukla et al. (2022); Sivasankar & Udhayakumar (2022) and Gokul & Udhayakumar (2024).

The main contribution of this manuscript lies in the efficient proof of the conjecture that controllability is preserved under sublinear perturbations. This result significantly generalizes existing works, as the sublinear condition encompasses a broad class of perturbations, including bounded functions. Moreover, our framework allows the perturbations to depend simultaneously on both the state and the control, adding complexity and novelty to the analysis.

The paper is organized as follows. In Section 2, we present preliminary results on the existence and uniqueness of mild solutions for system (1.1). Section 3 provides characterizations of the approximate controllability for the linear equation associated with system (1.1). In Section 4, we prove the main result of this paper, establishing the approximate controllability of the nonlinear system (1.1). Section 5 is devoted to an application, and the paper concludes with final remarks.

2. Preliminaries: existence and uniqueness

This section is devoted to present some conditions for ensure the existence and uniqueness of mild solutions for system (1.1); for details, we referred to Agarwal et al. (2022).

 

Definition 1 (See Agarwal et al. (2022)).
Let |$u\in L^{2}([0,\tau ],U)$|⁠. A function |$z\in PW_{p\beta }$| is said to be a mild solution of problem (1.1) if it satisfies the integral equation
(2.1)

In order to ensure existence and uniqueness of mild solutions of problem (1.1) the following conditions are required:

  • H1) There exist positive constants |$L_{h}$|⁠, |$\gamma ,$| and |$d_{k}$|⁠, |$k\in I_{p}$| such that

    • i) |$L_{h}qM<\gamma +M\sum _{k=1}^{p}d_{k}<\frac{1}{2}$|⁠,

    • ii) |$J_{k}(0)=0$| and |$\left \Vert J_{k}(z,u)-J_{k}(v,w)\right \Vert{}_\beta \leq d_{k}\big (\left \Vert z-w\right \Vert{}_\beta +\left \Vert u-w\right \Vert \big )$|⁠, |$z,v\in Z^\alpha $|⁠, |$u,w\in U$|⁠.

    • iii) |$h(0)=0$| and
      $$ \begin{align*} & \left\Vert[h(z)](t)-[h(w)](t)]\right\Vert{}_\beta\leq L_h\displaystyle\sum_{i=1}^q\left\Vert z_i(t)-w_i(t)\right\Vert{}_\beta, \quad z,w\in PW_{qp\beta}. \end{align*} $$

  • H2) The map |$g:[0,\tau ]\times PW_{r\beta }\to D(A)$| satisfies

    • i) |$\left \Vert Ag(t,\eta _{1})-Ag(t,\eta _{2})\right \Vert \leq{\mathcal{K}}(\left \Vert \eta _{1}\right \Vert{}_{r\beta },\left \Vert \eta _{2}\right \Vert{}_{r\beta })\left \Vert \eta _{1}-\eta _{2}\right \Vert{}_{r\beta }$|⁠, |$\ \eta _{1},\eta _{2}\in PW_{r\beta }$|⁠,

    • ii) |$ \left \Vert Ag(t,\eta )\right \Vert \leq \psi (\left \Vert \eta \right \Vert{}_{r\beta })$|⁠, |$\ \eta \in PW_{r\beta }$|⁠,

    • iii) |$\left \Vert g(t,\eta _{1})-g(t,\eta _{2})\right \Vert{}_\beta \leq L_{g}\left \Vert \eta _{1}-\eta _{2}\right \Vert{}_{r\beta }$|⁠, |$\ \eta _{1},\eta _{2}\in PW_{r\beta }$|⁠,

    • iv) |$\left \Vert f(t,\eta _{1},u)-f(t,\eta _{2},w)\right \Vert \leq{\mathcal{K}}(\left \Vert \eta _{1}\right \Vert{}_{r\beta },\left \Vert \eta _{2}\right \Vert{}_{r\beta })\left (\left \Vert \eta _{1}-\eta _{2}\right \Vert{}_{r\beta }+\left \Vert u-w\right \Vert \right )$|⁠, |$\ \eta _{1},\eta _{2}\in PW_{r\beta }$|⁠, |$u,w\in U$|⁠,

    • v) |$\left \Vert f(t,\eta )\right \Vert \leq \psi (\left \Vert \eta \right \Vert{}_{r\beta })$|⁠, |$\ \eta \in PW_{r\beta }$|⁠,where |${\mathcal{K}}\in C({{\mathbb{R}}}^{+}\times{{\mathbb{R}}}^{+},{{\mathbb{R}}}^{+})$| and |$\psi \in C({{\mathbb{R}}}^{+},{{\mathbb{R}}}^{+})$| are nondecreasing functions.

  • H3) There exists |$\rho>0$| such that
    $$ \begin{align*} M\psi(\left\Vert\eta\right\Vert+ & L_{h}q(\left\Vert\tilde{\eta}\right\Vert+\rho)+\left(ML_{h}q+M\displaystyle\sum_{k=1}^{q}d_{k}\right)(\left\Vert\tilde{\eta}\right\Vert+\rho)\\ + & \left(\displaystyle\frac{2M_\alpha}{1-\alpha}\tau^{1-\alpha}+1\right)\psi(\left\Vert\tilde{\eta}\right\Vert+\rho)<\rho \end{align*} $$
    where the function |$\tilde{\eta }$| is defined as follows:
    (2.2)
  • H4) Assume the following relation holds:
    $$ \begin{align*} & ML_hq(1+\gamma)+2M_\alpha{\mathcal{K}}(\left\Vert\tilde{\eta}\right\Vert+\rho,\left\Vert\tilde{\eta}\right\Vert+\rho)\displaystyle\frac{\tau^{1-\alpha}}{1-\alpha}<\displaystyle\frac{1}{2}. \end{align*} $$

 

Theorem 1 (See (Agarwal et al., 2022, Theorems 3,4)).

Suppose that (H|$_{1}$|⁠), (H|$_{2}$|⁠) and (H|$_{3}$|⁠) hold. Then problem (1.1) has at least one mild solution in |$PW_{p\beta }$|⁠. If, in addition, (H|$_{4}$|⁠) holds, then the mild solution is unique.

3. Approximate controllability of linear system

In this section, we present some characterization for the approximate controllability of the following linear evolution equation without impulses, delay and nonlocal conditions:

(3.1)

where |$z^{0}\in Z$| and |$u\in L^{2}([0,\tau ],U)$|⁠. System (3.1) admits only one mild solution that is given by

(3.2)

 

Definition 2 (See Leiva & Quintana (2009)).
System (3.1) is said to be approximately controllable on |$[0,\tau ]$| if for every |$z^{0},z^{1}\in Z$| and |$\varepsilon>0$| there exists a control |$u\in L^{2}([0,\tau ],U)$| such that the mild solution (3.2) corresponding to |$u$| verifies
$$ \begin{align*} & z(0)=z^0 \quad \mbox{and} \quad \left\Vert z(\tau)-z^1\right\Vert<\varepsilon. \end{align*} $$

 

Definition 3 (See Guevara & Leiva (2018)).

For system (3.1) we define the following concepts:

  • a) The controllability map
    (3.3)
    and the adjoint of this operator |$\mathscr{G}^{*}:Z\to L^{2}([0,\tau ],U)$| is given by
    $$ \begin{align*} & (\mathscr{G}^*z)(t)=B^*T^*(\tau-t)z, \quad t\in [0,\tau]. \end{align*} $$
  • b) The Gramian controllability operator is given by
    (3.4)

The following lemma holds in general for a linear bounded operator |$\mathscr{G}:W\to Z$| between Hilbert spaces |$W$| and |$Z$|⁠.

 

Lemma 1 (See Curtain & Zwart (1995); Leiva & Quintana (2009); Leiva & Uzcátegui (2011); Leiva et al. (2013)).

The equation (3.1) is approximately controllable on |$[0,\tau ]$| if and only if one of the following statements holds:

  • a) |$\overline{\operatorname{Range}(\mathscr{G})}=Z$|⁠.

  • b) |$\operatorname{Ker}(\mathscr{G}^{*})=\{0\}$|⁠.

  • c) |$\langle L_{_{\mathscr{G}}}z,z\rangle>0$| for |$z\neq 0$| in |$Z$|⁠.

  • d) |$B^{*}T^{*}(t)z=0$| on |$[0,\tau ]$| implies |$z=0$|⁠.

  • e) |$\lim _{\alpha \to 0^{+}}\alpha (\alpha I+L_{_{\mathscr{G}}})^{-1}z=0$|⁠.

  • f) |$\sup _{\alpha>0}\left \Vert \alpha (\alpha I+L_{\mathscr{G}})^{-1}\right \Vert \leq 1$|⁠.

Therefore, if the system (3.1) is approximately controllable on |$[0,\tau ]$|⁠, then for all |$z\in Z$|⁠, we have |$\mathscr{G} u_\alpha =z-\alpha (\alpha I+L_{_{\mathscr{G}}})^{-1}z$| where

$$ \begin{align*} & u_\alpha=\mathscr{G}^*(\alpha I+L_{_{\mathscr{G}}})^{-1}z, \quad \alpha\in (0,1]. \end{align*} $$

So, |$\lim _{\alpha \to 0^{+}}\mathscr{G} u_\alpha =z$| and the error |$\mathscr{E}_\alpha z$| of this approximation is given by the formula

$$ \begin{align*} & \mathscr{E}_\alpha z=\alpha(\alpha I+L_{_{\mathscr{G}}})^{-1}z, \quad \alpha\in(0,1]. \end{align*} $$

 

Remark 1.
The lemma 1 implies that the family of linear operators
$$ \begin{align*} & \varGamma_\alpha:Z\to L^2([0,\tau],U) \end{align*} $$
defined, for |$0<\alpha \leq 1$|⁠, by
(3.5)
is an approximate inverse for the right of the operator |$\mathscr{G}$| in the sense that
$$ \begin{align*} & \displaystyle\lim_{\alpha\to 0^+}\mathscr{G}\varGamma_\alpha=I, \end{align*} $$
in the strong topology.

 

Lemma 2.
Let |$Z$| and |$W$| be normed spaces and |$ \mathscr{G}:W\to Z$| a linear and bounded operator. If |$S\subset W$| is a linear subspace such that |$\overline{S}=W$|⁠, then
$$ \begin{align*} & \overline{\operatorname{Range}(\mathscr{G})}=Z \quad \mbox{if and only if} \quad \overline{\operatorname{Range}(\mathscr{G}\big|_S)}=Z. \end{align*} $$

 

Proof.
Suppose that |$\overline{\operatorname{Range}(\mathscr{G})}=Z$|⁠. Then, for a given |$z\in Z$| there exists a sequence |$\{w_{n}\}_{n\geq 1}\subset W$| such that
$$ \begin{align*} & \displaystyle\lim_{n\to\infty}\mathscr{G}(w_n)=z. \end{align*} $$
On the other hand, since |$\overline{S}=W$|⁠, for each |$n\in{{\mathbb{N}}}$| there exists |$w_{n}^{\prime}\in S$| such that |$\left \Vert w_{n}-w^{\prime}_{n}\right \Vert <\frac{1}{n}$|⁠. Then |$\{w^{\prime}_{n}\}_{n\geq 1}$| is a sequence in |$S$| and |$\mathscr{G}(w^{\prime}_{n})=\mathscr{G}(w_{n})+\mathscr{G}(w^{\prime}_{n}-w_{n})\to z$| as |$n\to \infty $|⁠. Therefore, |$\overline{\operatorname{Range}\left (\mathscr{G}\big |_{S}\right )}=Z$|⁠.

The reciprocal statement is trivial.

 

Remark 2.
According to the Lemma 2, if the system (3.1) is approximately controllable, it is approximately controllable with control functions in the following dense spaces of |$L^{2}([0,\tau ],U)$|⁠:
$$ \begin{align*} & S=C([0,\tau],U), \quad S=C^\infty([0,\tau],U), \quad S=PW([0,\tau],U). \end{align*} $$
Moreover, the operators |$\mathscr{G}$|⁠, |$L_{_{\mathscr{G}}}$| and |$\varGamma _\alpha $| given in (3.3), (3.4) and (3.5) are well defined in the space of continuous functions.

4. Approximate controllability of the semilinear system

In this section we shall prove the main result of this paper, the approximate controllability of the nonlinear system (1.1). In order to accomplish with this task, we will assume that the linear system (3.1) is approximately controllable on |$[0,\tau ]$|⁠, and, in addition, we shall need the following assumptions:

H5)

  • i) |$\left \Vert f(t,\phi ,u)\right \Vert \leq a_{0}\left \Vert \phi \right \Vert{}_{r\beta }^{\alpha _{0}}+b_{0}\left \Vert u\right \Vert{}^{\beta _{0}}+c_{0}$|⁠, |$\ \frac{1}{2}\leq \alpha _{0},\beta _{0}<1$|⁠.

  • ii) |$\left \Vert J_{k}(z,u)\right \Vert{}_\beta \leq a_{k}\left \Vert z\right \Vert{}_{\beta }^{\alpha _{k}}+b_{k}\left \Vert u\right \Vert{}^{\beta _{k}}+c_{k}$|⁠, |$\ \frac{1}{2}\leq \alpha _{k},\beta _{k}<1$|⁠, |$\ k=1,\dots ,p$|⁠.

  • iii) |$\left \Vert h(\phi _{1},\dots ,\phi _{q})\right \Vert{}_\beta \leq \sum _{i=1}^{q} e_{i}\left \Vert \phi _{i}\right \Vert{}_{r\beta }^{\gamma _{i}}+e_{0}$|⁠, |$\ \frac{1}{2}\leq \gamma _{i}<1$|⁠, |$\ i=1,\dots ,q$|⁠.

  • iv) |$\left \Vert g(t,\phi )\right \Vert{}_\beta \leq r_{1}\left \Vert \phi \right \Vert{}_{r\beta }^{\gamma _{0}}+r_{0}$|⁠, |$\ \frac{1}{2}\leq \gamma _{0}<1$|⁠.

  • v) |$\left\Vert Ag(t,\phi )\right\Vert \leq l_{1}\left\Vert \phi \right\Vert{}_{r\beta }^{\nu _{0}}+l_{0}$|⁠, |$\ \frac{1}{2}\leq \nu _{0}<1$|⁠.

 

Definition 4 (Approximate controllability. See Leiva (2023)).
The system (1.1) is said to be approximately controllable on |$[0,\tau ]$| if for every |$\eta \in PW_{r\beta }$|⁠, |$z^{1}\in Z^\beta $| and |$\varepsilon>0$|⁠, there exists |$u\in C([0,\tau ],U)$| such that the mild solution |$z(t)$| of (1.1) corresponding to |$u$| verifies:
$$ \begin{align*} & z(0)+\big(h(z_{\tau_1},\dots,z_{\tau_q})\big)(0)=\eta(0), \quad \mbox{and} \quad \left\Vert z(\tau)-z^1\right\Vert{}_\beta<\varepsilon. \end{align*} $$

The main tool to prove the approximate controllability of system (1.1) is the following result:

 

Theorem 2 (Rothe’s fixed point. See Smart (1974); Banas & Goebel (1980); Isac (2004)).

Let |$E$| be a Banach space. Let |$W\subset E$| be a closed convex subset such that the zero of |$E$| is contained in the interior of |$W$|⁠. Let |$\varPhi :W\to E$| be a continuous mapping with |$\varPhi (W)$| relatively compact in |$E$| and |$\varPhi (\partial W)\subset W$|⁠. Then there exists a point |$x^{*}\in W$| such that |$\varPhi (x^{*})=x^{*}$|⁠.

We define the following operators:

(4.1)

by the formula

(4.2)

where

(4.3)

and

(4.4)

with

$$ \begin{align*} & {\mathcal{L}}:PW_{p\beta}\times C([0,\tau],U)\to Z^\beta \end{align*} $$

given by

$$ \begin{align*} {\mathcal{L}}(z,u)= & z^{1}-T(\tau)\big[\eta(0)-\big(h(z_{\tau_{1}},\dots,z_{\tau_{q}})\big)(0)-g(0,\eta(0) -\big(h(z_{\tau_{1}},\dots,{\tau_{q}})\big)(0)\big]\\ & -g(\tau,z_\tau)+\displaystyle\int_{0}^\tau A T(\tau-s)g(s,z_{s})ds-\displaystyle\int_{0}^\tau T(\tau-s)f(s,z_{s},u(s))ds \\ & -\displaystyle\sum_{0<t_{k}<\tau}T(\tau-t_{k})J_{k}(z(t_{k}),u(t_{k})). \end{align*} $$

On |$PW_{p\beta }\times C([0,\tau ],U)$|⁠, we will consider the norm

$$ \begin{align*} & |||(z,u)|||=\left\Vert z\right\Vert{}_{p\beta}+\left\Vert u\right\Vert. \end{align*} $$

 

Lemma 3.

The operator |${\mathcal{S}}^{\, \alpha} $| is continuous.

 

Proof.
It suffices to demonstrate that the operators
$$ \begin{align*} & {\mathcal{S}}_1^{\, \alpha}:PW_{p\beta}\times C([0,\tau],U)\to PW_{p\beta} \end{align*} $$
and
$$ \begin{align*} & {\mathcal{S}}_2^{\, \alpha}:PW_{p\beta}\times C([0,\tau],U)\to C([0,\tau],U) \end{align*} $$
defined by (4.3) and (4.4) are continuous.
Let |$z,\tilde{z}\in PW_{p\beta }$| and |$u,\tilde{u}\in U$|⁠. First, note that
$$ \begin{align*} \left\Vert{\mathcal{L}}(z,u)-{\mathcal{L}}(\tilde{z},\tilde{u})\right\Vert{}_\beta \leq & \left\Vert T(\tau)\big[\big(h(z_{\tau_{1}}\dots,z_{\tau_{q}})\big)(0)-\big(h(\tilde{z}_{\tau_{1}}\dots,\tilde{z}_{\tau_{q}})\big)(0)\big]\right\Vert{}_\beta\\ & + ||T(\tau)\big[g\big(0,\eta(0)-\big(h(z_{\tau_{1}}\dots,z_{\tau_{q}})\big)(0)\big)\\ & \qquad \qquad \qquad -g\big(0,\eta(0)-\big(h(\tilde{z}_{\tau_{1}}\dots,\tilde{z}_{\tau_{q}})\big)(0)\big)\big]||_\beta\\ & +\left\Vert g(\tau,z_\tau)-g(\tau,\tilde{z}_\tau)\right\Vert{}_\beta+\displaystyle\int_{0}^\tau\left\Vert T(\tau-s)\big[Ag(s,z_{s})-A(g(s,\tilde{z}_{s})\big]\right\Vert{}_\beta ds \\ & +\displaystyle\int_{0}^\tau\left\Vert T(t-s)\big[f(s,z_{s},u(s))-f(s,\tilde{z}_{s},\tilde{u}(s))\big]\right\Vert{}_\beta ds\\ & +\displaystyle\sum_{0<t_{k}<\tau}\left\Vert T(\tau-t_{k})\big[J_{k}(z(t_{k}),u(t_{k}))-J_{k}(\tilde{z}(t_{k}),\tilde{u}(t_{k}))\big]\right\Vert{}_\beta. \end{align*} $$
By definition of |$\left \Vert \cdot \right \Vert{}_\beta $| and (1.4) we have that
$$ \begin{align*} \left\Vert{\mathcal{L}}(z,u)-{\mathcal{L}}(\tilde{z},\tilde{u})\right\Vert{}_\beta \leq & \ \left\Vert T(\tau)\right\Vert\left\Vert\big(h(z_{\tau_{1}}\dots,z_{\tau_{q}})\big)(0)-\big(h(\tilde{z}_{\tau_{1}}\dots,\tilde{z}_{\tau_{q}})\big)(0)\right\Vert{}_\beta\\ & + \left\Vert T(\tau)\right\Vert||g\big(0,\eta(0)-\big(h(z_{\tau_{1}}\dots,z_{\tau_{q}})\big)(0)\big)\\ & \qquad \qquad \qquad -g\big(0,\eta(0)-\big(h(\tilde{z}_{\tau_{1}}\dots,\tilde{z}_{\tau_{q}})\big)(0)\big)||_\beta\\ & +\left\Vert g(\tau,z_\tau)-g(\tau,\tilde{z}_\tau)\right\Vert{}_\beta+\displaystyle\int_{0}^\tau\left\Vert T(\tau-s)\right\Vert\left\Vert[Ag(s,z_{s})-A(g(s,\tilde{z}_{s})\right\Vert{}_\beta ds \\ & +\displaystyle\int_{0}^\tau\left\Vert A^\beta T(t-s)\right\Vert\left\Vert f(s,z_{s},u(s))-f(s,\tilde{z}_{s},\tilde{u}(s))\right\Vert ds\\ & +\displaystyle\sum_{0<t_{k}<\tau}\left\Vert T(\tau-t_{k})\right\Vert\left\Vert J_{k}(z(t_{k}),u(t_{k}))-J_{k}(\tilde{z}(t_{k}),\tilde{u}(t_{k}))\right\Vert{}_\beta. \end{align*} $$
From (1.2), (1.3), hypothesis (H|$_{1}$|⁠)-(ii), (iii) and (H|$_{2}$|⁠)-(iv), we get
(4.5)
where
$$ \begin{align*} & \varTheta=ML_h[1+L_g]q+L_g +{\mathcal{K}}(\left\Vert z\right\Vert{}_{p\beta},\left\Vert\tilde{z}\right\Vert{}_{p\beta})\left[M\tau+M_\beta\displaystyle\frac{\tau^{1-\beta}}{1-\beta}\right]+M\displaystyle\sum_{k=1}^p d_k. \end{align*} $$
Now, consider the difference |$\varPi _{0}:=||S_{1}^{\, \alpha} (z,u)(t)- S_{1}^{\, \alpha} (\tilde{z},\tilde{u})(t)||_\beta $|⁠. Then
$$ \begin{align*} \varPi_{0} \leq & \left\Vert T(t)\big[\big(h(z_{\tau_{1}},\dots,z_{\tau_{q}})\big)(0)-\big(h(\tilde{z}_{\tau_{1}},\dots,\tilde{z}_{\tau_{q}})\big)(0)\big]\right\Vert{}_\beta\\ & + \left\Vert T(t)\big[g\big(0,\eta(0)-\big(h(z_{\tau_{1}},\dots,z_{\tau_{q}})\big)(0)\big) -g\big(0,\eta(0)-\big(h(\tilde{z}_{\tau_{1}},\dots,\tilde{z}_{\tau_{q}})\big)(0)\big)\big]\right\Vert{}_\beta \\[-3pt] & \left\Vert g(t,z_{t})-g(t,\tilde{z}_{t})\right\Vert{}_\beta + \displaystyle\int_{0}^{t}\left\Vert AT(t-s)[g(s,z_{s})-g(s,\tilde{z}_{s})]\right\Vert{}_\beta ds \\[-3pt] & +\displaystyle\int_{0}^{t}\left\Vert T(t-s)B\varGamma_\alpha[{\mathcal{L}}(z,u)-{\mathcal{L}}(\tilde{z},\tilde{u})](s)]\right\Vert{}_\beta ds\\ & +\displaystyle\int_{0}^{t}\left\Vert T(t-s)\big[f(s,z_{s},u(s))-f(s,\tilde{z}_{s},\tilde{u}(s))\big]\right\Vert{}_\beta ds\\[-3pt] & + \displaystyle\sum_{0<t_{k}<t}\left\Vert T(t-t_{k})\big[J_{k}(z(t_{k}),u(t_{k}))-J_{k}(\tilde{z}(t_{k}),\tilde{u}(t_{k}))\big]\right\Vert{}_\beta. \end{align*} $$
By definition of |$\left \Vert \cdot \right \Vert{}_\beta $| and (1.4), we get that
$$ \begin{align*} \varPi_{0} \leq & \left\Vert T(t)\right\Vert\left\Vert\big(h(z_{\tau_{1}},\dots,z_{\tau_{q}})\big)(0)-\big(h(\tilde{z}_{\tau_{1}},\dots,\tilde{z}_{\tau_{q}})\big)(0)\right\Vert{}_\beta\\ & + \left\Vert T(t)\right\Vert\left\Vert g\big(0,\eta(0)-\big(h(z_{\tau_{1}},\dots,z_{\tau_{q}})\big)(0)\big) -g\big(0,\eta(0)-\big(h(\tilde{z}_{\tau_{1}},\dots,\tilde{z}_{\tau_{q}})\big)(0)\big)\right\Vert{}_\beta\\[-3pt] & +\left\Vert g(t,z_{t})-g(t,\tilde{z}_{t})\right\Vert{}_\beta + \displaystyle\int_{0}^{t}\left\Vert A^\beta T(t-s)\right\Vert\left\Vert Ag(s,z_{s})-Ag(s,\tilde{z}_{s})\right\Vert ds\\[-3pt] & +\displaystyle\int_{0}^{t}\left\Vert T(t-s)\right\Vert\left\Vert B\varGamma_\alpha\big({\mathcal{L}}(z,u)-{\mathcal{L}}(\tilde{z},\tilde{u})\big)(s)\right\Vert{}_\beta ds\\ & +\displaystyle\int_{0}^{t}\left\Vert A^\beta T(t-s)\right\Vert\left\Vert f(s,z_{s},u(s))-f(s,\tilde{z}_{s},\tilde{u}(s))\right\Vert ds\\ & + \displaystyle\sum_{0<t_{k}<t}\left\Vert T(t-t_{k})\right\Vert\left\Vert J_{k}(z(t_{k}),u(t_{k}))-J_{k}(\tilde{z}(t_{k}),\tilde{u}(t_{k}))\right\Vert{}_\beta. \end{align*} $$
Hypotheses (H|$_{1}$|⁠)-(ii), (iii), (H|$_{2}$|⁠)-(i), (ii), (iii), (1.3) and (4.5) yield the following estimate
$$ \begin{align*} \varPi_{0} \leq &\ ML_{h}\left\Vert z-\tilde{z}\right\Vert{}_{p\beta} + ML_{g}L_{h}q\left\Vert z-\tilde{z}\right\Vert{}_{p\beta}+L_{g}\left\Vert z-\tilde{z}\right\Vert{}_{p\beta}\\ & + \displaystyle\int_{0}^{t}\displaystyle\frac{M_\beta}{(t-s)^\beta}{\mathcal{K}}(\left\Vert z_{s}\right\Vert{}_{r\beta},\left\Vert\tilde{z}\right\Vert{}_{r\beta})\left\Vert z_{s}-\tilde{z}_{s}\right\Vert{}_{r\beta}ds \\ & +\displaystyle\int_{0}^{t} M\left\Vert B\right\Vert\left\Vert\varGamma_\alpha\right\Vert\varTheta\big[\left\Vert z-\tilde{z}\right\Vert{}_{p\beta}+\left\Vert u-\tilde{u}\right\Vert\big]ds\\ & + \displaystyle\int_{0}^{t} \displaystyle\frac{M_\beta}{(t-s)^\beta}{\mathcal{K}}(\left\Vert z_{s}\right\Vert{}_{r\beta},\left\Vert\tilde{z}\right\Vert{}_{r\beta})\big[\left\Vert z_{s}-\tilde{z}_{s}\right\Vert{}_{r\beta} +\left\Vert u-\tilde{u}\right\Vert\big]ds\\ &+ M\displaystyle\sum_{0<t_{k}<t}d_{k}\big[\left\Vert z-\tilde{z}\right\Vert{}_\beta+\left\Vert u-\tilde{u}\right\Vert\big] \end{align*} $$
After computing the integral, we finally obtain
$$ \begin{align*} \varPi_{0}\leq \bigg[ML_{h}q+ & ML_{g}L_{h}q+ L_{g}+2M_\beta{\mathcal{K}}(\left\Vert z\right\Vert{}_{p\beta},\left\Vert\tilde{z}\right\Vert{}_{p\beta})\displaystyle\frac{\tau^{1-\beta}}{1-\beta}\\ +& M\tau\left\Vert B\right\Vert\left\Vert\varGamma_\alpha\right\Vert\varTheta+M\displaystyle\sum_{k=1}^{p} d_{k}\bigg]\big[\left\Vert z-\tilde{z}\right\Vert{}_{p\beta}+\left\Vert u-\tilde{u}\right\Vert\big]. \end{align*} $$
So,
$$ \begin{align*} \left\Vert{\mathcal{S}}_{1}^{\, \alpha}(z,u)-{\mathcal{S}}_{1}^{\, \alpha}(\tilde{z},\tilde{u})\right\Vert{}_{p\beta}\leq \bigg[ML_{h}q + & ML_{g}L_{h}q+ L_{g}+2M_\beta{\mathcal{K}}(\left\Vert z\right\Vert{}_{p\beta},\left\Vert\tilde{z}\right\Vert{}_{p\beta})\displaystyle\frac{\tau^{1-\beta}}{1-\beta}\\ +& M\tau\left\Vert B\right\Vert\left\Vert\varGamma_\alpha\right\Vert\varTheta+M\displaystyle\sum_{k=1}^{p} d_{k}\bigg]\big[\left\Vert z-\tilde{z}\right\Vert{}_{p\beta}+\left\Vert u-\tilde{u}\right\Vert\big]. \end{align*} $$
Analogously,
$$ \begin{align*} & \left\Vert{\mathcal{S}}_2^{\, \alpha}(z,u)-{\mathcal{S}}_2^{\, \alpha}(\tilde{z},\tilde{u})\right\Vert \leq M\left\Vert B\right\Vert\left\Vert(\alpha I+L_{\mathscr{G}})^{-1}\right\Vert \varTheta \big[\left\Vert z-\tilde{z}\right\Vert{}_{p\beta}+\left\Vert u-\tilde{u}\right\Vert\big]. \end{align*} $$
Thereby leading |${\mathcal{S}}_{1}^{\, \alpha} $| and |${\mathcal{S}}_{2}^{\, \alpha} $| are continuous and therefore |${\mathcal{S}}^{\, \alpha} $| is continuous.

 

Lemma 4.

The operator |${\mathcal{S}}^{\, \alpha} $| is compact.

 

Proof.

Let |$\mathscr{B}$| a bounded subset of |$PW_{p\beta }\times C([0,\tau ],U)$|⁠, then, for any |$(z,u)\in \mathscr{B}$|⁠, we have that |$|||(z,u)|||\leq R$| for some |$R>0$|⁠.

For any |$(z,u)\in \mathscr{B}$|⁠, we get, as a consequence of hypotheses (H|$_{1}$|⁠)-(ii), (iii) and (H|$_{2}$|⁠)-(ii), (iv) that
$$ \begin{align*} \left\Vert{\mathcal{L}}(z,u)\right\Vert{}_\beta\leq & \left\Vert z^{1}\right\Vert{}_\beta+\left\Vert T(\tau)\big[\eta(0)-\big(h(z_{\tau_{1}},\dots,z_{\tau_{q}})\big)(0)\big]\right\Vert{}_\beta\\ & +\left\Vert T(\tau)g\big(0,\eta(0)-\big(h(z_{\tau-1},\dots,z_{\tau_{q}})\big)(0)\big)\right\Vert{}_\beta\\[-3pt] & +\left\Vert g(\tau,z_\tau)\right\Vert+\displaystyle\int_{0}^\tau\left\Vert T(\tau-s)Ag(s,z_{s})\right\Vert{}_\beta ds\end{align*} $$
 
$$ \begin{align*} &+\displaystyle\int_{0}^\tau\left\Vert T(\tau-s)f(s,z_{s},u(s))\right\Vert{}_\beta ds+\displaystyle\sum_{0<t_{k}<\tau}\left\Vert T(\tau-t_{k})J_{k}(z(t_{k}),u(t_{k}))\right\Vert{}_\beta\\ \leq & \left\Vert z^{1}\right\Vert{}_\beta+\left\Vert T(\tau)\right\Vert\big[1+L_{g}\big]\big[\left\Vert\eta(0)\right\Vert{}_\beta+L_{h}q\left\Vert z\right\Vert{}_{p\beta}\big]+L_{g}\left\Vert z\right\Vert{}_{p\beta}\\ & \left[M\tau+M_\beta\displaystyle\frac{\tau^{1-\beta}}{1-\beta}\right]\psi(\left\Vert z\right\Vert{}_{p\beta}) +M\displaystyle\sum_{k=1}^{p} d_{k}\big [\left\Vert z\right\Vert{}_{p\beta}+\left\Vert u\right\Vert\big]\\ \leq & M_{1} \end{align*} $$
where
$$ \begin{align*} M_{1}= & \left\Vert z^{1}\right\Vert{}_\beta+M\big[1+L_{g}\big]\big[\left\Vert\eta(0)\right\Vert{}_\beta+L_{h} qR\big]+ \left[M\tau+M_\beta\displaystyle\frac{\tau^{1-\beta}}{1-\beta}\right]\psi(R)\\[-3pt] & + 2MR\displaystyle\sum_{k=1}^{p} d_{k}. \end{align*} $$
Also,
$$ \begin{align*} \left\Vert S_{1}^{\, \alpha}(z,u)(t)\right\Vert{}_\beta\leq & \left\Vert T(t)\right\Vert\big[\left\Vert\eta(0)\right\Vert{}_\beta +\left\Vert\big(h(z_{\tau_{1}},\dots,z_{\tau_{q}})\big)(0)\big]\right\Vert{}_\beta\\ & +\left\Vert g(0,\eta(0)-\big(h(z_{\tau_{1}},\dots,z_{\tau_{q}})\big)(0))\right\Vert{}_\beta+\left\Vert g(t,z_{t})\right\Vert{}_\beta\\ & + \displaystyle\int_{0}^{t}\left\Vert AT(t-s)g(s,z_{s})\right\Vert{}_\beta ds+\displaystyle\int_{0}^{t}\left\Vert T(t-s)B\big(\varGamma_\alpha{\mathcal{L}}(z,u)\big)(s)\right\Vert{}_\beta ds\\[-3pt] & + \displaystyle\int_{0}^{t}\left\Vert T(t-s)f(s,z_{s},u(s))\right\Vert{}_\beta ds+\displaystyle\sum_{0<t_{k}<t}\left\Vert T(t-t_{k})J_{k}(z(t_{k}),u(t_{k}))\right\Vert{}_\beta\\[-3pt] \leq & M\big[\left\Vert\eta(0)\right\Vert{}_\beta+L_{h}q\left\Vert z\right\Vert{}_{p\beta}+L_{g}(\left\Vert\eta(0)\right\Vert{}_\beta+L_{h}q\left\Vert z\right\Vert{}_{p\beta})\big]+ L_{g}\left\Vert z_{t}\right\Vert{}_{r\beta}\\ & + \displaystyle\int_{0}^{t}\left\Vert A^\beta T(t-s)\right\Vert\left\Vert Ag(s,z_{s})\right\Vert{}_\beta ds + \displaystyle\int_{0}^{t}\left\Vert T(t-s)\right\Vert\left\Vert B(\varGamma_\alpha{\mathcal{L}}(z,u))(s)\right\Vert{}_\beta ds\\[-3pt] & + \displaystyle\int_{0}^{t}\left\Vert T(t-s)\right\Vert\left\Vert f(s,z_{s},u(s))\right\Vert{}_\beta ds+\displaystyle\sum_{0<t_{k}<t}\left\Vert T(t-t_{k})\right\Vert \left\Vert J_{k}(z(t_{k}),u(t_{k}))\right\Vert{}_\beta\\[-3pt] \leq & M\big[\left\Vert\eta(0)\right\Vert{}_\beta+L_{h}q\left\Vert z\right\Vert{}_{p\beta}+L_{g}(\left\Vert\eta(0)\right\Vert{}_\beta+L_{h}q\left\Vert z\right\Vert{}_{p\beta})\big]+ L_{g}\left\Vert z\right\Vert{}_{p\beta}\\ &+ \displaystyle\int_{0}^{t}\displaystyle\frac{M_\beta}{(t-s)^\beta}\psi(\left\Vert z_{s}\right\Vert{}_{r\beta})ds+\displaystyle\int_{0}^{t} M\left\Vert B\right\Vert\left\Vert\varGamma_\alpha\right\Vert M_{1}ds +\displaystyle\int_{0}^{t} M\psi(\left\Vert z_{s}\right\Vert{}_{r\beta})ds\\ & + \displaystyle\sum_{0<t<t_{k}}Md_{k}\left\Vert z(\tau_{k})\right\Vert{}_\beta. \end{align*} $$
Therefore, |$\left \Vert S_{1}^{\, \alpha} (z,u)\right \Vert{}_{p\beta }\leq M_{2}$|⁠, where
$$ \begin{align*} M_{2}=&M\big[\left\Vert\eta(0)\right\Vert{}_\beta+L_{h}qR+L_{g}(\left\Vert\eta(0)\right\Vert{}_\beta+L_{h}qR)\big]+ L_{g}R + M_\beta\displaystyle\frac{\tau^{1-\beta}}{1-\beta}\psi(R)\\[-9pt] & + MM_{1}\left\Vert B\right\Vert\left\Vert\varGamma_\alpha\right\Vert\tau+M\psi(R)\tau+ MR\displaystyle\sum_{k=1}^{p}d_{k}, \end{align*} $$
and
$$ \begin{align*} & \left\Vert{\mathcal{S}}_2^{\, \alpha}(z,u)\right\Vert\leq \left\Vert B\right\Vert M\left\Vert(\alpha I+L_{\mathscr{G}})^{-1}\right\Vert M_1. \end{align*} $$
Hence, |${\mathcal{S}}^{\, \alpha} (\mathscr{B})$| is bounded.
Now, without loss of generality, we assume that |$0<\sigma _{1}<\sigma _{2}<\tau $|⁠. If
$$ \begin{align*} & \varPi_1:= \left\Vert{\mathcal{S}}_1^{\, \alpha}(z,u)(\sigma_2)-{\mathcal{S}}_1^{\, \alpha}(z,u)(\sigma_1)\right\Vert{}_\beta \end{align*} $$
then
$$ \begin{align*} \varPi_{1}\leq & \left\Vert T(\sigma_{2})-T(\sigma_{1})\right\Vert\big[\left\Vert\eta(0)\right\Vert{}_\beta+\left\Vert\big(h(z_{\tau_{1}},\dots,z_{\tau_{q}})\big)(0)\right\Vert{}_\beta \\ & + \left\Vert g(0,\eta(0)-\big((h(z_{\tau_{1}},\dots,z_{\tau_{q}})\big)(0))\right\Vert{}_\beta\big] + \left\Vert g(\sigma_{2},z_{\sigma_{2}})-g(\sigma_{1},z_{\sigma_{1}})\right\Vert{}_\beta \\ & + \displaystyle\int_{0}^{\sigma_{1}}\left\Vert A\big[T(\sigma_{2}-s)-T(\sigma_{1}-s)\big]g(s,z_{s})\right\Vert{}_\beta ds + \displaystyle\int_{\sigma_{1}}^{\sigma_{2}}\left\Vert AT(\sigma_{2}-s)g(s,z_{s})\right\Vert{}_\beta ds\\ & + \displaystyle\int_{0}^{\sigma_{1}}\left\Vert\big[T(\sigma_{2}-s)-T(\sigma_{1}-s)\big]B(\varGamma_\alpha{\mathcal{L}}(z,u))(s)\right\Vert{}_\beta ds \\ & + \displaystyle\int_{\sigma_{1}}^{\sigma_{2}}\left\Vert T(\sigma_{2}-s)B(\varGamma_\alpha{\mathcal{L}}(z,u))(s)\right\Vert{}_\beta ds \\ & + \displaystyle\int_{0}^{\sigma_{1}}\left\Vert\big[T(\sigma_{2}-s)-T(\sigma_{1}-s)\big]f(s,z_{s},u(s))\right\Vert{}_\beta ds\\ & + \displaystyle\int_{\sigma_{1}}^{\sigma_{2}}\left\Vert T(\sigma_{2}-s)f(s,z_{s},u(s))\right\Vert{}_\beta ds\\ & + \displaystyle\sum_{0<t_{k}<\sigma_{1}}\left\Vert\big[T(\sigma_{2}-t_{k})-T(\sigma_{1}-t_{k})\big]J_{k}(z(t_{k}),u(t_{k}))\right\Vert{}_\beta \\ & + \displaystyle\sum_{\sigma_{1}<t_{k}<\sigma_{2}}\left\Vert T(\sigma_{2}-t_{k})J_{k}(z(t_{k}),u(t_{k}))\right\Vert{}_\beta. \end{align*} $$
Let |${\mathcal{I}}_{i}$|⁠, |$i\in \{1,\dots ,6\}$| denote each of the above six integrals and |${\mathcal{E}}_{1},{\mathcal{E}}_{2}$| denote each of the above sum, respectively. Then
$$ \begin{align*} \varPi_{1}\leq & \left\Vert T(\sigma_{2})-T(\sigma_{1})\right\Vert\big[\left\Vert\eta(0)\right\Vert{}_\beta+\left\Vert\big(h(z_{\tau_{1}},\dots,z_{\tau_{q}})\big)(0)\right\Vert{}_\beta \\ & + \left\Vert g\big(0,\eta(0)-\big((h(z_{\tau_{1}},\dots,z_{\tau_{q}})\big)(0)\big)\right\Vert{}_\beta\big] +\displaystyle\sum_{i=1}^{6}{\mathcal{I}}_{i}+{\mathcal{E}}_{1}+{\mathcal{E}}_{2} \end{align*} $$
Let |$\varepsilon>0$| be small sufficient, by the semigroup property, we get that
$$ \begin{align*} & T(\sigma_2-s)-T(\sigma_1-s)=\big[T(\varepsilon)-T(\sigma_2-\sigma_1+\varepsilon)\big]T(\sigma_1-s-\varepsilon). \end{align*} $$
Next, we are going to manipulate each integral |${\mathcal{I}}_{i}$| and each sum |${\mathcal{E}}_{i}$|⁠.
$$ \begin{align*} {\mathcal{I}}_{1}= & \displaystyle\int_{0}^{\sigma_{1}-\varepsilon}\left\Vert A\big[T(\sigma_{2}-s)-T(\sigma_{1}-s)\big]g(s,z_{s})\right\Vert{}_\beta ds\\[4pt] &+ \displaystyle\int_{\sigma_{1}-\varepsilon}^{\sigma_{1}}\left\Vert A\big[T(\sigma_{2}-s)-T(\sigma_{1}-s)\big]g(s,z_{s})\right\Vert{}_\beta ds\\[4pt] \leq & \left\Vert T(\varepsilon)-T(\sigma_{2}-\sigma_{1}+\varepsilon)\right\Vert\displaystyle\int_{0}^{\sigma_{1}-\varepsilon}\left\Vert A^\beta T(\sigma_{1}-s-\varepsilon)\right\Vert\left\Vert Ag(s,z_{s})\right\Vert ds\\[4pt] & + \displaystyle\int_{\sigma_{1}-\varepsilon}^{\sigma_{1}}\left\Vert A^\beta T(\sigma_{2}-s)-A^\beta T(\sigma_{1}-s)\right\Vert\left\Vert Ag(s,z_{s})\right\Vert ds \\[4pt] \leq & \left\Vert T(\varepsilon)-T(\sigma_{2}-\sigma_{1}+\varepsilon)\right\Vert\displaystyle\int_{0}^{\sigma_{1}-\varepsilon}\displaystyle\frac{M_\beta}{(\sigma_{1}-s-\varepsilon)^\beta}\left\Vert\psi(z)\right\Vert{}_{p\beta}ds \\[4pt] & + \displaystyle\int_{\sigma_{1}-\varepsilon}^{\sigma_{1}}M_\beta\left[\displaystyle\frac{1}{(\sigma_{2}-s)^\beta}+\displaystyle\frac{1}{(\sigma_{1}-s)^\beta}\right]\psi(\left\Vert z\right\Vert{}_{p\beta})ds\\ \leq & \left\Vert T(\varepsilon)-T(\sigma_{2}-\sigma_{1}+\varepsilon)\right\Vert M_\beta\psi(R)\displaystyle\frac{(\sigma_{1}-\varepsilon)^{1-\beta}}{1-\beta}\\ & + \displaystyle\frac{M_\beta\psi(R)}{1-\beta}\big[(\sigma_{2}-\sigma_{1}+\varepsilon)^{1-\beta}-(\sigma_{2}-\sigma_{1})^{1-\beta}+\varepsilon^{1-\beta}\big],\\{\mathcal{I}}_{2}\leq & \displaystyle\int_{\sigma_{1}}^{\sigma_{2}}\left\Vert A^\beta T(\sigma_{2}-s)\right\Vert\left\Vert Ag(s,z_{s})\right\Vert ds \leq \displaystyle\int_{\sigma_{1}}^{\sigma_{2}}\displaystyle\frac{M_\beta}{(\sigma_{2}-s)^\beta}\psi(\left\Vert z\right\Vert{}_{p\beta})ds\\ \leq & \displaystyle\frac{M_\beta}{1-\beta}\psi(R)(\sigma_{2}-\sigma_{1})^{1-\beta}, \\{\mathcal{I}}_{3}= & \displaystyle\int_{0}^{\sigma_{1}-\varepsilon}\left\Vert\big[T(\sigma_{2}-s)-T(\sigma_{1}-s)\big]B(\varGamma_\alpha{\mathcal{L}}(z,u))(s)\right\Vert{}_\beta ds \\ & + \displaystyle\int_{\sigma_{1}-\varepsilon}^{\sigma_{1}}\left\Vert\big[T(\sigma_{2}-s)-T(\sigma_{1}-s)\big]B(\varGamma_\alpha{\mathcal{L}}(z,u))(s)\right\Vert{}_\beta ds \\ \leq & \left\Vert T(\varepsilon)-T(\sigma_{2}-\sigma_{1}+\varepsilon)\right\Vert\displaystyle\int_{0}^{\sigma_{1}-\varepsilon}\left\Vert T(\sigma_{1}-s-\varepsilon)\right\Vert\left\Vert B(\varGamma_\alpha{\mathcal{L}}(z,u))(s)\right\Vert{}_\beta\\ & + \displaystyle\int_{\sigma_{1}-\varepsilon}^{\sigma_{1}}\big[\left\Vert T(\sigma_{2}-s)\right\Vert+\left\Vert T(\sigma_{1}-s)\right\Vert\big]\left\Vert B\varGamma_\alpha({\mathcal{L}}(z,u))(s)\right\Vert{}_\beta ds\\ \leq & \left\Vert T(\varepsilon)-T(\sigma_{2}-\sigma_{1}+\varepsilon)\right\Vert M\left\Vert B\right\Vert\left\Vert\varGamma_\alpha\right\Vert M_{1}(\sigma_{1}-\varepsilon)+2M\left\Vert B\right\Vert\left\Vert\varGamma_\alpha\right\Vert M_{1}\varepsilon, \\{\mathcal{I}}_{4}\leq & \displaystyle\int_{\sigma_{1}}^{\sigma_{2}}\left\Vert T(\sigma_{2}-s)\right\Vert\left\Vert B(\varGamma_\alpha{\mathcal{L}}(z,u))(s)\right\Vert{}_\beta ds\\ \leq &\ M\left\Vert B\right\Vert\left\Vert\varGamma_\alpha\right\Vert M_{1}(\sigma_{2}-\sigma_{1}), \end{align*} $$
 
$$ \begin{align*} {\mathcal{I}}_{5}= & \displaystyle\int_{0}^{\sigma_{1}-\varepsilon}\left\Vert\big[T(\sigma_{2}-s)-T(\sigma_{1}-s)\big]f(s,z_{s},u(s))\right\Vert ds\\ & + \displaystyle\int_{\sigma_{1}-\varepsilon}^{\sigma_{1}}\left\Vert\big[T(\sigma_{2}-s)-T(\sigma_{1}-s)\big]f(s,z_{s},u(s))\right\Vert ds\\ \leq & \left\Vert T(\varepsilon)-T(\sigma_{2}-\sigma_{1}+\varepsilon)\right\Vert\displaystyle\int_{0}^{\sigma_{1}-\varepsilon}\left\Vert A^\beta T(\sigma_{1}-s-\varepsilon)\right\Vert\left\Vert f(s,z_{s},u(s))\right\Vert ds\\ & + \displaystyle\int_{\sigma_{1}-\varepsilon}^{\sigma_{1}}\left\Vert A^\beta T(\sigma_{2}-s)-A^\beta T(\sigma_{1}-s)\right\Vert\left\Vert f(s,z_{s},u(s))\right\Vert ds \\ \leq & \left\Vert T(\varepsilon)-T(\sigma_{2}-\sigma_{1}+\varepsilon)\right\Vert\displaystyle\int_{0}^{\sigma_{1}-\varepsilon}\displaystyle\frac{M_\beta}{(\sigma_{1}-s-\varepsilon)^{\beta}}\psi(\left\Vert z\right\Vert{}_{p\beta})ds\\[4pt] & + \displaystyle\int_{\sigma_{1}-\varepsilon}^{\sigma_{1}}M_\beta\left[\displaystyle\frac{1}{(\sigma_{2}-s)^\beta}+\displaystyle\frac{1}{(\sigma_{1}-s)^\beta}\right]\psi(\left\Vert z\right\Vert{}_{p\beta})ds\\[4pt] \leq & \left\Vert T(\varepsilon)-T(\sigma_{2}-\sigma_{1}+\varepsilon)\right\Vert M_\beta\psi(R)\displaystyle\frac{(\sigma_{1}-\varepsilon)^{1-\beta}}{1-\beta}\\ & + \displaystyle\frac{M_\beta\psi(R)}{1-\beta}\big[(\sigma_{2}-\sigma_{1}+\varepsilon)^{1-\beta}-(\sigma_{2}-\sigma_{1})^{1-\beta}+\varepsilon^{1-\beta}\big], \\[4pt] {\mathcal{I}}_{6}\leq & \displaystyle\int_{\sigma_{1}}^{\sigma_{2}}\left\Vert A^\beta T(\sigma_{2}-s)\right\Vert\left\Vert f(s,z_{s},u(s))\right\Vert ds \leq \displaystyle\int_{\sigma_{1}}^{\sigma_{2}}\displaystyle\frac{M_\beta}{(\sigma_{2}-s)^\beta}\psi(\left\Vert z\right\Vert{}_{p\beta})ds\\ \leq & \displaystyle\frac{M_\beta}{1-\beta}\psi(R)(\sigma_{2}-\sigma_{1})^{1-\beta}, \\{\mathcal{E}}_{1}= & \displaystyle\sum_{0<t_{k}<\sigma_{1}-\varepsilon}\left\Vert\big[T(\sigma_{2}-t_{k})-T(\sigma_{1}-t_{k})]J_{k}(z(t_{k}),u(t_{k}))\right\Vert{}_\beta + \\[2pt] & +\displaystyle\sum_{\sigma_{1}-\varepsilon<t_{k}<\sigma_{1}}\left\Vert\big[T(\sigma_{2}-t_{k})-T(\sigma_{1}-t_{k})]J_{k}(z(t_{k}),u(t_{k}))\right\Vert{}_\beta\\ \leq & \left\Vert T(\varepsilon)-T(\sigma_{2}-\sigma_{1}+\varepsilon)\right\Vert\displaystyle\sum_{0<t_{k}<\sigma_{1}-\varepsilon}\left\Vert T(\sigma_{1}-t_{k}-\varepsilon)\right\Vert\left\Vert J_{k}(z(t_{k})u(t_{k}))\right\Vert{}_\beta\\ & + \displaystyle\sum_{\sigma_{1}-\varepsilon<t_{k}<\sigma-1}\left\Vert T(\sigma_{2}-t-k)-T(\sigma_{1}-t_{k})\right\Vert\left\Vert J_{k}(z(t_{k}),u(t_{k}))\right\Vert{}_\beta\\[4pt] \leq & \left\Vert T(\varepsilon)-T(\sigma_{2}-\sigma_{1}+\varepsilon)\right\Vert 2MR\displaystyle\sum_{0<t_{k}<\sigma_{1}-\varepsilon}d_{k} + 2MR\displaystyle\sum_{\sigma_{1}-\varepsilon<t_{k}<\sigma_{1}}d_{k}, \end{align*} $$
and
$$ \begin{align*} {\mathcal{E}}_{2} \leq & \displaystyle\sum_{\sigma_{1}<t_{k}<\sigma_{2}}\left\Vert T(\sigma_{2}-\sigma_{1})\right\Vert\left\Vert J_{k}(z(t_{k}),u(t_{k}))\right\Vert{}_\beta \leq 2MR\displaystyle\sum_{\sigma_{1}<t_{k}<\sigma_{2}}d_{k}. \end{align*} $$
Therefore, the following estimate holds:
$$ \begin{align*} \varPi_{1}\leq & \left\Vert T(\sigma_{2})-T(\sigma_{1})\right\Vert\big[\left\Vert\eta(0)\right\Vert{}_\beta+L_{h}qR+L_{g}\psi(\left\Vert\eta\right\Vert+L_{h}qR)\big] \\[4pt] & +\left\Vert T(\varepsilon)-T(\sigma_{2}-\sigma_{1}+\varepsilon)\right\Vert2M_\beta\psi(R)\displaystyle\frac{(\sigma_{1}-\varepsilon)^{1-\beta}}{1-\beta}\\[4pt] & + 2\displaystyle\frac{M_\beta\psi(R)}{1-\beta}\big[(\sigma_{2}-\sigma_{1}+\varepsilon)^{1-\beta}-(\sigma_{2}-\sigma_{1})^{1-\beta}+\varepsilon^{1-\beta}\big]\\[4pt] & +2\displaystyle\frac{M_\beta}{1-\beta}\psi(R)(\sigma_{2}-\sigma_{1})^{1-\beta} \\ & + \left\Vert T(\varepsilon)-T(\sigma_{2}-\sigma_{1}+\varepsilon)\right\Vert M\left\Vert B\right\Vert\left\Vert\varGamma_\alpha\right\Vert M_{1}(\sigma_{1}-\varepsilon)+2M\left\Vert B\right\Vert\left\Vert\varGamma_\alpha\right\Vert M_{1}\varepsilon\\[4pt] & + \left\Vert T(\varepsilon)-T(\sigma_{2}-\sigma_{1}+\varepsilon)\right\Vert 2MR\displaystyle\sum_{0<t_{k}<\sigma_{1}-\varepsilon}d_{k} + 2MR\displaystyle\sum_{\sigma_{1}-\varepsilon<t_{k}<\sigma_{1}}d_{k}\\[4pt] & + 2MR\displaystyle\sum_{\sigma_{1}<t_{k}<\sigma_{2}}d_{k}. \end{align*} $$
Also,
$$ \begin{align*} & \left\Vert{\mathcal{S}}_2^{\, \alpha}(z,u)(\sigma_2)-{\mathcal{S}}_2^{\, \alpha}(z,u)(\sigma_1)\right\Vert\leq \left\Vert B^*\right\Vert\left\Vert T^*(\tau-\sigma_2)-T^*(\tau-\sigma_1)\right\Vert \left\Vert(\alpha I+L_{\mathscr{G}})^{-1}\right\Vert M_1. \end{align*} $$

Since |$T(t)$| is a compact opertaror for |$t>0$|⁠, |$T(t)$| is a uniformly continuous semigroup away from zero, which implies that |$\left \Vert{\mathcal{S}}^{\, \alpha} (z,u)(\sigma _{2})-{\mathcal{S}}^{\, \alpha} (z,u)(\sigma _{1})\right \Vert{}_\beta $| goes to zero uniformly on |$(z,u)$| as |$\sigma _{2}-\sigma _{1}\to 0$|⁠, and therefore |${\mathcal{S}}^{\, \alpha} (\mathscr{B})$| is equicontinuous.

Consequently, is we take a sequence |$\{\varphi _{j}:j=1,2,\dots \}$| on |${\mathcal{S}}^{\, \alpha} (\mathscr{B})$|⁠, this sequence is uniformly bounded and equicontinuous on the interval |$[-r,t_{1}]$| and, by the Arzelà theorem, there is a sub-sequence |$\{\varphi _{j}^{1}:j=1,2\dots \}$| of |$\{\varphi _{j}:j=1,2,\dots \}$|⁠, which is uniformly convergent on |$[-r,t_{1}]$|⁠.

Consider the sequence |$\{\varphi _{j}^{1}:j=1,2\dots \}$| on the interval |$(t_{1},t_{2}]$|⁠. On this interval, the sequence |$\{\varphi _{j}^{1}:j=1,2\dots \}$| is uniformly bounded and equicontinuous, and for the same reason, it has a sub-sequence |$\{\varphi _{j}^{2}:j=1,2\dots \}$| uniformly convergent on |$[-r,t_{2}]$|⁠.

Continuing this process for the intervals |$(t_{2},t_{3}],\ (t_{3},t_{4}], \dots , (t_{p},\tau ]$|⁠, we see that the sequence |$\{\varphi _{j}^{p+1}:j=1,2\dots \}$| converges uniformly on the interval |$[-r,\tau ]$|⁠. This means that |$\overline{{\mathcal{S}}^{\, \alpha} (\mathscr{B})}$| is compact, which implies the operator |${\mathcal{S}}^{\, \alpha} $| is compact.

 

Lemma 5.
$$ \begin{align*} & \displaystyle\lim_{{\substack{\left\Vert z\right\Vert{}_{p\beta}\to \infty\\\left\Vert u\right\Vert\to \infty}}}\displaystyle\frac{|||{\mathcal{S}}^{\, \alpha}(z,u)|||}{|||(z,u)|||}=0. \end{align*} $$

 

Proof.
Let us consider the following estimates:
$$ \begin{align*} \left\Vert{\mathcal{L}}(z,u)\right\Vert{}_\beta\leq &\ M_{3}+M\big[1+L_{g}\big]\left[\displaystyle\sum_{i=1}^{q} e_{i}\left\Vert z\right\Vert{}^{\gamma_{i}}_\beta +e_{0}\right]+ r_{1}\left\Vert z\right\Vert{}_{p\beta}^{\gamma_{0}}+r_{0} \\[4pt] &+M_\beta\displaystyle\frac{\tau^{1-\beta}}{1-\beta}\big[l_{1}\left\Vert z\right\Vert{}_{p\beta}^{\nu_{0}}+l_{0}\big] + M_\beta\displaystyle\frac{\tau^{1-\beta}}{1-\beta}\big[a_{0}\left\Vert z\right\Vert{}^{\alpha_{0}}_{p\beta}+b_{0}\left\Vert u\right\Vert{}^{\beta_{0}}+c_{0}\big]\\[4pt] & + M\displaystyle\sum_{0<t_{k}<\tau}\big[a_{k}\left\Vert z\right\Vert{}^{\alpha_{k}}_{p\beta}+b_{k}\left\Vert u\right\Vert{}^{\beta_{k}}+c_{k}\big] \end{align*} $$
where |$M_{3}=\left \Vert z^{1}\right \Vert{}_\beta +M\big [1+L_{g}\big ]\left \Vert \eta (0)\right \Vert{}_\beta $|⁠.
$$ \begin{align*} \left\Vert{\mathcal{S}}_{1}^{\, \alpha}(z,u)\right\Vert{}_{p\beta}\leq &\ M\big[1+L_{g}\big]\left\Vert\eta(0)\right\Vert{}_\beta+M_{3}M_{4}\\[4pt] & + M\big[1+L_{g}\big]\big[1+M_{4}\big]\left[\displaystyle\sum_{i=1}^{q} e_{i}\left\Vert z\right\Vert{}_{p\beta}^{\gamma_{i}}+e_{0}\right] + \big[1+M_{4}\big]\big[r_{1}\left\Vert z\right\Vert{}_{p\beta}^{\gamma_{0}}+r_{0}\big]\\[4pt] & + M_\beta\displaystyle\frac{\tau^{1-\beta}}{1-\beta}\big[1+M_{4}\big]\big[l_{1}\left\Vert z\right\Vert{}_{p\beta}^{\nu_{0}}+l_{0}\big]\\[4pt] & +M_\beta\displaystyle\frac{\tau^{1-\beta}}{1-\beta} \big[1+M_{4}\big]\big[a_{0}\left\Vert z\right\Vert{}_{p\beta}^{\alpha_{0}}+b_{0}\left\Vert u\right\Vert{}^{\beta_{k}}+c_{0}\big]\\[4pt] & + M\big[1+M_{4}\big]\displaystyle\sum_{0<\tau_{k}<\tau}\big[a_{k}\left\Vert z\right\Vert{}_{p\beta}^{\alpha_{k}}+b_{k}\left\Vert u\right\Vert{}^{\beta_{k}}+c_{k}\big], \end{align*} $$
where |$M_{4}=M\tau \left \Vert B\right \Vert \left \Vert \varGamma _\alpha \right \Vert $|⁠, and
$$ \begin{align*} \left\Vert{\mathcal{S}}_{2}^{\, \alpha}(z,u)\right\Vert\leq \left\Vert B\right\Vert & \left\Vert M\right\Vert\left\Vert(\alpha I+L_{\mathscr{G}})^{-1}\right\Vert\\[4pt] &\bigg\{ M_{3}+M\big[1+L_{g}\big]\left[\displaystyle\sum_{i=1}^{q} e_{i}\left\Vert z\right\Vert{}^{\gamma_{i}}_\beta +e_{0}\right]+ r_{1}\left\Vert z\right\Vert{}_{p\beta}^{\gamma_{0}}+r_{0} \\[4pt] &+M_\beta\displaystyle\frac{\tau^{1-\beta}}{1-\beta}\big[l_{1}\left\Vert z\right\Vert{}_{p\beta}^{\nu_{0}}+l_{0}\big] + M_\beta\displaystyle\frac{\tau^{1-\beta}}{1-\beta}\big[a_{0}\left\Vert z\right\Vert{}^{\alpha_{0}}_{p\beta}+b_{0}\left\Vert u\right\Vert{}^{\beta_{0}}+c_{0}\big]\\[4pt] & + M\displaystyle\sum_{0<t_{k}<\tau}\big[a_{k}\left\Vert z\right\Vert{}^{\alpha_{k}}_{p\beta}+b_{k}\left\Vert u\right\Vert{}^{\beta_{k}}+c_{k}\big]\bigg\} \end{align*} $$
Since
$$ \begin{align*} \displaystyle\frac{\left\Vert{\mathcal{S}}_{1}^{\, \alpha}(z,u)\right\Vert{}_{p\beta}}{|||(z,u)|||} \leq & \displaystyle\frac{M\big[1+L_{g}\big]\left\Vert\eta(0)\right\Vert{}_\beta}{|||(z,u)|||}+\displaystyle\frac{M_{3}M_{4}}{|||(z,u)|||}\\[4pt] & + M\big[1+L_{g}\big]\big[1+M_{4}\big]\left[\displaystyle\sum_{i=1}^{q} e_{i}\left\Vert z\right\Vert{}_{p\beta}^{\gamma_{i}-1}+\displaystyle\frac{e_{0}}{|||(z,u)|||}\right]\\[4pt] & + \big[1+M_{4}\big]\left[r_{1}\left\Vert z\right\Vert{}_{p\beta}^{\gamma_{0}-1}+\displaystyle\frac{r_{0}}{|||(z,u)|||}\right]\\[4pt] & + M_\beta\displaystyle\frac{\tau^{1-\beta}}{1-\beta}\big[1+M_{4}\big]\left[l_{1}\left\Vert z\right\Vert{}_{p\beta}^{\nu_{0}-1}+\displaystyle\frac{l_{0}}{|||(z,u)|||}\right]\\ & +M_\beta\displaystyle\frac{\tau^{1-\beta}}{1-\beta} \big[1+M_{4}\big]\left[a_{0}\left\Vert z\right\Vert{}_{p\beta}^{\alpha_{0}-1}+b_{0}\left\Vert u\right\Vert{}^{\beta_{k}-1}+\displaystyle\frac{c_{0}}{|||(z,u)|||}\right]\\ & + M\big[1+M_{4}\big]\displaystyle\sum_{0<\tau_{k}<\tau}\left[a_{k}\left\Vert z\right\Vert{}_{p\beta}^{\alpha_{k}-1}+b_{k}\left\Vert u\right\Vert{}^{\beta_{k}-1}+\displaystyle\frac{c_{k}}{|||(z,u)|||}\right] \end{align*} $$
and
$$ \begin{align*} \displaystyle\frac{\left\Vert{\mathcal{S}}_{2}^{\, \alpha}(z,u)\right\Vert}{|||(z,u)|||}\leq & \left\Vert B\right\Vert\left\Vert M\right\Vert\left\Vert(\alpha I+L_{\mathscr{G}})^{-1}\right\Vert\\ &\bigg\{\displaystyle\frac{M_{3}}{|||(z,u)|||}+M\big[1+L_{g}\big]\left[\displaystyle\sum_{i=1}^{q} e_{i}\left\Vert z\right\Vert{}^{\gamma_{i}-1}_\beta +\displaystyle\frac{e_{0}}{|||(z,u)|||}\right]\\ & + r_{1}\left\Vert z\right\Vert{}_{p\beta}^{\gamma_{0}-1}+\displaystyle\frac{r_{0}}{|||(z,u)|||} +M_\beta\displaystyle\frac{\tau^{1-\beta}}{1-\beta}\left[l_{1}\left\Vert z\right\Vert{}_{p\beta}^{\nu_{0}-1}+\displaystyle\frac{l_{0}}{|||(z,u)|||}\right]\\ & + M_\beta\displaystyle\frac{\tau^{1-\beta}}{1-\beta}\left[a_{0}\left\Vert z\right\Vert{}^{\alpha_{0}-1}_{p\beta}+b_{0}\left\Vert u\right\Vert{}^{\beta_{0}-1}+\displaystyle\frac{c_{0}}{|||(z,u)|||}\right]\\ & + M\displaystyle\sum_{0<t_{k}<\tau}\left[a_{k}\left\Vert z\right\Vert{}^{\alpha_{k}-1}_{p\beta}+b_{k}\left\Vert u\right\Vert{}^{\beta_{k}-1}+\displaystyle\frac{c_{k}}{|||(z,u)|||}\right]\bigg\} \end{align*} $$
Then
$$ \begin{align*} & \displaystyle\lim_{{\substack{\left\Vert z\right\Vert{}_{p\beta}\to \infty\\\left\Vert u\right\Vert\to \infty}}}\displaystyle\frac{|||{\mathcal{S}}^{\, \alpha}(z,u)|||}{|||(z,u)|||}= \displaystyle\lim_{{\substack{\left\Vert z\right\Vert{}_{p\beta}\to \infty\\\left\Vert u\right\Vert\to \infty}}}\displaystyle\frac{\left\Vert{\mathcal{S}}_1^{\, \alpha}(z,u)\right\Vert{}_{p\beta}}{|||(z,u)|||}+\displaystyle\lim_{{\substack{\left\Vert z\right\Vert{}_{p\beta}\to \infty\\\left\Vert u\right\Vert\to \infty}}}\displaystyle\frac{\left\Vert{\mathcal{S}}_2^{\, \alpha}(z,u)\right\Vert}{|||(z,u)|||}=0. \end{align*} $$

Next theorem is the main result of this paper.

 

Theorem 3.
The nonlinear system (3.1) is approximately controllable on |$[0,\tau ]$|⁠. Moreover, a sequence of controls steering the system (1.1) from initial state |$\eta $| to an |$\varepsilon $|-neighborhood of the final state |$z^{1}$| at time |$\tau>0$| is given by
$$ \begin{align*} & u_\alpha(t)=B^*T^*(\tau-t)(\alpha I+L_{\mathscr{G}})^{-1}{\mathcal{L}}(z^{\, \alpha},u_\alpha), \quad t\in [0,\tau], \end{align*} $$
and the error of this approximation |$\mathscr{E}_\alpha z$| is given by
$$ \begin{align*} & \mathscr{E}_\alpha z= \alpha(\alpha I+L_{\mathscr{G}})^{-1}{\mathcal{L}}(z^{\, \alpha},u_\alpha), \end{align*} $$
with
$$ \begin{align*} z^\alpha(t)= & T(t)\big[\eta(0)-\big(h(z_{\tau_{1}},\dots,z_{\tau_{q}})\big)(0)-g\big(0,\eta(0)-(h(z_{\tau_{1}},\dots,z_{\tau_{q}})(0)\big)\big]\\ & + \ g(t,z^\alpha_{t}) -\displaystyle\int_{0}^{t}AT(t-s)g(s,z^\alpha_{s})ds+\displaystyle\int_{0}^{t}T(t-s)Bu_\alpha(s)ds\\ &+ \displaystyle\int_{0}^{t}T(t-s)f(s,z^\alpha_{s},u_\alpha(s))ds +\displaystyle\sum_{0<t_{k}<t}T(t-t_{k})J_{k}(z^\alpha(t_{k}),u_\alpha(t_{k})), \ t\in [0,\tau],\\ z^\alpha(t) & +\big(h(z_{\tau_{1}},\dots,z_{\tau_{q}})\big)(t)=\eta(t), \ t\in [-r,0]. \end{align*} $$

 

Proof.
Let us consider the operator defined by (4.2). By lemma 5, for a fixed |$0<\rho <1$|⁠, there exists |$R>0$| big enough such that
$$ \begin{align*} & |||{\mathcal{S}}^{\, \alpha}(z,u)|||\leq \rho|||(z,u)||| \quad \mbox{with} \quad |||(z,u)|||=R. \end{align*} $$
Hence, if we denote by |$B_{_{R}}(0)$| the ball of center zero and radius |$R$|⁠, we get that |${\mathcal{S}}^{\, \alpha} (\partial B_{_{R}}(0))\subset B_{_{R}}(0)$|⁠. Since |${\mathcal{S}}^{\, \alpha} $| is compact and maps the sphere |$\partial B_{_{R}}(0)$| into the interior of the ball |$B_{_{R}}(0)$|⁠, we can apply Rothe’s fixed point Theorem 2 to ensure the existence of a point |$(z^\alpha ,u_\alpha )\in B_{_{{\mathbb{R}}}}(0)\subset C_{\mathrm{rd}}([0,\tau ]_{_{{\mathbb{T}}}},Z)\times C_{\mathrm{rd}}([0,\tau ]_{_{{\mathbb{T}}}},U)$| such that
$$ \begin{align*} & (z^\alpha,u_\alpha)={\mathcal{S}}^{\, \alpha}(z^\alpha,u_\alpha). \end{align*} $$
We claim that the sequence |$\{(z^\alpha ,u_\alpha )\}_{_{\alpha \in (0,1]}}$| is bounded. In fact, suppose that |$\{(z^\alpha ,u_\alpha )\}_{_{\alpha \in (0,1]}}$| is unbounded, then there exists a subsequence |$\{(z^{\alpha _{n}},u_ {\alpha _{n}})\}_{_{\alpha _{_{n}}\in (0,1]}}\subset \{(z^\alpha ,u_\alpha )\}_{_{\alpha \in (0,1]}}$| such that
$$ \begin{align*} & \displaystyle\lim_{n\to\infty}|||(z^{\alpha_n},u_{\alpha_n})|||=\infty. \end{align*} $$
On the other hand, from lemma 5, for |$\alpha \in (0,1]$|⁠, we get
$$ \begin{align*} & \displaystyle\lim_{n\to\infty}\displaystyle\frac{|||{\mathcal{S}}^{\, \alpha}(z^{\alpha_n},u_{\alpha_n})|||}{|||(z^{\alpha_n},u_{\alpha_n})|||}=0. \end{align*} $$
Particularly, we have the following situation:
$$ \begin{align*} & \begin{matrix} \displaystyle\frac{|||{\mathcal{S}}^{\alpha_1}(z^{\alpha_1},u_{\alpha_1})|||}{|||(z^{\alpha_1},u_{\alpha_1})|||} & \displaystyle\frac{|||{\mathcal{S}}^{\alpha_1}(z^{\alpha_2},u_{\alpha_2})|||}{|||(z^{\alpha_2},u_{\alpha_2})|||} & \dots & \displaystyle\frac{|||{\mathcal{S}}^{\alpha_1}(z^{\alpha_n},u_{\alpha_n})|||}{|||(z^{\alpha_n},u_{\alpha_n})|||}\to 0.\\\\ \displaystyle\frac{|||{\mathcal{S}}^{\alpha_2}(z^{\alpha_1},u_{\alpha_1})|||}{|||(z^{\alpha_1},u_{\alpha_1})|||} & \displaystyle\frac{|||{\mathcal{S}}^{\alpha_2}(z^{\alpha_2},u_{\alpha_2})|||}{|||(z^{\alpha_2},u_{\alpha_2})|||} & \dots & \displaystyle\frac{|||{\mathcal{S}}^{\alpha_2}(z^{\alpha_n},u_{\alpha_n})|||}{|||(z^{\alpha_n},u_{\alpha_n})|||} \to 0. \\\\ \vdots & \vdots & \vdots & \vdots \\ \\ \displaystyle\frac{|||{\mathcal{S}}^{\alpha_k}(z^{\alpha_1},u_{\alpha_1})|||}{|||(z^{\alpha_1},u_{\alpha_1})|||} & \displaystyle\frac{|||{\mathcal{S}}^{\alpha_k}(z^{\alpha_2},u_{\alpha_2})|||}{|||(z^{\alpha_2},u_{\alpha_2})|||} & \dots & \displaystyle\frac{|||{\mathcal{S}}^{\alpha_k}(z^{\alpha_n},u_{\alpha_n})|||}{|||(z^{\alpha_n},u_{\alpha_n})|||} \to 0. \end{matrix} \end{align*} $$
Now, applying Cantor’s diagonalization process, we obtain that
$$ \begin{align*} & \displaystyle\lim_{n\to\infty}\displaystyle\frac{|||{\mathcal{S}}^{\alpha_n}(z^{\alpha_n},u_{\alpha_n})|||}{|||(z^{\alpha_n},u_{\alpha_n})|||}=0. \end{align*} $$
but
$$ \begin{align*} & \displaystyle\frac{|||{\mathcal{S}}^{\alpha_n}(z^{\alpha_n},u_{\alpha_n})|||}{|||(z^{\alpha_n},u_{\alpha_n})|||}=1, \end{align*} $$
which is a contradiction. Then, the claim is true and hence there exists |$\gamma>0$| such that
$$ \begin{align*} & |||(z^\alpha,u_\alpha)|||\leq \gamma. \end{align*} $$
Therefore, without loss of generality, we can assume that the sequence |${\mathcal{L}}(z^\alpha ,u_\alpha )$| converges to |$y\in Z^\beta $|⁠. So, if
$$ \begin{align*} & u_\alpha=\varGamma_\alpha{\mathcal{L}}(z^\alpha,u_\alpha)=\mathscr{G}^*(\alpha I+L_{\mathscr{G}})^{-1}{\mathcal{L}}(z^\alpha,u_\alpha), \end{align*} $$
then
$$ \begin{align*} \mathscr{G} u_\alpha= & L_{\mathscr{G}}(\alpha I+L_{\mathscr{G}})^{-1}{\mathcal{L}}(z^\alpha,u_\alpha)\\ = & (\alpha I+L_{\mathscr{G}}-\alpha I)(\alpha I+L_{\mathscr{G}})^{-1}{\mathcal{L}}(z^\alpha,u_\alpha)\\ = & {\mathcal{L}}(z^\alpha,u_\alpha)-\alpha(\alpha I+L_{\mathscr{G}})^{-1}{\mathcal{L}}(z^\alpha,u_\alpha). \end{align*} $$
Hence
$$ \begin{align*} & \mathscr{G} u_\alpha-{\mathcal{L}}(z^\alpha,u_\alpha)=-\alpha(\alpha I+L_{\mathscr{G}})^{-1}{\mathcal{L}}(z^\alpha,u_\alpha). \end{align*} $$
To conclude the proof of this theorem, it is enough to prove that
$$ \begin{align*} & \displaystyle\lim_{\alpha\to 0}\{-\alpha(\alpha I+L_{\mathscr{G}})^{-1}\}{\mathcal{L}}(z^\alpha,u_\alpha)=0. \end{align*} $$
From Theorem 1-(e), we get
$$ \begin{align*} & \displaystyle\lim_{\alpha\to 0}\alpha(\alpha I+L_{\mathscr{G}})^{-1}{\mathcal{L}}(z^\alpha,u_\alpha)= \displaystyle\lim_{\alpha\to 0}\alpha(\alpha I+L_{\mathscr{G}})^{-1}({\mathcal{L}}(z^\alpha,u_\alpha)-y) \end{align*} $$
On the other hand, by Theorem 1-(f),
$$ \begin{align*} & \left\Vert\alpha(\alpha I+L_{\mathscr{G}})^{-1}({\mathcal{L}}(z^\alpha,u_\alpha)-y)\right\Vert\leq \left\Vert{\mathcal{L}}(z^\alpha,u_\alpha)-y\right\Vert. \end{align*} $$
Therefore, since |${\mathcal{L}}(z^\alpha ,u_\alpha )$| converges to |$y$|⁠, we get that
$$ \begin{align*} & \displaystyle\lim_{\alpha\to 0}\{\alpha(\alpha I+L_{\mathscr{G}})^{-1}({\mathcal{L}}(z^\alpha,u_\alpha)-y)=0. \end{align*} $$
So
$$ \begin{align*} & \displaystyle\lim_{\alpha\to 0}(\mathscr{G} u_\alpha-{\mathcal{L}}(z^\alpha,u_\alpha))=\displaystyle\lim_{\alpha\to 0}\{-\alpha(\alpha I+L_{\mathscr{G}})^{-1}\}{\mathcal{L}}(z^\alpha,u_\alpha)=0 \end{align*} $$
Therefore
$$ \begin{align*} \displaystyle\lim_{\alpha\to 0}\bigg\{ & T(t)\big[\eta(0)-\big(h(z_{\tau_{1}},\dots,z_{\tau_{q}})\big)(0)-g\big(0,\eta(0)-(h(z_{\tau_{1}},\dots,z_{\tau_{q}})(0)\big)\big]\\ & + \ g(t,z^\alpha_{t}) -\displaystyle\int_{0}^{t}AT(t-s)g(s,z^\alpha_{s})ds+\displaystyle\int_{0}^{t}T(t-s)Bu_\alpha(s)ds\\ &+ \displaystyle\int_{0}^{t}T(t-s)f(s,z^\alpha_{s},u_\alpha(s))ds +\displaystyle\sum_{0<t_{k}<t}T(t-t_{k})J_{k}(z^\alpha(t_{k}),u_\alpha(t_{k}))\bigg\}=z^{1}, \end{align*} $$
and the proof of the theorem is completed.

As a consequence of the foregoing theorem, we can prove the following characterization:

 

Theorem 4.
The system (1.1) is approximately controllable on |$[0,\tau ]$| if for all states |$\eta $| and a final state |$z^{1}$| and |$\alpha \in (0,1]$| the operator |${\mathcal{S}}^{\, \alpha} $| given by (4.2) has a fixed point and the sequence |$\{{\mathcal{L}}(z^\alpha ,u_\alpha )\}_{\alpha \in (0,1]}$| converges, i.e.,
$$ \begin{align*} & (z^\alpha,u_\alpha)={\mathcal{S}}^{\, \alpha}(z^\alpha,u_\alpha), \quad \displaystyle\lim_{\alpha\to 0}{\mathcal{L}}(z^\alpha,u_\alpha)=y\in Z^\beta. \end{align*} $$

5. Application

A significant specific class of unbounded operator |$A$|⁠, encompassing numerous examples, is as follows: consider the strongly continuous semigroup |$\{T(t)\}_{t\geq 0}$| generated by |$-A$|⁠, which satisfies the following spectral decomposition:

$$ \begin{align*} & Az=\displaystyle\sum_{i=1}^\infty\lambda_j\displaystyle\sum_{k=1}^{\gamma_j}\langle z,\phi_{j,k}\rangle\phi_{j,k}, \quad z\in Z, \end{align*} $$

with the eigenvalues |$0<\lambda _{1}<\lambda _{2}<\cdots <\lambda _{n}\to \infty $| of |$A$| having finite multiplicity |$\gamma _{j}$| equal to the dimension of the corresponding eigenspaces, and |$\{\phi _{j,k}\}$| is a complete orthonormal set of eigenfunctions of |$A$|⁠. So, the strongly continuous semigroup is given by

$$ \begin{align*} & T(t)z=\displaystyle\sum_{j=1}^\infty\displaystyle\sum_{k=1}^{\gamma_j}\langle z,\phi_{j,k}\rangle\phi_{j,k}, \quad z\in Z, \end{align*} $$

and

$$ \begin{align*} & A^\beta z=\displaystyle\sum_{j=1}^\infty\lambda_j^\beta\displaystyle\sum_{k=1}^{\gamma_j}\langle z,\phi_{j,k}\rangle\phi_{j,k}, \quad z\in Z^\beta. \end{align*} $$

As a consequence, we have the following estimate:

$$ \begin{align*} & \left\Vert T(t)\right\Vert\leq Ke^{-\mu t}, \quad t\geq 0, \ \mu>0. \end{align*} $$

 

Theorem 5 (See (Leiva et al., 2012, Theorem 2.4)).
If the vectors |$B^{*}\phi _{j,k}$| are linearly independent in |$Z$|⁠, then the system (3.1) is approximately controllable on |$[0,\tau ]$|⁠. Moreover, a sequence of controls steering the system (3.1) from initial state |$z^{0}$| to an |$\varepsilon $|-neighborhood of the final state |$z^{1}$| at time |$\tau>0$| is given by
(5.1)
and the error of this approximation |$\mathscr{E}_\alpha $| is given by
$$ \begin{align*} & \mathscr{E}_\alpha=\alpha(\alpha I+L_{\mathscr{G}})^{-1}(z^1-T(\tau)z^0). \end{align*} $$

 

Remark 3.
The hypothesis vectors |$B^{*}\phi _{j,k}$| being linearly independent in |$Z$| can be replaced by the following weaker condition:
$$ \begin{align*} & \displaystyle\sum_{k=1}^{\gamma_k}a_{j,k}B^*\phi_{j,k}=0 \quad \Rightarrow \quad a_{j,k}=0, \ j=1,2,\dots \end{align*} $$

For example, we will investigate the controllability for a class of heat’s equation of neutral type with impulses and nonlocal conditions of the form

(5.2)

where |$\varOmega =[0,\pi ]$|⁠, |$\gamma :{{\mathbb{R}}}^{+}\to{{\mathbb{R}}}^{+}$| is a nondecreasing function such that |$\gamma (0)=0$| and |$\gamma (t)\leq \min \left \{4\left \Vert \xi \right \Vert{}_{_{L^\infty [0,\tau ]}},L\right \}$|⁠. Here |$\eta :[-r,0]\times \varOmega \to{{\mathbb{R}}}$| is a piecewise continuous function, |$\omega $| is an open nonempty subset of |$\varOmega $|⁠, |$1_\omega $| denotes the characteristic function of the set |$\omega $|⁠, the distributed control |$u$| belongs to |$L^{2}([0,\tau ],L^{2}(\varOmega ))$|⁠. We assume that there exists |$L>0$|⁠, and |$\alpha ,\beta , \xi \in L^\infty [0,\tau ]$| such that

$$ \begin{align*} & \left\vert p(t,x,u)-p(t,y,v)\right\vert\leq L\big(\left\vert x-y\right\vert+\left\vert u-v\right\vert\big), \ t\in [0,\tau], \ x,y,u,v\in{{\mathbb{R}}}, \end{align*} $$

and

$$ \begin{align*} & \left\vert p(t,x,u)\right\vert\leq \xi(t)\left\vert x\right\vert{}^{\alpha_0}+\alpha(t)\left\vert u\right\vert{}^{\beta_0}+\beta(t), \ \frac{1}{2}\leq \alpha_0,\beta_0<1. \end{align*} $$

We also assume that |$h:{{\mathbb{R}}}^{q}\to{{\mathbb{R}}}$| and |$J_{k}:{{\mathbb{R}}}^{2}\to{{\mathbb{R}}}$| when formulated abstractly, satisfy (H|$_{1}$|⁠)-(iii),(ii) and (H|$_{5}$|⁠)-(ii), (iii), respectively.

Let |$Z=U=L^{2}(\varOmega )$| and consider the linear operator |$A:D(A)\subset Z\to Z$| defined by |$A\phi =-\phi _{xx}$|⁠, where |$D(A)=H_{0}^{1}(\varOmega )\cap H^{2}(\varOmega )$|⁠. The properties of the operator |$A$| are well-known in the literature (see, for instance Henry (1981)). The spectrum of |$A$| consists only discrete eigenvalues |$\lambda _{n}=n^{2}$|⁠, |$n\in{{\mathbb{N}}}$|⁠. Their corresponding normalized eigenvectors are given by |$w_{n}=\left (\frac{2}{\pi }\right )^{1/2}\sin (nx)$|⁠, |$x\in [0,\pi ]$|⁠. The collection of these functions |$\{w_{n}:n\in{{\mathbb{N}}}\}$| constitutes and orthonormal basis for |${{\mathbb{Z}}}$|⁠. For all |$z\in D(A)$|⁠, the operator |$A$| has the representation

$$ \begin{align*} & Az=\displaystyle\sum_{n=1}^\infty\lambda_n\langle z,w_n\rangle w_n, \end{align*} $$

where |$\langle \cdot ,\cdot \rangle $| is the inner product in |$Z$|⁠. It is also well-known that |$A$| is a sectorial operator (see Henry (1981)) and therefore, |$-A$| generates a compact analytic semigroup |$T(t)$| of uniformly bounded linear operators on |$Z$| given by

$$ \begin{align*} & T(t)z=\displaystyle\sum_{n=1}^\infty e^{-\lambda_n t}\langle z,w_n\rangle w_n, \end{align*} $$

and satisfying |$\left \Vert T(t)\right \Vert \leq e^{-\lambda _{1}t}$| for |$t\geq 0$|⁠, moreover, |$T(t)$| is compact for |$t>0$| (see Bárcenas et al. (2005)).

Since |$A$| is a sectorial operator |$0\in \rho (A)$|⁠, it is possible to define fractional powers of |$A$|⁠. In particular, the operator |$A^{1/2}$| is given by

$$ \begin{align*} & A^{1/2}z=\displaystyle\sum_{n=1}^\infty\lambda_n^{1/2}\langle z,w_n\rangle w_n, \ z\in D(A^{1/2}), \end{align*} $$

where (see Mokkedem & Fu (2014))

$$ \begin{align*} D(A^{1/2})=& \left\{z\in Z:\displaystyle\sum_{n=1}^\infty\lambda_{n}^{1/2}\langle z,w_{n}\rangle w_{n}\in{{\mathbb{Z}}}\right\}\\ = & \big\{z\in Z:z^{\prime}\in Z, z \mbox{ is absolutely continuous and }z(0)=z(\pi)=0\big\}. \end{align*} $$

This mean (see Henry (1981)) that |$Z^{1/2}=H_{0}^{1}(\varOmega )$| with norm |$\left \Vert \cdot \right \Vert{}_{1/2}$|⁠. Therefore, system (5.2) can be written abstractly as (for details, see Agarwal et al. (2022))

(5.3)

where |$0<t_{1}<\cdots <t_{p}<\tau$|⁠, |$0<\tau _{1}<\dots <\tau _{q}<r<\tau $|⁠, |$B_\omega =1_\omega $| is a bounded linear operator, the control |$u$| belongs to |$L^{2}([0,\tau ],U)$|⁠, |$z_{t}$| is the time history function |$[-r,0]\ni \theta \mapsto z_{t}(\theta )=z(t+\theta )\in Z^{1/2}$| and the functions |$g:[0,\tau ]\times PW_{r1/2}\to Z$|⁠, |$f:[0,\tau ]\times PW_{r1/2}\times U\to Z$|⁠, |$h:PW_{qp1/2}\to PW_{r1/2}$|⁠, |$J_{k}:Z^{1/2}\to Z^{1/2}$|⁠, |$\eta \in PW_{r1/2}$| are defined by

$$ \begin{align*} & \begin{array}{rcl} \big[g(t,\phi)\big](x) & = & -\displaystyle\int_0^x\gamma(t)\phi(-r,s)ds, \ x\in\varOmega\\ \big[f(t,\phi,u)](x) & = & p(t,\phi(-r,x),u(x)), \ x\in\varOmega\\ \big[\big(h(z_{\tau_1},\dots,z_{\tau_q})\big)(\theta)\big](x) & = & h\big(v(\tau_1+\theta,x),\dots,v(\tau_q+\theta,x)\big), \ x\in\varOmega, \\ \big[J_k(z(t_k),u(t_k))\big](x) & = & J_k(v(t_k,x),u(t_k,x)), \ k\in I_p, \ x\in\varOmega,\\ \big[\eta(\theta)\big](x) & = & \eta(\theta,x), \ x\in\varOmega, \ \theta\in [-r,0], \end{array} \end{align*} $$

accordingly. Following Agarwal et al. (2022), we can assume that the functions |$g,f,h$| and |$J_{k}$| satisfy the hypotheses (H|$_{1}$|⁠)–(H|$_{5}$|⁠).

By (Leiva & Quintana, 2009, Theorem 3.4), we have that the linear system

(5.4)

is approximately controllable on |$[0,\tau ]$|⁠, so by Theorem 3, we can conclude

 

Theorem 6.

The system (5.2) is approximate controllable on |$[0,\tau ]$|⁠.

6. Final remark

In this work, we have demonstrated that impulses, nonlocal conditions and delays in the state variable, when viewed as perturbations of a linear system that is approximately controllable, do not disrupt the controllability of the semi-linear system under certain conditions. This result aligns with the natural conjecture that, in real-world scenarios, most mechanical systems are subject to such influences, even if they are often omitted in their representative models.

Acknowledgements

We would like to thank the two anonymous referees for their valuable suggestions and comments which led to the improvement of this article.

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