An outlook on the controllability of non-instantaneous impulsive neutral fractional nonlocal systems via Atangana–Baleanu–Caputo derivative

Abstract

In this article, we tackle the optimal control and controllability of neutral fractional nonlocal integro-differential equations (NFNIE) of Atangana–Baleanu–Caputo with non-instantaneous impulses. Semigroup theory, noncompactness measurements and fixed point approaches were used to attain the results. An illustration is provided to support the theoretical findings.

Keywords:

• Atangana–Baleanu–Caputo (ABC) controllability
• fixed point theorem
• impulsive equation
• optimal control
• semigroup theory

1. Introduction

Fractional calculus offers a variety of ways to explain how real and complex numbers work. The fractional derivative outperforms the integral derivative in the model of contemporary issues (Abdeljawad & Baleanu, 2016; Anurag Shukla & Pandey, 2015; Banas, 1980; Caputo & Fabrizio, 2015; Gupta, Jarad, Valliammal, Ravichandran & Nisar, 2022; Kilbas, Srivastava, & Trujillo, 2006; Miller & Ross, 1993; Podlubny, 1999; Vijayakumar et al., 2022; Yang, 2019). The intricate connection between the M-L function and fractional calculus takes on a new dimension (Bahaa & Hamiaz, 2018; Baleanu & Fernandez, 2018; Kaliraj, Thilakraj, Ravichandran, & Nisar, 2021; Ma et al., 2023; Shukla, Nagarajan, & Pandey, 2016; Shukla, Sukavanam, & Pandey, 2014). Newton and Leibnitz first proposed the idea of fractional theory in 1695. Both engineering and physical modelling have been used in this procedure (Abdeljawad & Baleanu, 2017; Aimene, Baleanu, & Seba, 2019; Atangana & Baleanu, 2016; Bahaa & Hamiaz, 2018; Baleanu & Fernandez, 2018; Devi & Kumar, 2021; Jothimani, Kaliraj, Panda, Nisar & Ravichandran, 2021; Jarad, Abdeljawad, & Hammouch, 2018; Kumar & Pandey, 2020; Kumar, Kostic, & Pinto, 2023; Mahmud, Tanriverdi & Muhamad, 2023; Nisar, Jothimani, Kaliraj, & Ravichandran, 2021; Ruhil & Malik, 2023; Yang, Agarwal, & Liang, 2016).

The capacity of a control system to transition from a fixed state to a specific state in a finite amount of time is known as controllability. The principle of controllability was first put forth by R. Kalman in 1960. It is regarded as one of the fundamental and important ideas in control systems. Controllability is a dynamic feature that describes the control performance that can be achieved. In Bedi, Kumar, and Khan (2021), the results of controllability are addressed for the system, $$\begin{array}{l}{}^{\mathit{ABC}}{\mathcal{D}}^{\psi }\{\nu (q)-\nu (q,\nu (q))\}=\mathbb{A}\nu (q)+\mathfrak{I}\nu (q)\\ +\sigma (q,\nu (q)), q\in \underset{i=0}{\overset{m}{\cup }}(s_i,q_{i+1}],\\ \nu (q)={\kappa }_i(q,\nu (q)), q\in \underset{i=1}{\overset{m}{\cup }}(s_i,q_i],\\ \nu (0)-\varsigma (0,\nu (0))={\nu }_0,\end{array}$$ where $\mathfrak{I}$ signifies a bounded linear operator with a control function $u(q).$ Also, Balasubramaniam (2021) made a deep study on controllability via ABC derivative. Further, in Aimene et al. (2019), the controllability results of semilinear impulsive systems have been examined by AB fractional derivative. In addition, Kumar and Pandey (2020) proposed the existing results of a non-instantaneous impulsive system supported by AB derivative.

Fractional systems are used to describe optimum control issues with various specified limits (Abbas, 2021; Agrawal, 2004, 2008; Aimene et al., 2019; Benchohra & Seba, 2009; Chalishajar, Ravichandran, Dhanalakshmi & Murugesu, 2019; Dineshkumar & Hoon Joo, 2023; Dineshkumar, Udhayakumar, Vijayakumar, Shukla, & Sooppy Nisar, 2023; Enes & Onur Kıymaz, 2023; Dineshkumar, Udhayakumar, Vijayakumar, Nisar, & Shukla, 2022; Dineshkumar, Udhayakumar, Vijayakumar, Sooppy Nisar, et al., 2022; Tajadodi, Khan, Francisco Gómez-Aguilar, & Khan, 2021). Recently, researchers give more attention to fractional derivatives, especially when several applications in biology, economics, science and engineering have appeared (Goufo, Ravichandran & Birajdar, 2021; Kavitha Williams, Vijayakumar, Shukla, & Nisar, 2022; Kavitha Williams, Vijayakumar, Nisar, & Shukla, 2023). The noninstantaneous and instantaneous fractional differential systems of their qualitative behaviours have been debated comprehensively (Gautam & Dabas, 2015; Ravichandran, jothimani, Nisar, Mahmoud & Yahia, 2022; Kumar & Pandey, 2020; Pandey, Das, & Sukavanam, 2014).

Motivated by the above literature, we widen to procure some abundant conditions for controllability and optimal controls of the form: (1.1) $$\begin{array}{l}{}^{\mathit{ABC}}{\mathcal{D}}^p[v(q)-h(q,v(q),{\int }_0^q\mathcal{K}(q,s,v(s))ds)]\\ =\mathbb{A}[v(q)-h(q,v(q),{\int }_0^q\mathcal{K}(q,s,v(s))ds)]+\mathcal{B}u(q)\\ +f(q,v(q),{\int }_0^q\xi (q,s,v(s))ds), q\in \underset{i=0}{\overset{m}{\cup }}(s_i,q_{i+1}]\\ v(q)=g_i(q,v(q)), q\in \underset{i=1}{\overset{m}{\cup }}(s_i,q_i]\\ v(0)+\varsigma (v)=v_0\in V,\end{array}$$(1.1) where ${}^{\mathit{ABC}}{\mathcal{D}}^p$ ABC is the derivative of order $p\&i=\overline{0,m}.$ In Banach space $\mathcal{V},$ the $p-$ resolvent infinitesimal generator of $\mathbb{A}:D\subset \mathcal{V}\to \mathcal{V}$ is ${\left\{T_p(q)\right\}}_{q\ge 0}.$ For $s_i\in (q_i, q_{i+1}), i=1,2,\dots m ,s_i$ meets, $0=q_0=s_0<q_1\le s_1\le q_2\cdots <q_m\le s_m<q_{m+1}=b.$ The term ${\left\{S_p(t)\right\}}_{q\ge 0}$ acts as a solution operator on $(\mathcal{V},\Vert \cdot \Vert ).$ Let $I=[0,b]$ and $\mathcal{B}:\mathcal{U}\to \mathcal{V}$ is a bounded linear operator with control $u(\cdot )\in L^2(I,\mathcal{U}).$ Furthermore, $h, f$ defined over $(s_i, q_{i+1})\times \Omega \times \mathcal{V}\to \mathcal{V}.$ and $\varsigma :\mathcal{P}\mathcal{C}(I, \mathcal{V})\to \mathcal{V}$ are given functions with certain assumptions. In addition, the non-instantaneous impulsive functions are defined as $g_i:(q_i, s_i]\times V\to \mathcal{V}, i=1,2,3,\dots m.$

We assume that, $\mathcal{K}v(q)={\int }_0^q\mathcal{K}(q,s,v(s))ds$ and $\xi v(q)={\int }_0^q\xi (q,s,v(s))ds.$

2. Basic results

Definition 2.1.

The Kuratowski noncompact measure is $m\in \mathcal{V}$ be a bounded set (Banas, 1980), $$\zeta (m)=\mathit{inf}\left\{\delta >0:m\subseteq \underset{i=1}{\overset{n}{\cup }}{m}_i \text{and diameter} (m_i)\le \delta \right\}.$$

Lemma 2.2.

Let $m, m_1,m_2$ be bounded subsets on Banach space (Banas, 1980),

1. $\zeta (m_1)=0\iff \overline{m_1}$ is compact.

2. $m_1\subset m_2\Rightarrow \zeta (m_1)\le \zeta (m_2).$

3. $\zeta (m+v)=\zeta (m),$ for any $v\in \mathcal{V}.$

4. $\zeta (m_1+m_2)\le \zeta (m_1)+\zeta (m_2).$

5. $\zeta (K{m}_1)=\left|K\right|\zeta (m_1);K\in \mathbb{R}.$

6. $\zeta (\overline{m})=\zeta (m),$ $\overline{m}$ is closed set of $m.$

7. $\zeta (m_1\cup m_2)=\max \left\{\zeta (m_1), \zeta (m_2)\right\}.$

Lemma 2.3.

For the uniform integration of ${\mathcal{E}}_0={\left\{{\mathcal{U}}_n\right\}}_{n=1}^{\infty }\subset L^1(I,\mathcal{V}) \exists \mathcal{U}\in L^1(I,{\mathbb{R}}^n)$ with $\Vert {\mathcal{U}}_n\Vert \le \mathcal{U}(q),$ a.e $q\in I$ (Aimene et al., 2019), $$\zeta ({\left\{{\int }_I{\mathcal{U}}_n(q)dq\right\}}_{n=1}^{\infty })\le 2{\int }_I\zeta ({\left\{{\mathcal{U}}_n(q)\right\}}_{n=1}^{\infty })dt.$$

Definition 2.4.

The AB FD with $p\in (0,1)$ of the Caputo is $q\in (u,v)\&\mathfrak{I}\in {\mathbb{H}}^{\prime }(u,v),u<v$ as (Atangana & Baleanu, 2016), (2.1) $${}^{\mathit{ABC}}{\mathcal{D}}_{a^{+}}^p\mathfrak{I}(q)=\frac{Q(p)}{1-p}{\int }_a^q{\mathfrak{I}}^{\prime }(s)E_p(-\gamma {(q-s)}^p)ds,$$(2.1) and $\gamma =\frac{p}{1-p}, E_p(\cdot ),$ $Q(p)=(1-p)+\frac{p}{\Gamma (p)}.$

Definition 2.5.

The AB differentiation of $\mathfrak{I}$ in the view of RL sense and for $\mathfrak{I}\in {\mathbb{H}}^{\prime }(u,v),u<v$ of order $p\in (0,1)$ as (Atangana & Baleanu, 2016), (2.2) $${}^{\mathit{ABR}}{\mathcal{D}}_{a^{+}}^p\mathfrak{I}(q)=\frac{Q(p)}{1-p}\frac{d}{dp}{\int }_a^q\mathfrak{I}(s)E_p(-\gamma {(q-s)}^p)ds.$$(2.2) AB integral is, (2.3) $${}^{AB}{I}_{a^{+}}^p=\frac{1-p}{Q(p)}\mathfrak{I}(q)+\frac{p}{Q(p)\Gamma (p)}{\int }_a^q{(q-s)}^{p-1}\mathfrak{I}(s)ds.$$(2.3)

Remark 2.6.

Here, the two-parameter generalizations of M-L function: $$\begin{array}{c}{E}_p(v):=\sum _{k=0}^{\infty }\frac{v^k}{\Gamma (pk+1)},\hspace{1em}v\in \mathbb{C}, p>0.\\ E_{p,\mathrm{ }q}(v):=\sum _{k=0}^{\infty }\frac{v^k}{\Gamma (pk+q)},\hspace{1em}p>0, q>0.\end{array}$$

In particular, $$\begin{array}{l}{E}_{p,1}(v)=E_p(v),\hspace{1em}v\in \mathbb{C}.\\ L{(}^{\mathit{ABC}}{D}_{a^{+}}^{p}g(q))=\frac{Q(p)}{1-p}L(E_p(-\gamma q^p))(sL(g(q))-g(0))\\ L(q^{q-1}{E}_{p,\mathrm{ }q}(-\gamma q^p))=\frac{s^{p-q}}{s^p+\gamma }\\ L(q^p)=\frac{\Gamma (p+1)}{s^{p+1}}.\end{array}$$

Definition 2.7.

The sectorial operator $\mathbb{A}$ is (Pazy, 1983):

$(1)$ $\mathbb{A}$ is closed & linear.

$(2)$ Let $\varsigma \in \mathbb{R} \& \alpha \in [\frac{\pi }{2}, \pi ], \exists R>0,$ s.t.,

1. $\Vert R(\delta , \mathbb{A})\Vert \le \frac{R}{\left|\delta -\varsigma \right|}, \delta \in \sum _{\alpha ,\mathrm{ }\varsigma }.$

2. $\sum _{\alpha ,\mathrm{ }\varsigma }=\left\{\delta \in \mathbb{C}:\delta \ne \varsigma ,\mathit{arg}(\delta -\varsigma )<\alpha \right\}\subset \mathbb{A}.$

Remark 2.8.

For the mild solution of (1.1)(1.1) $$\begin{array}{l}{}^{\mathit{ABC}}{\mathcal{D}}^p[v(q)-h(q,v(q),{\int }_0^q\mathcal{K}(q,s,v(s))ds)]\\ =\mathbb{A}[v(q)-h(q,v(q),{\int }_0^q\mathcal{K}(q,s,v(s))ds)]+\mathcal{B}u(q)\\ +f(q,v(q),{\int }_0^q\xi (q,s,v(s))ds), q\in \underset{i=0}{\overset{m}{\cup }}(s_i,q_{i+1}]\\ v(q)=g_i(q,v(q)), q\in \underset{i=1}{\overset{m}{\cup }}(s_i,q_i]\\ v(0)+\varsigma (v)=v_0\in V,\end{array}$$(1.1) , consider a subsequent Cauchy problem (Pazy, 1983) $$\begin{array}{c}{}^{\mathit{ABC}}{D}_q^{p}v(q)=\mathbb{A}v(q)+f(q),\hfill \\ v(0)=v_0\in \mathcal{V},\hfill & \hfill \end{array}$$ and mild solution is, $$\mathcal{V}(q)=\mathcal{G}{T}_p(q)v_0+\frac{\mathcal{K}\mathcal{G}(1-p)}{Q(p)\Gamma (p)}{\int }_0^q{(q-s)}^{p-1}f(s)ds+\frac{P{\mathcal{G}}^2}{Q(p)}{\int }_0^q{S}_p(q-s)f(s)ds,$$ where $0<p<1$ and $q\in I. \mathcal{G}=\zeta {(\zeta I-\mathbb{A})}^{-1}, \mathcal{K}=-\gamma \mathbb{A}{(\zeta I-\mathbb{A})}^{-1}$ are linear operators with $\zeta =\frac{Q(p)}{1-p}$ and, $$\begin{array}{c}{T}_p(q)=E_p(-\mathcal{K}{q}^p)=\frac{1}{2\pi i}{\int }_{\Gamma }{e}^{\lambda q}{\lambda }^{p-1}{({\lambda }^{p}I-\mathcal{K})}^{-1}d\lambda \\ S_p(q)=q^{p-1}{E}_{p,p}(-\mathcal{K}{q}^p)=\frac{1}{2\pi i}{\int }_{\Gamma }{e}^{\lambda q}{({\lambda }^{p}I-\mathcal{K})}^{-1}d\lambda .\end{array}$$

Definition 2.9.

The problem (1.1)(1.1) $$\begin{array}{l}{}^{\mathit{ABC}}{\mathcal{D}}^p[v(q)-h(q,v(q),{\int }_0^q\mathcal{K}(q,s,v(s))ds)]\\ =\mathbb{A}[v(q)-h(q,v(q),{\int }_0^q\mathcal{K}(q,s,v(s))ds)]+\mathcal{B}u(q)\\ +f(q,v(q),{\int }_0^q\xi (q,s,v(s))ds), q\in \underset{i=0}{\overset{m}{\cup }}(s_i,q_{i+1}]\\ v(q)=g_i(q,v(q)), q\in \underset{i=1}{\overset{m}{\cup }}(s_i,q_i]\\ v(0)+\varsigma (v)=v_0\in V,\end{array}$$(1.1) is called controllable if for all $\varsigma \in \Omega$ and $v_1\in V,$ and a control $u\in L^2(I,\mathcal{U})$ shows the mild solution of $v(\cdot )$ of $(1.1)\&v(b)=v_1$ on I (Aimene et al., 2019).

Theorem 2.10.

Let $\mathcal{S}\subset \mathcal{V},$ is a nonempty, closed, convex $\&$ bounded set. Let $\Phi :\mathcal{S}\to \mathcal{S}$ is continuously called $\mathcal{K}-$ set contraction (Banas, 1980), (2.4) $$\zeta (\Phi (\mathcal{D}))\le \mathcal{K}\zeta (\mathcal{D}),$$(2.4)

Then $\Phi$ has a fixed point.

Definition 2.11.

The mild solution of (1.1)(1.1) $$\begin{array}{l}{}^{\mathit{ABC}}{\mathcal{D}}^p[v(q)-h(q,v(q),{\int }_0^q\mathcal{K}(q,s,v(s))ds)]\\ =\mathbb{A}[v(q)-h(q,v(q),{\int }_0^q\mathcal{K}(q,s,v(s))ds)]+\mathcal{B}u(q)\\ +f(q,v(q),{\int }_0^q\xi (q,s,v(s))ds), q\in \underset{i=0}{\overset{m}{\cup }}(s_i,q_{i+1}]\\ v(q)=g_i(q,v(q)), q\in \underset{i=1}{\overset{m}{\cup }}(s_i,q_i]\\ v(0)+\varsigma (v)=v_0\in V,\end{array}$$(1.1) is $v(\cdot )\in {\Omega }_b$ if $v(q)=\varsigma (q)$ on $v(q)\in \mathcal{V},q\in I$ shows $\mathcal{V}(q)=g_i(q,v(q),q\in (q_i,s_i],i=1,2,\dots m,$ if $$\begin{array}{c}\begin{array}{c}v(q)=\{\begin{array}{c}\mathcal{G}{T}_p(q)[v_0-\varsigma (v)-h(0, v(0),0)]+h(q,v(q),\mathcal{K}v(q))\hfill \\ +\frac{\mathcal{K}\mathcal{G}(1-p)}{Q(p)\Gamma (p)}{\int }_0^q{(q-s)}^{p-1}[\mathcal{B}u(s)+f(s,v(s),\xi v(s))]ds\hfill \\ +\frac{P{\mathcal{G}}^2}{Q(p)}{\int }_0^q{S}_p(q-s)[\mathcal{B}u(s)+f(s,v(s),\xi v(s))]ds, \hspace{1em}q\in [0,q_1]\hfill \\ \mathcal{G}{T}_p(q-s_i)[g_i(s_i,v(s_i))-h(s_i,v(s_i),\mathcal{K}v(s_i))]+h(q,v(q),\mathcal{K}v(q))\hfill \\ +\frac{\mathcal{K}\mathcal{G}(1-p)}{Q(p)\Gamma (p)}{\int }_{s_i}^q{(q-s)}^{p-1}[\mathcal{B}u(s)+f(s,v(s),\xi v(s))]ds\hfill \\ +\frac{P{\mathcal{G}}^2}{Q(p)}{\int }_{s_i}^q{S}_p(q-s)[\mathcal{B}u(s)+f(s,v(s),\xi v(s))]ds, \hspace{1em}q\in (s_i,q_{i+1}], i=\overline{1,m}.\hfill \end{array}\hfill \end{array}\hfill \end{array}$$

3. Main sequels

If $A\in {\mathbb{A}}^q({\alpha }_0,{\Xi }_0),$ $\mathcal{R} \& C$ are $+\text{ve}$ constants, provided $\Vert T_p(q)\Vert \le \mathcal{R}{e}^{\Xi q},$ $\Vert S_p(q)\Vert \le C{e}^{\Xi q}(1+q^{p-1}),$ for all $q>0,\Xi >{\Xi }_0.$ Let ${\mathcal{R}}_T=\underset{q\ge 0}{\sup }\Vert T_p(q)\Vert ,{\mathcal{R}}_S=\underset{q\ge 0}{\sup }C{e}^{\Xi q}(1+q^{p-1}).$

Let $\Vert T_p(q)\Vert \le {\mathcal{R}}_T, \Vert S_p(q)\Vert \le q^{p-1}{\mathcal{R}}_S:$

• H0 For ${\mathcal{R}}_h, {\mathcal{R}}_h^{*}>0$ and $h:I\times \Omega \times \mathcal{V}\to \mathcal{V}$ is continuous,

1. $\Vert h(q,v_1,v_2)-h(q,v_3,v_4)\Vert \le {\mathcal{R}}_h\Vert v_1-v_3\Vert +{\mathcal{R}}_h^{*}\Vert v_2-v_4\Vert ,$ For every $v_1,v_3\in \Omega$ and $v_2,v_4\in \mathcal{V}.$

2. $\Vert h(q,u,v)\Vert \le {\mathcal{R}}_h^{**}(1+n)\eta ,$ ${\mathcal{R}}_h^{**}>0.$

• H1 Let $f:I\times \Omega \times \mathcal{V}\to \mathcal{V}.$ For ${\mathcal{R}}_f, {\mathcal{R}}_f^{*}>0,$

$\Vert f(q,v_1,v_2)-f(q,v_3,v_4)\Vert \le {\mathcal{R}}_f\Vert v_1-v_3\Vert +{\mathcal{R}}_f^{*}\Vert v_2-v_4\Vert ,$ for every $v_1,v_3\in \Omega$ and $v_2,v_4\in \mathcal{V}.$

• H2 Let ${\mathcal{R}}_{\mathcal{K}}>0$ with $\Vert \mathcal{K}(q,s,y_1)-\mathcal{K}(q,s,y_2)\Vert \le {\mathcal{R}}_{\mathcal{K}}\Vert y_1-y_2\Vert ,$ $\forall$ $y_1,y_2\in \Omega .$

• H3 Let ${\mathcal{R}}_{\xi }>0$ with $\Vert \xi (q,s,u)-\xi (q,s,v)\Vert \le {\mathcal{R}}_{\xi }\Vert u-v\Vert ,$ $\forall$ $u,v\in \Omega .$

• H4 For $f:I\times \Omega \times \mathcal{V}\to \mathcal{V}$ fulfils $\Vert f(q,v,\xi )\Vert \le {\mathcal{R}}_f^{**},$ $q\in I,v\in \Omega$ and $\xi \in \mathcal{V}$ where ${\mathcal{R}}_f^{**}>0.$

• H5 $W:L^2(I,\mathcal{U})\to \mathcal{V}$ and $$Wu=\frac{\mathcal{K}\mathcal{G}(1-p)}{Q(p)\Gamma (p)}{\int }_{s_i}^{q_{i+1}}{(b-s)}^{p-1}\mathcal{B}u(s)ds+\frac{P{\mathcal{G}}^2}{Q(p)}{\int }_{s_i}^{q_{i+1}}{S}_p(b-s)\mathcal{B}u(s)ds.$$

• H6 Let $g_i:(q_i,s_i]\times \Omega \to V,$ for each $i=\overline{1,m},$

1. $\Vert g_i(q,v_1)-g_i(q,v_2)\Vert \le {\mathcal{R}}_{g_i}\Vert v_1-v_2\Vert ,$ for each $v_1,v_2\in \Omega , q\in (q_i,s_i],$ for constants ${\mathcal{R}}_{g_i}>0.$

2. $\Vert g_i(q,v)\Vert \le {\mathcal{R}}_{g_i}^{*}\eta +\delta ,$ for some constants ${\mathcal{R}}_{g_i}^{*}>0.$

• H7 Let $\mathcal{G}$&$\mathcal{K}$ are bounded linear operators, $\Vert \mathcal{G}\Vert \le {\mathcal{R}}_{\mathcal{G}}$ and $\Vert \mathcal{K}\Vert \le {\mathcal{R}}_{\mathcal{K}},$ ${\mathcal{R}}_{\mathcal{G}}>0, {\mathcal{R}}_{\mathcal{K}}>0.$

• H8 Let $L_f>0$ and $e_1,e_2\subseteq \Omega ,$ any bounded sets, we have $\zeta (f(q,e_1,e_2))\le L_f(\zeta (e_1)+\zeta (e_2)).$

Further, let us assume: $$\begin{array}{l}{\mathcal{R}}_p=\frac{{\mathcal{R}}_{\mathcal{G}}}{Q(p)}{b}^p[\frac{{\mathcal{R}}_{\mathcal{K}}(1-p)}{\Gamma (p+1)}+{\mathcal{R}}_{\mathcal{G}}{\mathcal{R}}_S]\\ {\mathcal{R}}_g={\max }_{i=\overline{1,m}}{\mathcal{R}}_{g_i}\\ \kappa ={\max }_{i=\overline{1,m}}\underset{q\in I}{\sup }\Vert g_i(q,0)\Vert .\end{array}$$

Theorem 3.1.

Let $\mathbf{H}\mathbf{0}-\mathbf{H}\mathbf{8}$ holds, then (1.1)(1.1) $$\begin{array}{l}{}^{\mathit{ABC}}{\mathcal{D}}^p[v(q)-h(q,v(q),{\int }_0^q\mathcal{K}(q,s,v(s))ds)]\\ =\mathbb{A}[v(q)-h(q,v(q),{\int }_0^q\mathcal{K}(q,s,v(s))ds)]+\mathcal{B}u(q)\\ +f(q,v(q),{\int }_0^q\xi (q,s,v(s))ds), q\in \underset{i=0}{\overset{m}{\cup }}(s_i,q_{i+1}]\\ v(q)=g_i(q,v(q)), q\in \underset{i=1}{\overset{m}{\cup }}(s_i,q_i]\\ v(0)+\varsigma (v)=v_0\in V,\end{array}$$(1.1) is controllable on I shows: (3.1) $$L_b(1+n)+2({\mathcal{R}}_{\mathcal{B}}^{*}{\mathcal{N}}_p+L_f(1+n){\mathcal{R}}_p)<1,$$(3.1) and ${\mathcal{N}}_p=L_b(1+n)+2{\mathcal{R}}_W^{*}+L_f(1+n){\mathcal{R}}_p.$

Proof.

Let $\Phi :{\Omega }_b\to {\Omega }_b$ is, $$\begin{array}{c}\begin{array}{c}\Phi (v)(q)=\{\begin{array}{c}\mathcal{G}{T}_p(q)[v_0-\varsigma (v)-h(0,v(0),0)]+h(q),v(q),\mathcal{K}v(q))\hfill \\ +\frac{\mathcal{K}\mathcal{G}(1-p)}{Q(p)\Gamma (p)}{\int }_0^q{(q-s)}^{p-1}[\mathcal{B}{u}_v(s)+f(s,v(s),\xi v(s))]ds\hfill \\ +\frac{P{\mathcal{G}}^2}{Q(p)}{\int }_0^q{S}_p(q-s)[\mathcal{B}{u}_v(s)+f(s,v(s),\xi v(s))]ds, \text{for} q\in [0,q_1], i=0\hfill \\ g_i(q,v(q)), \text{for} q\in (q_i,s_i], i\ge 1\hfill \\ \mathcal{G}{T}_p(q-s_i)[g_i(s_i,v(s_i))-h(s_i,v(s_i),\mathcal{K}v(s_i))]+h(q),v(q),\mathcal{K}v(q))\hfill \\ +\frac{\mathcal{K}\mathcal{G}(1-p)}{Q(p)\Gamma (p)}{\int }_{s_i}^q{(q-s)}^{p-1}[\mathcal{B}{u}_v(s)+f(s,v(s),\xi v(s))]ds\hfill \\ +\frac{P{\mathcal{G}}^2}{Q(p)}{\int }_{s_i}^q{S}_p(q-s)[\mathcal{B}{u}_v(s)+f(s,v(s),\xi v(s))]ds, \text{for} q\in (s_i,q_{i+1}], i=\overline{1,m}.\hfill \end{array}\hfill \end{array}\hfill \end{array}$$

By using H5, we get (3.2) $$\begin{array}{c}u(v)(q)=W^{-1}\{\begin{array}{c}v(b)-\mathcal{G}{T}_p(b)[v_0-\varsigma (v)-h(0,v(0),0)]-h(b,v(b),\mathcal{K}v(b))\hfill \\ -\frac{\mathcal{K}\mathcal{G}(1-p)}{Q(p)\Gamma (p)}{\int }_0^l{(l-s)}^{p-1}f(s,v(s),\xi v(s))ds\hfill \\ -\frac{P{\mathcal{G}}^2}{Q(p)}{\int }_0^l{S}_p(l-s)f(s,v(s),\xi v(s))ds, \text{for} q\in [0,q_1], i=0\hfill \\ 0, \text{for} q\in (q_i,s_i] \hfill \\ v(b)-\mathcal{G}{T}_p(l-s_i)[g_i(s_i,v(s_i))-h(s_i,v(s_i),\mathcal{K}v(s_i))]-h(b,v(b),\mathcal{K}v(b))\hfill \\ -\frac{\mathcal{K}\mathcal{G}(1-p)}{Q(p)\Gamma (p)}{\int }_{s_i}^l{(l-s)}^{p-1}f(s,v(s),\xi v(s))ds\hfill \\ -\frac{P{\mathcal{G}}^2}{Q(p)}{\int }_{s_i}^l{S}_p(l-s)f(s,v(s),\xi v(s))ds, \text{for} q\in (s_i,q_{i+1}],i=\overline{1,m}.\hfill \end{array}\hfill \end{array}$$(3.2)

Set ${\Omega }_{\kappa }=\{v\in {\Omega }_b:{\Vert v\Vert }_{{\Omega }_b}\le \kappa \}.$ So, ${\Omega }_{\kappa }$ is convex and closed subset of ${\Omega }_b.$

Step 1: $\Phi :{\Omega }_{\kappa }\to {\Omega }_{\kappa }$ meets the assumption of Theorem 2.10.

By $(3.2)$ and $\mathbf{H}\mathbf{4}, \mathbf{H}\mathbf{6}, \mathbf{H}\mathbf{7},$ we get $$\begin{array}{c}\begin{array}{c}\Vert u(v)(q)\Vert \le \Vert W^{-1}\Vert \{\begin{array}{c}\Vert v(b)\Vert +\Vert \mathcal{G}\Vert \Vert T_p(b)\Vert [\Vert v_0\Vert +\Vert \varsigma (v)\Vert +\Vert h(0,v(0),0)\Vert ]+\Vert h(b,v(b),\mathcal{K}v(b))\Vert \hfill \\ +\frac{\Vert \mathcal{K}\Vert \Vert \mathcal{G}\Vert (1-p)}{Q(p)\Gamma (p)}{\int }_0^l{(l-s)}^{p-1}\Vert f(s,v(s),\xi v(s))\Vert ds\hfill \\ +\frac{p\Vert {\mathcal{G}}^2\Vert }{Q(p)}{\int }_0^l\Vert S_p(l-s)\Vert \Vert f(s,v(s),\xi v(s))\Vert ds, t\in [0,t_1]\hfill \\ \Vert v_b\Vert +\Vert \mathcal{G}\Vert \Vert T_p(l-s_i)\Vert [\Vert g_i(s_i,v(s_i))\Vert +\Vert h(s_i,v(s_i),\mathcal{K}v(s_i))\Vert ]\hfill \\ +\Vert h(b,v(b),\mathcal{K}v(b))\Vert +\frac{\Vert \mathcal{K}\Vert \Vert \mathcal{G}\Vert (1-p)}{Q(p)\Gamma (p)}{\int }_{s_i}^l{(l-s)}^{p-1}\Vert f(s,v(s),\xi v(s))\Vert ds\hfill \\ +\frac{p\Vert {\mathcal{G}}^2\Vert }{Q(p)}{\int }_{s_i}^l\Vert S_p(l-s)\Vert \Vert f(s,v(s),\xi v(s))\Vert ds, q\in (s_i,q_{i+1}], i=\overline{1,m}.\hfill \end{array}\hfill \end{array}\hfill \end{array}$$

Thus, $$\begin{array}{c}\Vert u(v)(q)\Vert \le {\mathcal{R}}_W^{*}\{\begin{array}{c}\Vert v(b)\Vert +{\mathcal{R}}_{\mathcal{G}}{\mathcal{R}}_T[\Vert v_0\Vert +\Vert \varsigma (v)\Vert +\Vert h(0,v(0),0)\Vert ]+\Vert h(b,v(b),\mathcal{K}v(b))\Vert \hfill \\ +\frac{{\mathcal{R}}_{\mathcal{K}}{\mathcal{R}}_{\mathcal{G}}(1-p)}{Q(p)\Gamma (p+1)}{\mathcal{R}}_f^{**}{b}^p+\frac{{\mathcal{R}}_{\mathcal{G}}^{2}p}{Q(p)}{\mathcal{R}}_f^{**}{\mathcal{R}}_S{\int }_0^l{(l-s)}^{p-1}ds,q\in [0,q_1]\hfill \\ \Vert v(b)\Vert +{\mathcal{R}}_{\mathcal{G}}{\mathcal{R}}_T[({\mathcal{R}}_h^{**}(1+n)+{\mathcal{R}}_{g_i^{*}})\eta +\delta ]+{\mathcal{R}}_h^{**}(1+n)\eta \hfill \\ +\frac{{\mathcal{R}}_{\mathcal{G}}{\mathcal{R}}_f^{**}{b}^p}{Q(p)}[\frac{{\mathcal{R}}_{\mathcal{K}}(1-p)}{\Gamma (p+1)}+{\mathcal{R}}_{\mathcal{G}}{\mathcal{R}}_S], q\in (s_i,q_{i+1}], i=\overline{1,m}.\hfill \end{array}\hfill \end{array}$$

Let ${\mathcal{R}}_0={\mathcal{R}}_{\mathcal{G}}{\mathcal{R}}_T[\Vert \varsigma (0)\Vert +\Vert h(0,\varsigma (0),0)\Vert ],\exists {\Xi }_1, {\Xi }_2$ as positive number, s.t., (3.3) $$\begin{array}{c}\Vert u(v)(q)\Vert \le {\mathcal{R}}_W^{*}\{\begin{array}{c}\Vert v(b)\Vert +{\mathcal{R}}_0+{\mathcal{R}}_h^{**}(1+n)\eta +{\mathcal{R}}_f^{**}{\mathcal{R}}_p:={\Xi }_1, q\in [0,q_1],\hfill \\ \Vert v(b)\Vert +{\mathcal{R}}_{\mathcal{G}}{\mathcal{R}}_T[({\mathcal{R}}_h^{**}(1+n)+{\mathcal{R}}_{g_i^{*}})\eta +\delta ]+{\mathcal{R}}_h^{**}(1+n)\eta +{\mathcal{R}}_f^{**}{\mathcal{R}}_p:={\Xi }_2,\hfill \\ q\in (s_i,q_{i+1}], i=\overline{1,m}.\hfill \end{array}\hfill \end{array}$$(3.3)

Then for each $v\in {\Omega }_{\kappa },$ (3.3)(3.3) $$\begin{array}{c}\Vert u(v)(q)\Vert \le {\mathcal{R}}_W^{*}\{\begin{array}{c}\Vert v(b)\Vert +{\mathcal{R}}_0+{\mathcal{R}}_h^{**}(1+n)\eta +{\mathcal{R}}_f^{**}{\mathcal{R}}_p:={\Xi }_1, q\in [0,q_1],\hfill \\ \Vert v(b)\Vert +{\mathcal{R}}_{\mathcal{G}}{\mathcal{R}}_T[({\mathcal{R}}_h^{**}(1+n)+{\mathcal{R}}_{g_i^{*}})\eta +\delta ]+{\mathcal{R}}_h^{**}(1+n)\eta +{\mathcal{R}}_f^{**}{\mathcal{R}}_p:={\Xi }_2,\hfill \\ q\in (s_i,q_{i+1}], i=\overline{1,m}.\hfill \end{array}\hfill \end{array}$$(3.3) yields $$\begin{array}{c}\begin{array}{c}\Vert \Phi (v)(q)\Vert \le \{\begin{array}{c}{\mathcal{R}}_0+{\mathcal{R}}_h^{**}(1+n)\eta +\frac{{\mathcal{R}}_{\mathcal{G}}}{Q(p)}{b}^p[{\mathcal{R}}_{\mathcal{B}}^{*}{\Xi }_1+{\mathcal{R}}_f^{**}](\frac{{\mathcal{R}}_{\mathcal{K}}(1-p)}{\Gamma (p+1)}+{\mathcal{R}}_{\mathcal{G}}{\mathcal{R}}_S)\le \kappa , q\in [0,q_1]\hfill \\ {\mathcal{R}}_{\mathcal{G}}{\mathcal{R}}_T[({\mathcal{R}}_h^{**}(1+n)+{\mathcal{R}}_{g_i^{*}})\eta +\delta ]+{\mathcal{R}}_h^{**}(1+n)\eta \hfill \\ +\frac{{\mathcal{R}}_{\mathcal{G}}}{Q(p)}{b}^p[{\mathcal{R}}_{\mathcal{B}}^{*}{\Xi }_2+{\mathcal{R}}_f^{**}](\frac{{\mathcal{R}}_{\mathcal{K}}(1-p)}{\Gamma (p+1)}+{\mathcal{R}}_{\mathcal{G}}{\mathcal{R}}_S)\le \kappa , q\in (s_i,q_{i+1}], i=\overline{1,m}.\hfill \end{array}\hfill \end{array}\hfill \end{array}$$ $\therefore \Vert \Phi (v)\Vert \le \kappa$ implies $\Phi ({\Omega }_{\kappa })\subset {\Omega }_{\kappa }.$

Step 2: $\Phi$ on ${\Omega }_{\kappa }$ is continuous. For $t\in [0,q_1]$ and ${\{v^n\}}_{n\in \mathbb{N}}$ with $v^n\to v$ in ${\Omega }_{\kappa },$ provided $$\begin{array}{c}\Vert \Phi (v^n)(q)-\Phi (v)(q)\Vert \hfill \\ \le \Vert h(q,v^n(q),\mathcal{K}{v}^n(q))-h(q,v(q),\mathcal{K}v(q))\Vert \hfill \\ +\frac{\Vert \mathcal{K}\Vert \Vert \mathcal{G}\Vert (1-p)}{Q(p)\Gamma (p)}{\int }_0^q{(q-s)}^{p-1}[\Vert \mathcal{B}\Vert \Vert u_{v^n}(s)-u_v(s)\Vert +\Vert f(s,v^n(s),\xi v^n(s))-f(s,v(s),\xi v(s))\Vert ]ds\hfill \\ +\frac{P\Vert {\mathcal{G}}^2\Vert }{Q(p)}{\int }_0^q\Vert S_p(q-s)\Vert [\Vert \mathcal{B}\Vert \Vert u_{v^n}(s)-u_v(s)\Vert +\Vert f(s,v^n(s),\xi v^n(s))-f(s,v(s),\xi v(s))\Vert ]ds\hfill \end{array}$$ $$\begin{array}{c}\le \Vert h(q,v^n(q),\mathcal{K}{v}^n(q))-h(q,v(q),\mathcal{K}v(q))\Vert +\frac{{\mathcal{R}}_{\mathcal{K}}{\mathcal{R}}_{\mathcal{G}}(1-p)}{Q(p)\Gamma (p)}\hfill \\ (\times ){\int }_0^q{(q-s)}^{p-1}({\mathcal{R}}_{\mathcal{B}}^{*}(\frac{{\mathcal{R}}_{\mathcal{K}}{\mathcal{R}}_{\mathcal{G}}(1-p)}{Q(p)\Gamma (p)}{\int }_0^b{(b-\Xi )}^{p-1}\Vert f(\Xi ,v^n(\Xi ),\xi v^n(\Xi ))-f(\Xi ,v(\Xi ),\xi v(\Xi ))\Vert d\Xi \hfill \\ +\frac{p{\mathcal{R}}_{\mathcal{G}}^2}{Q(p)}{\mathcal{R}}_S{\int }_0^b{(b-\Xi )}^{p-1}\Vert f(\Xi ,v^n(\Xi ),\xi v^n(\Xi ))-f(\Xi ,v(\Xi ),\xi v(\Xi ))\Vert d\Xi )\hfill \\ +\Vert f(s,v^n(s),\xi v^n(s))-f(s,v(s),\xi v(s))\Vert )ds+\frac{p{\mathcal{R}}_{\mathcal{G}}^2}{Q(p)}{\mathcal{R}}_S\hfill \\ (\times ){\int }_0^q{(q-s)}^{p-1}({\mathcal{R}}_{\mathcal{B}}^{*}(\frac{M_{\mathcal{K}}{M}_{\mathcal{G}}(1-p)}{Q(p)\Gamma (p)}{\int }_0^b{(b-\Xi )}^{p-1}\Vert f(\Xi ,v^n(\Xi ),\xi v^n(\Xi ))-f(\Xi ,v(\Xi ),\xi v(\Xi ))\Vert d\Xi \hfill \\ +\frac{p{\mathcal{R}}_{\mathcal{G}}^2}{Q(p)}{\mathcal{R}}_S{\int }_0^b{(b-\Xi )}^{p-1}\Vert f(\Xi ,v^n(\Xi ),\xi v^n(\Xi ))-f(\Xi ,v(\Xi ),\xi v(\Xi ))\Vert d\Xi )\hfill \\ +\Vert f(s,v^n(s),\xi v^n(s))-f(s,v(s),\xi v(s))\Vert )ds.\hfill \end{array}$$

For $q\in (q_i,s_i], i\ge 1,$ $$\Vert \Phi (v^n)(q)-\Phi (v)(q)\Vert \le \Vert g_i(q,v^n(q))-g_i(q,v(q))\Vert \le {\mathcal{R}}_{g_i}\Vert v^n(q)-v(q)\Vert .$$

Further, for every $q\in (s_i,q_{i+1}],i=\overline{1,m},$ $$\begin{array}{l}\Vert \Phi (v^n)(q)-\Phi (v)(q)\Vert \\ \le {\mathcal{R}}_{\mathcal{G}}{\mathcal{R}}_T\Vert g_i(s_i,v^n(s_i))-h(s_i,v^n(s_i),\mathcal{K}{v}^n(s_i))-g_i(s_i,v(s_i))-h(s_i,v(s_i),\mathcal{K}v(s_i))\Vert \\ +\Vert h(q,v^n(q),\mathcal{K}{v}^n(q))-h(q,v(q),\mathcal{K}v(q))\Vert +\frac{\Vert \mathcal{K}\Vert \Vert \mathcal{G}\Vert (1-p)}{Q(p)\Gamma (p)}\\ (\times ){\int }_{q_i}^q{(q-s)}^{p-1}[\Vert \mathcal{B}\Vert \Vert u_{v^n}(s)-u_v(s)\Vert +\Vert f(s,v^n(s),\xi v^n(s))-f(s,v(s),\xi v(s))\Vert ]ds\\ +\frac{P\Vert {\mathcal{G}}^2\Vert }{Q(p)}{\int }_{q_i}^q\Vert S_p(q-s)\Vert [\Vert \mathcal{B}\Vert \Vert u_{v^n}(s)-u_v(s)\Vert +\Vert f(s,v^n(s),\xi v^n(s))-f(s,v(s),\xi v(s))\Vert ]ds\\ \le {\mathcal{R}}_{\mathcal{G}}{\mathcal{R}}_T\Vert g_i(s_i,v^n(s_i))-h(s_i,v^n(s_i),\mathcal{K}{v}^n(s_i))-g_i(s_i,v(s_i))-h(s_i,v(s_i),\mathcal{K}v(s_i))\Vert \\ +\Vert h(q,v^n(q),\mathcal{K}{v}^n(q))-h(q,v(q),\mathcal{K}v(q))\Vert +\frac{{\mathcal{R}}_{\mathcal{K}}{\mathcal{R}}_{\mathcal{G}}(1-p)}{Q(p)\Gamma (p)}\\ (\times ){\int }_{q_i}^q{(q-s)}^{p-1}({\mathcal{R}}_{\mathcal{B}}^{*}(\frac{{\mathcal{R}}_{\mathcal{K}}{\mathcal{R}}_{\mathcal{G}}(1-p)}{Q(p)\Gamma (p)}{\int }_{q_i}^b{(b-\Xi )}^{p-1}\Vert f(\Xi ,v^n(\Xi ),\xi v^n(\Xi ))-f(\Xi ,v(\Xi ),\xi v(\Xi ))\Vert d\Xi \\ +\frac{p{\mathcal{R}}_{\mathcal{G}}^2}{Q(p)}{\mathcal{R}}_S{\int }_{q_i}^b{(b-\Xi )}^{p-1}\Vert f(\Xi ,v^n(\Xi ),\xi v^n(\Xi ))-f(\Xi ,v(\Xi ),\xi v(\Xi ))\Vert d\Xi )\\ +\Vert f(s,v^n(s),\xi v^n(s))-f(s,v(s),\xi v(s))\Vert )ds+\frac{p{\mathcal{R}}_{\mathcal{G}}^2}{Q(p)}{\mathcal{R}}_S\\ (\times ){\int }_{q_i}^q{(q-s)}^{p-1}({\mathcal{R}}_{\mathcal{B}}^{*}(\frac{{\mathcal{R}}_{\mathcal{K}}{\mathcal{R}}_{\mathcal{G}}(1-p)}{Q(p)\Gamma (p)}{\int }_{q_i}^b{(b-\Xi )}^{p-1}\Vert f(\Xi ,v^n(\Xi ),\xi v^n(\Xi ))-f(\Xi ,v(\Xi ),\xi v(\Xi ))\Vert d\Xi \\ +\frac{p{\mathcal{R}}_{\mathcal{G}}^2}{Q(p)}{\mathcal{R}}_S{\int }_{q_i}^q{(b-\Xi )}^{p-1}\Vert f(\Xi ,v^n(\Xi ),\xi v^n(\Xi ))-f(\Xi ,v(\Xi ),\xi v(\Xi ))\Vert d\Xi )\\ +\Vert f(s,v^n(s),\xi v^n(s))-f(s,v(s),\xi v(s))\Vert )ds.\end{array}$$

Hence, $\underset{n\to \infty }{\lim }\Phi (v_n)=\Phi (v)$ in ${\Xi }_{\kappa }$ implies continuous of $\Phi .$

Step 3: $\{\Phi v:v\in {\Omega }_{\kappa }\}$ is equicontinuous.

Choose $v\in {\Omega }_{\kappa }\&{\lambda }_1, {\lambda }_2\in [0,q_1]$ with ${\lambda }_1<{\lambda }_2,$ $$\begin{array}{l}\Vert (\Phi v)({\lambda }_2)-(\Phi v)({\lambda }_1)\Vert \hfill \\ \le {\mathcal{R}}_{\mathcal{G}}\Vert T_p({\lambda }_2)-T_p({\lambda }_1)\Vert \Vert v_0-\varsigma (v)-h(0,v(0),0)\Vert +\Vert h({\lambda }_2,v({\lambda }_2),\mathcal{K}v({\lambda }_2))-h({\lambda }_1,v({\lambda }_1),\mathcal{K}v({\lambda }_1))\Vert \hfill \\ +\frac{{\mathcal{R}}_{\mathcal{K}}{\mathcal{R}}_{\mathcal{G}}(1-p)}{Q(p)\Gamma (p)}({\mathcal{R}}_{\mathcal{B}}^{**}{\Xi }_1+{\mathcal{R}}_f^{**}){\int }_0^{{\lambda }_1}(({\lambda }_2-s{)}^{p-1}-{({\lambda }_1-s)}^{p-1})ds\hfill \\ +\frac{p{\mathcal{R}}_{\mathcal{G}}^2}{Q(p)}({\mathcal{R}}_{\mathcal{B}}^{**}{\Xi }_1+{\mathcal{R}}_f^{**}){\int }_0^{{\lambda }_1}\Vert (S_p({\lambda }_2-s)-S_p({\lambda }_1-s))\Vert ds\hfill \end{array}$$ $$\begin{array}{l}+\frac{{\mathcal{R}}_{\mathcal{K}}{\mathcal{R}}_{\mathcal{G}}(1-p)}{Q(p)\Gamma (p)}({\mathcal{R}}_{\mathcal{B}}^{**}{\Xi }_1+{\mathcal{R}}_f^{**}){\int }_{{\lambda }_1}^{{\lambda }_2}{({\lambda }_2-s)}^{p-1}ds\\ +\frac{p{\mathcal{R}}_{\mathcal{G}}^2}{Q(p)}({\mathcal{R}}_{\mathcal{B}}^{**}{\Xi }_1+{\mathcal{R}}_f^{**}){\int }_{{\lambda }_1}^{{\lambda }_2}\Vert S_p({\lambda }_2-s)\Vert ds.\end{array}$$

Further, for ${\lambda }_1, {\lambda }_2\in (s_i,q_{i+1}], i=\overline{1,m},$ $$\begin{array}{c}\Vert (\Phi v)({\lambda }_2)-(\Phi v)({\lambda }_1)\Vert \hfill \\ \le {\mathcal{R}}_{\mathcal{G}}\Vert T_p({\lambda }_2)-T_p({\lambda }_1)\Vert \Vert g_i(s_i,v(s_i))-h(s_i,v(s_i),\mathcal{K}v(s_i))\Vert \hfill \\ +\Vert h({\lambda }_2,v({\lambda }_2),\mathcal{K}v({\lambda }_2))-h({\lambda }_1,v({\lambda }_1),\mathcal{K}v({\lambda }_1))\Vert \hfill \\ +\frac{{\mathcal{R}}_{\mathcal{K}}{\mathcal{R}}_{\mathcal{G}}(1-p)}{\Gamma (p)Q(p)}({\mathcal{R}}_{\mathcal{B}}^{**}{\Xi }_2+{\mathcal{R}}_f^{**}){\int }_{q_i}^{{\lambda }_1}(({\lambda }_2-s{)}^{p-1}-{({\lambda }_1-s)}^{p-1})ds\hfill \\ +\frac{p{\mathcal{R}}_{\mathcal{G}}^2}{Q(p)}({\mathcal{R}}_f^{**}+{\mathcal{R}}_{\mathcal{B}}^{**}{\Xi }_2){\int }_{q_i}^{{\lambda }_1}\Vert (S_p({\lambda }_2-s)-S_p({\lambda }_1-s))\Vert ds\hfill \\ +\frac{{\mathcal{R}}_{\mathcal{K}}{\mathcal{R}}_{\mathcal{G}}(1-p)}{\Gamma (p)Q(p)}({\mathcal{R}}_{\mathcal{B}}^{**}{\Xi }_2+{\mathcal{R}}_f^{**}){\int }_{{\lambda }_1}^{{\lambda }_2}{({\lambda }_2-s)}^{p-1}ds\hfill \\ +\frac{p{\mathcal{R}}_{\mathcal{G}}^2}{Q(p)}({\mathcal{R}}_{\mathcal{B}}^{**}{\Xi }_2+{\mathcal{R}}_f^{**}){\int }_{{\lambda }_1}^{{\lambda }_2}\Vert S_p({\lambda }_2-s)\Vert ds,\hfill \end{array}$$ $\to 0 \text{as} {\lambda }_2\to {\lambda }_1,$ equicontinuity of $\Phi ({\Omega }_{\kappa }).$

Step 4: $\Phi :{\Omega }_{\kappa }\to {\Omega }_{\kappa },$a set contraction.

For any $\mathcal{D}\subseteq {\Omega }_{\kappa },$a countable set ${\{v^n\}}_{n=1}^{\infty }\subset \mathcal{D}$ exists, then $\zeta (\Phi (\mathcal{D})(q))\le 2\zeta ({\{v^n\}}_{n=1}^{\infty })$ by the equicontinuity on ${\Omega }_{\kappa }.$ By $\mathbf{H}\mathbf{8},$ and for any $q\in [0,q_1],$ (3.4) $$\begin{array}{c}\zeta ({\{\frac{\mathcal{K}\mathcal{G}(1-p)}{Q(p)\Gamma (p)}{\int }_0^q{(q-s)}^{p-1}f(s,v^n(s),\xi v^n(s))ds\}}_{n=1}^{\infty })\hfill \\ \le 2\frac{{\mathcal{R}}_{\mathcal{K}}{\mathcal{R}}_{\mathcal{G}}(1-p)}{Q(p)\Gamma (p)}{\int }_0^q{(q-s)}^{p-1}(L_f(\underset{0\le \tau \le s}{\sup }\zeta {(\{v^n(\tau )\})}_{n=1}^{\infty }+b\zeta {(\{v^n(\tau )\})}_{n=1}^{\infty }))ds\hfill \\ \le 2\frac{{\mathcal{R}}_{\mathcal{K}}{\mathcal{R}}_{\mathcal{G}}(1-p)}{Q(p)\Gamma (p)}{b}^p(L_f(1+n)\zeta ({\{v^n(\tau )\}}_{n=1}^{\infty })).\hfill \end{array}$$(3.4)

Similarly, (3.5) $$\begin{array}{l}\zeta ({\{\frac{p{\mathcal{R}}_{\mathcal{G}}^2}{Q(p)}{\int }_0^q{S}_p(q-s)f(s,v^n(s),\xi v^n(s))ds\}}_{n=1}^{\infty })\\ \le 2\frac{{\mathcal{R}}_{\mathcal{G}}^2}{Q(p)}{b}^p{\mathcal{R}}_S{L}_f(1+n)\zeta ({\{v^n\}}_{n=1}^{\infty }).\end{array}$$(3.5)

Using (3.4)(3.4) $$\begin{array}{c}\zeta ({\{\frac{\mathcal{K}\mathcal{G}(1-p)}{Q(p)\Gamma (p)}{\int }_0^q{(q-s)}^{p-1}f(s,v^n(s),\xi v^n(s))ds\}}_{n=1}^{\infty })\hfill \\ \le 2\frac{{\mathcal{R}}_{\mathcal{K}}{\mathcal{R}}_{\mathcal{G}}(1-p)}{Q(p)\Gamma (p)}{\int }_0^q{(q-s)}^{p-1}(L_f(\underset{0\le \tau \le s}{\sup }\zeta {(\{v^n(\tau )\})}_{n=1}^{\infty }+b\zeta {(\{v^n(\tau )\})}_{n=1}^{\infty }))ds\hfill \\ \le 2\frac{{\mathcal{R}}_{\mathcal{K}}{\mathcal{R}}_{\mathcal{G}}(1-p)}{Q(p)\Gamma (p)}{b}^p(L_f(1+n)\zeta ({\{v^n(\tau )\}}_{n=1}^{\infty })).\hfill \end{array}$$(3.4) , (3.5)(3.5) $$\begin{array}{l}\zeta ({\{\frac{p{\mathcal{R}}_{\mathcal{G}}^2}{Q(p)}{\int }_0^q{S}_p(q-s)f(s,v^n(s),\xi v^n(s))ds\}}_{n=1}^{\infty })\\ \le 2\frac{{\mathcal{R}}_{\mathcal{G}}^2}{Q(p)}{b}^p{\mathcal{R}}_S{L}_f(1+n)\zeta ({\{v^n\}}_{n=1}^{\infty }).\end{array}$$(3.5) , we get $$\zeta ({\{u_{v^n}(q)\}}_{n=1}^{\infty })$$ $$=\zeta (\{w^{-1}[v(b)-\mathcal{G}{T}_p(b)[v_0-\varsigma (v)-h(0,v(0),0)]-h(b,v(b),\mathcal{K}v(b))$$ $$-\frac{\mathcal{K}G(1-p)}{\Gamma (p)Q(p)}{\int }_0^l{(l-s)}^{p-1}f(s,v(s),\xi v(s))ds-\frac{p{\mathcal{G}}^2}{Q(p)}{\int }_0^l{S}_p(l-s)f(s,v^n(s),\xi v^n(s))ds]{\}}_{n=1}^{\infty })$$ $$\le 2{\mathcal{R}}_W^{*}{b}^p(\frac{{\mathcal{R}}_{\mathcal{K}}{\mathcal{R}}_{\mathcal{G}}(1-p)}{Q(p)\Gamma (1+p)}+\frac{{\mathcal{R}}_{\mathcal{G}}^2}{Q(p)}{\mathcal{R}}_S)L_f(1+n)\zeta ({\{v^n\}}_{n=1}^{\infty })$$ (3.6) $$\cong 2{\mathcal{R}}_W^{*}{\mathcal{R}}_p{L}_f(1+n)\zeta ({\{v^n\}}_{n=1}^{\infty })$$(3.6) (3.7) $$\cong {\mathcal{N}}_p\zeta (\mathcal{D}).$$(3.7)

By (3.4)–(3.6)(3.6) $$\cong 2{\mathcal{R}}_W^{*}{\mathcal{R}}_p{L}_f(1+n)\zeta ({\{v^n\}}_{n=1}^{\infty })$$(3.6) , we get (3.8) $$\begin{array}{c}\zeta (\Phi (\mathcal{D})(q))=\zeta ({\{\Phi (v^n)(q)\}}_{n=1}^{\infty })\hfill \\ =\zeta (\{\mathcal{G}{T}_p(q)[v_0-\varsigma (v)-h(0,v(0),0)]+h(q,v^n(q),\mathcal{K}{v}^n(q))+\frac{\mathcal{K}\mathcal{G}(1-p)}{Q(p)\Gamma (p)}{\int }_0^q{(q-s)}^{p-1}[\mathcal{B}{u}_{v^n}(s)\hfill \\ +f(s,v^n(s),\xi v^n(s))]ds+\frac{p{\mathcal{G}}^2}{Q(p)}{\int }_0^q{S}_p(q-s)[\mathcal{B}{u}_{v^n}(s)+f(s,v^n(s),\xi v^n(s))]{\}}_{n=1}^{\infty })\hfill \\ \le L_b(1+n)\zeta ({\{v^n(q)\}}_{n=1}^{\infty })+2(\frac{{\mathcal{R}}_{\mathcal{K}}{\mathcal{R}}_{\mathcal{G}}(1-p)}{Q(p)\Gamma (p)}{\int }_0^q{(q-s)}^{p-1}[{\mathcal{R}}_{\mathcal{B}}^{*}\zeta ({\{u_{v^n}(s)\}}_{n=1}^{\infty })\hfill \\ +\zeta ({\{f(s,v_s^n,\xi v^n(s))\}}_{n=1}^{\infty }ds)]ds+\frac{p{M}_{\mathcal{G}}^2}{Q(p)}{\mathcal{R}}_S{\int }_0^q{(q-s)}^{p-1}[{\mathcal{R}}_{\mathcal{B}}^{*}\zeta ({\{u_{v^n}(s)\}}_{n=1}^{\infty })\hfill \\ +\zeta ({\{f(s,v_s^n,\xi v^n(s))\}}_{n=1}^{\infty }ds)]ds)\hfill \\ \le [L_b(1+n)+2b^p({\mathcal{R}}_{\mathcal{B}}^{*}{\mathcal{N}}_p+L_f(1+n))(\frac{{\mathcal{R}}_{\mathcal{K}}{\mathcal{R}}_{\mathcal{G}}(1-p)}{Q(p)\Gamma (p+1)}+\frac{M_{\mathcal{G}}^2}{Q(p)}{\mathcal{R}}_S)]\nu (\mathcal{D}).\hfill \end{array}$$(3.8)

Also, for $q\in (q_i,s_i] ,\zeta (g_i(s_i,v(s_i))\le L_{g_i}\zeta (\mathcal{D}), i=\overline{1,m}$ is obviously hold.

Therefore, $\Phi v(b)=v_1,$ $(1.1)$ is controllable. □

4. Optimal control

Let us focus the Lagrange problem [LP] (Liu & Wang, 2017): a control $(v^0,u^0)\in C_{1-v}(I,Y)\times U_{ad}$ s.t. $J(v^0,u^0)\le J(v,u),$ $\forall u\in U_{ad}$ and (4.1) $$J(v,u)={\int }_0^T\mathfrak{L}(q,v(q),u(q))dt.$$(4.1)

Consider the multivalued maps $u(\cdot ):I\overrightarrow{\to }{2}^Y/\left\{\varphi \right\},$ where Y be a separable reflexive Banach space. Let us denote that, $$\Psi u=\{v^n\in {\Omega }_{\mathcal{K}};v^n\text{is the solution to u}\in U_{ad}\}$$

We assume,

H9

1. $\mathfrak{L}(q,\cdot ,\cdot )$ is semi-continuous on $Y\times U$ for $q\in I.$

2. $\mathfrak{L}(q,v,\cdot )$ is convex on U for all $v_1,v_2\in Y,$ $q\in I.$

3. $\mathfrak{L}:I\times Y\times U\to R\cup \left\{\infty \right\}$ is Borel measurable.

4. Let $\mathcal{K}\ge 0,$ $l>0,$ $\zeta$ is non-negative $\text{\&} \zeta \in {\mathcal{L}}^{{}^{\prime }}(I,\mathcal{R}),$

$$\mathfrak{L}(q,v,u)\ge \zeta (q)+\mathcal{K}\Vert v{\Vert }_Y+l\Vert u{\Vert }_U.$$

Theorem 4.1.

If [H1–H9] holds. Then, the [LP] has optimal pair.

Proof.

If $\inf \{J(v,u)|(v,u)\in C_{1-v}(I,Y)\times U_{ad}\}=+\infty ,$ then the proof is: $$\inf \{J(v,u)|(v,u)\in C_{1-v}(I,Y)\times U_{ad}\}=\gamma <+\infty$$

By [H9] (iv), $\gamma >-\infty .$ By infimum defn $\exists$a minimizing sequence $\{(v^n,u^n)\}\subset A_{ad}\equiv \{(v,u)|v \text{is a soln of}(1.1)\text{to} u \in U_{ad}\},$ s. t. $J(v^n,u^n)\to \gamma$ as $n\to +\infty .$

Since $\left\{u^n\right\}\subseteq U_{ad}\in L^P(I,\mathcal{U})\forall n\in N$ is bounded, $\exists$a subsequence, $\left\{u^n\right\},$ $u^0\in L^P(I,\mathcal{U})$ s. t. $u^n\to u^0$ in $L^P(I,\mathcal{U}).$ Since $U_{ad}$ is convex, closed and $u^0\in U_{ad},$ by Mazur lemma.

From Step 1 and Step 3 in Theorem 3.1, the $\left\{v^n\right\}$ states that uniformly bounded and equicontinuous. Now, we need to prove ${\left\{v^n\right\}}_{n=1}^{\infty }$ is relatively compact. From (1.1)(1.1) $$\begin{array}{l}{}^{\mathit{ABC}}{\mathcal{D}}^p[v(q)-h(q,v(q),{\int }_0^q\mathcal{K}(q,s,v(s))ds)]\\ =\mathbb{A}[v(q)-h(q,v(q),{\int }_0^q\mathcal{K}(q,s,v(s))ds)]+\mathcal{B}u(q)\\ +f(q,v(q),{\int }_0^q\xi (q,s,v(s))ds), q\in \underset{i=0}{\overset{m}{\cup }}(s_i,q_{i+1}]\\ v(q)=g_i(q,v(q)), q\in \underset{i=1}{\overset{m}{\cup }}(s_i,q_i]\\ v(0)+\varsigma (v)=v_0\in V,\end{array}$$(1.1) , we have $$\begin{array}{cc}{v}^n(q)\hfill & =\{\begin{array}{c}\mathcal{G}{T}_p(q)[v_0-\varsigma (v)-h(0,v(0),0)]+h(q,v^n(q),\mathcal{K}{v}^n(q))+\frac{\mathcal{K}\mathcal{G}(1-p)}{Q(p)\Gamma (p)}{\int }_0^q{(q-s)}^{p-1}\hfill \\ \times [\mathcal{B}{u}_{v^n}(s)+f(s,v^n(s),\xi v^n(s))]ds\hfill \\ +\frac{p{\mathcal{G}}^2}{Q(p)}{\int }_0^q{S}_p(q-s)[\mathcal{B}{u}_{v^n}(s)+f(s,v^n(s),\xi v^n(s))]ds,q\in [0,q_1],i=0\hfill \\ \mathcal{G}{T}_p(q-s_i)[g_i(s_i,v^n(s_i))-h(s_i,v^n(s_i),\mathcal{K}{v}^n(s_i))]+h(q,v^n(q),\mathcal{K}{v}^n(q))\hfill \\ +\frac{\mathcal{K}\mathcal{G}(1-p)}{Q(p)\Gamma (p)}{\int }_{s_i}^q{(q-s)}^{p-1}[\mathcal{B}{u}_{v^n}(s)+f(s,v^n(s),\xi v^n(s))]ds\hfill \\ +\frac{p{\mathcal{G}}^2}{Q(p)}{\int }_{s_i}^q{S}_p(q-s)[\mathcal{B}{u}_{v^n}(s)+f(s,v^n(s),\xi v^n(s))]ds,q\in (s_i,q_{i+1}],i=\overline{0,m}.\hfill \end{array}\hfill \end{array}$$ where, $$v^n(q)=({\Phi }_1{v}^n)(q)+({\Phi }_2{v}^n)(q)=(\Phi v^n)(q)$$

To prove $(\Xi v^n)(q)=\{{\Phi }_2{v}^n(q);v^n(q)\in {\Omega }_k\}$ is relatively compact $\mathcal{P}\mathcal{C}(I,\mathcal{V})\forall q\in I.$ For any $u\in U,q\in I,v^n\in {\Omega }_{\mathcal{K}},$ and $ϵ\in (0,q_i-s_i),$ we define $$\begin{array}{c}({\Phi }_2{v}^n)(q)=\{\begin{array}{c}\frac{\mathcal{K}\mathcal{G}(1-p)}{Q(p)\Gamma (p)}{\int }_0^q{(q-s)}^{p-1}[\mathcal{B}{u}_{v^n}(s)+f(s,v^n(s),\xi v^n(s))]ds\hfill \\ +\frac{p{\mathcal{G}}^2}{Q(p)}{\int }_0^q{S}_p(q-s)[\mathcal{B}{u}_{v^n}(s)+f(s,v^n(s),\xi v^n(s))]ds,q\in [0,q_1],i=0\hfill \\ \frac{\mathcal{K}\mathcal{G}(1-p)}{Q(p)\Gamma (p)}{\int }_{s_i}^q{(q-s)}^{p-1}[\mathcal{B}{u}_{v^n}(s)+f(s,v^n(s),\xi v^n(s))]ds\hfill \\ +\frac{p{\mathcal{G}}^2}{Q(p)}{\int }_{s_i}^q{S}_p(q-s)[\mathcal{B}{u}_{v^n}(s)+f(s,v^n(s),\xi v^n(s))]ds,q\in (s_i,q_{i+1}],i=\overline{0,m}.\hfill \end{array}\hfill \end{array}$$

By the property of $U_{ad},$ [H1], [H3], [H4] the set ${\Pi }_{ϵ}=\{\mathcal{B}{u}_{v^n}(s)+f(s,v^n(s),\xi v^n(s));0\le s_i\le q_i-ϵ\}$ is relatively compact. By Lemma 2.3, we get $({\Phi }_2^{ϵ}{v}^n)(q)\in \overline{{\Pi }_{ϵ}}$ for every $q\in I.$ Hence ${\Xi }_{ϵ}(q)=\{({\Phi }_2^{ϵ}{v}^n)(q):v^n\in {\Omega }_{\mathcal{K}}\}$ is relatively compact in $\mathcal{P}\mathcal{C}(I,\mathcal{V}).$

From $[H0]-[H8],$ for $q\in (0,q_1]$ $$\begin{array}{c}\Vert ({\Phi }_2{v}^n)(q)-({\Phi }_2^{ϵ}{v}^n)(q)\Vert \hfill \\ \le \frac{\Vert \mathcal{K}\Vert \Vert \mathcal{G}\Vert (1-p)}{Q(p)\Gamma (p)}{\int }_0^q{(q-s)}^{p-1}[\Vert \mathcal{B}\Vert \Vert u_{v^n}(s)\Vert +\Vert f(s,v^n(s),\xi v^n(s))\Vert ]ds\hfill \\ \hspace{1em}-\frac{\Vert \mathcal{K}\Vert \Vert \mathcal{G}\Vert (1-p)}{Q(p)\Gamma (p)}{\int }_0^{q-ϵ}{(q-s)}^{p-1}[\Vert \mathcal{B}\Vert \Vert u_{v^n}(s)\Vert +\Vert f(s,v^n(s),\xi v^n(s))\Vert ]ds\hfill \end{array}$$ $$\begin{array}{c}\hspace{1em}+\frac{P\Vert {\mathcal{G}}^2\Vert }{Q(p)}{\int }_0^q\Vert S_p(q-s)\Vert [\Vert \mathcal{B}\Vert \Vert u_{v^n}(s)\Vert +\Vert f(s,v^n(s),\xi v^n(s))\Vert ]ds\hfill \\ \hspace{1em}-\frac{P\Vert {\mathcal{G}}^2\Vert }{Q(p)}{\int }_0^q\Vert S_p(q-s)\Vert [\Vert \mathcal{B}\Vert \Vert u_{v^n}(s)\Vert +\Vert f(s,v^n(s),\xi v^n(s))\Vert ]ds\hfill \\ \le \frac{\Vert \mathcal{K}\Vert \Vert \mathcal{G}\Vert (1-p)}{Q(p)\Gamma (p)}{\int }_{q-ϵ}^q{(q-s)}^{p-1}[\Vert \mathcal{B}\Vert \Vert u_{v^n}(s)\Vert +\Vert f(s,v^n(s),\xi v^n(s))\Vert ]ds\hfill \\ \hspace{1em}+\frac{P\Vert {\mathcal{G}}^2\Vert }{Q(p)}{\int }_{q-ϵ}^q\Vert S_p(q-s)\Vert [\Vert \mathcal{B}\Vert \Vert u_{v^n}(s)\Vert +\Vert f(s,v^n(s),\xi v^n(s))\Vert ]ds\hfill \\ \le (\frac{{\mathcal{R}}_{\mathcal{K}}{\mathcal{R}}_{\mathcal{G}}(1-p)}{Q(p)\gamma (p)}+\frac{p{\mathcal{R}}_{\mathcal{G}}^2}{Q(p)}{\mathcal{R}}_s)[{\mathcal{R}}_{\mathcal{B}}^{\ast }{w}_1+{\mathcal{R}}_f^{\ast \ast }]{\int }_{q-ϵ}^q{(q-s)}^{p-1}ds\hfill \end{array}$$

For $q\in (q_i,s_i], i\ge 1,$ $$\begin{array}{c}\Vert ({\Phi }_2{v}^n)(q)-({\Phi }_2^{ϵ}{v}^n)(q)\Vert \hfill \\ \le \frac{\Vert \mathcal{K}\Vert \Vert \mathcal{G}\Vert (1-p)}{Q(p)\Gamma (p)}{\int }_{q_i}^q{(q-s)}^{p-1}[\Vert \mathcal{B}\Vert \Vert u_{v^n}(s)\Vert +\Vert f(s,v^n(s),\xi v^n(s))\Vert ]ds\hfill \\ \hspace{1em}-\frac{\Vert \mathcal{K}\Vert \Vert \mathcal{G}\Vert (1-p)}{Q(p)\Gamma (p)}{\int }_{q_i}^{q-ϵ}{(q-s)}^{p-1}[\Vert \mathcal{B}\Vert \Vert u_{v^n}(s)\Vert +\Vert f(s,v^n(s),\xi v^n(s))\Vert ]ds\hfill \\ \hspace{1em}+\frac{P\Vert {\mathcal{G}}^2\Vert }{Q(p)}{\int }_{q_i}^q\Vert S_p(q-s)\Vert [\Vert \mathcal{B}\Vert \Vert u_{v^n}(s)\Vert +\Vert f(s,v^n(s),\xi v^n(s))\Vert ]ds\hfill \\ \hspace{1em}-\frac{P\Vert {\mathcal{G}}^2\Vert }{Q(p)}{\int }_{q_i}^q\Vert S_p(q-s)\Vert [\Vert \mathcal{B}\Vert \Vert u_{v^n}(s)\Vert +\Vert f(s,v^n(s),\xi v^n(s))\Vert ]ds\hfill \\ \le \frac{\Vert \mathcal{K}\Vert \Vert \mathcal{G}\Vert (1-p)}{Q(p)\Gamma (p)}{\int }_{q-ϵ}^q{(q-s)}^{p-1}[\Vert \mathcal{B}\Vert \Vert u_{v^n}(s)\Vert +\Vert f(s,v^n(s),\xi v^n(s))\Vert ]ds\hfill \\ \hspace{1em}+\frac{P\Vert {\mathcal{G}}^2\Vert }{Q(p)}{\int }_{q-ϵ}^q\Vert S_p(q-s)\Vert [\Vert \mathcal{B}\Vert \Vert u_{v^n}(s)\Vert +\Vert f(s,v^n(s),\xi v^n(s))\Vert ]ds\hfill \\ \le (\frac{{\mathcal{R}}_{\mathcal{K}}{\mathcal{R}}_{\mathcal{G}}(1-p)}{Q(p)\gamma (p)}+\frac{p{\mathcal{R}}_{\mathcal{G}}^2}{Q(p)}{\mathcal{R}}_s)[{\mathcal{R}}_{\mathcal{B}}^{\ast }{w}_2+{\mathcal{R}}_f^{\ast \ast }]{\int }_{q-ϵ}^q{(q-s)}^{p-1}ds\hfill \end{array}$$ leads to $\underset{ϵ\to 0}{\lim }\Vert ({\Phi }_2{v}^n)(q)-({\Phi }_2^{ϵ}{v}^n)(q)\Vert =0.$ Hence $\Xi (q)$ is relatively compact in $\mathcal{P}\mathcal{C}(I,\mathcal{V})\forall q\in I.$ From [H0] and [H6], ${\Phi }_1{v}^n$ is bounded and equicontinuous in ${\Omega }_{\mathcal{K}}.$ By (3.8)(3.8) $$\begin{array}{c}\zeta (\Phi (\mathcal{D})(q))=\zeta ({\{\Phi (v^n)(q)\}}_{n=1}^{\infty })\hfill \\ =\zeta (\{\mathcal{G}{T}_p(q)[v_0-\varsigma (v)-h(0,v(0),0)]+h(q,v^n(q),\mathcal{K}{v}^n(q))+\frac{\mathcal{K}\mathcal{G}(1-p)}{Q(p)\Gamma (p)}{\int }_0^q{(q-s)}^{p-1}[\mathcal{B}{u}_{v^n}(s)\hfill \\ +f(s,v^n(s),\xi v^n(s))]ds+\frac{p{\mathcal{G}}^2}{Q(p)}{\int }_0^q{S}_p(q-s)[\mathcal{B}{u}_{v^n}(s)+f(s,v^n(s),\xi v^n(s))]{\}}_{n=1}^{\infty })\hfill \\ \le L_b(1+n)\zeta ({\{v^n(q)\}}_{n=1}^{\infty })+2(\frac{{\mathcal{R}}_{\mathcal{K}}{\mathcal{R}}_{\mathcal{G}}(1-p)}{Q(p)\Gamma (p)}{\int }_0^q{(q-s)}^{p-1}[{\mathcal{R}}_{\mathcal{B}}^{*}\zeta ({\{u_{v^n}(s)\}}_{n=1}^{\infty })\hfill \\ +\zeta ({\{f(s,v_s^n,\xi v^n(s))\}}_{n=1}^{\infty }ds)]ds+\frac{p{M}_{\mathcal{G}}^2}{Q(p)}{\mathcal{R}}_S{\int }_0^q{(q-s)}^{p-1}[{\mathcal{R}}_{\mathcal{B}}^{*}\zeta ({\{u_{v^n}(s)\}}_{n=1}^{\infty })\hfill \\ +\zeta ({\{f(s,v_s^n,\xi v^n(s))\}}_{n=1}^{\infty }ds)]ds)\hfill \\ \le [L_b(1+n)+2b^p({\mathcal{R}}_{\mathcal{B}}^{*}{\mathcal{N}}_p+L_f(1+n))(\frac{{\mathcal{R}}_{\mathcal{K}}{\mathcal{R}}_{\mathcal{G}}(1-p)}{Q(p)\Gamma (p+1)}+\frac{M_{\mathcal{G}}^2}{Q(p)}{\mathcal{R}}_S)]\nu (\mathcal{D}).\hfill \end{array}$$(3.8) , we have $$\begin{array}{cc}\zeta {(v^n)}_{\mathcal{P}\mathcal{C}(I,\mathcal{V})}\hfill & \le [L_b(1+n)+2b^p({\mathcal{R}}_{\mathcal{B}}^{*}{\mathcal{N}}_p+L_f(1+n))(\frac{{\mathcal{R}}_{\mathcal{K}}{\mathcal{R}}_{\mathcal{G}}(1-p)}{Q(p)\Gamma (p+1)}+\frac{{\mathcal{R}}_{\mathcal{G}}^2}{Q(p)}{\mathcal{R}}_S)]\zeta {(v^n)}_{\mathcal{P}\mathcal{C}(I,\mathcal{V})}\hfill \end{array}$$ which implies that $\zeta ({\left\{v^n\right\}}_{n=1}^{\infty })=0$ by using (3.1)(3.1) $$L_b(1+n)+2({\mathcal{R}}_{\mathcal{B}}^{*}{\mathcal{N}}_p+L_f(1+n){\mathcal{R}}_p)<1,$$(3.1) .

$\therefore {\left\{v^n\right\}}_{n=1}^{\infty }$ is relatively compact in $\mathcal{P}\mathcal{C}(I, \mathcal{V})$ and $u\in U_{ad}.$ $\exists$ a subsequence $v^0\in {\Omega }_{\mathcal{K}}$ of ${\left\{v^n\right\}}_{n=1}^{\infty }$ such that $\underset{n\to \infty }{\lim }{v}^n\to v^0.$

By [H0]–[H4], [H6] and Lebesgue theorem, $$\begin{array}{cc}\tilde{v}(q)\hfill & =\{\begin{array}{c}\mathcal{G}{T}_p(q)[v_0-\varsigma (v)-h(0,v(0),0)]+h(q,\tilde{v}(q),\mathcal{K}\tilde{v}(q))+\frac{\mathcal{K}\mathcal{G}(1-p)}{Q(p)\Gamma (p)}{\int }_0^q{(q-s)}^{p-1}\hfill \\ \times [\mathcal{B}{u}_{\tilde{v}}(s)+f(s,\tilde{v}(s),\xi \tilde{v}(s))]ds\hfill \\ +\frac{p{\mathcal{G}}^2}{Q(p)}{\int }_0^q{S}_p(q-s)[\mathcal{B}{u}_{\tilde{v}}(s)+f(s,\tilde{v}(s),\xi \tilde{v}(s))]ds,q\in [0,q_1],i=0\hfill \\ \mathcal{G}{q}_p(q-s_i)[g_i(s_i,\tilde{v}(s_i))-h(s_i,\tilde{v}(s_i),\mathcal{K}\tilde{v}(s_i))]+h(q,\tilde{v}(q),\mathcal{K}\tilde{v}(q))\hfill \\ +\frac{\mathcal{K}\mathcal{G}(1-p)}{Q(p)\Gamma (p)}{\int }_{s_i}^q{(q-s)}^{p-1}[\mathcal{B}{u}_{\tilde{v}}(s)+f(s,\tilde{v}(s),\xi \tilde{v}(s))]ds\hfill \\ +\frac{p{\mathcal{G}}^2}{Q(p)}{\int }_{s_i}^q{S}_p(q-s)[\mathcal{B}{u}_{\tilde{v}}(s)+f(s,\tilde{v}(s),\xi \tilde{v}(s))]ds,q\in (s_i,q_{i+1}],i=\overline{0,m}.\hfill \end{array}\hfill \end{array}$$

So, $\tilde{v}\in \Psi (u)$ is continuously embedded in ${\mathfrak{L}}^{{}^{\prime }}(I,\mathcal{R}),$ by [H9] & Balder’s theorem (Balder, 1987), $$J(u)=\underset{n\to \infty }{\lim }{\int }_0^T\mathfrak{L}(q,v^n(q),u(q))dt\ge {\int }_0^T\mathfrak{L}(q,\tilde{v}(q),u(q))dt=J(\tilde{v},u)\ge J(u).$$ which shows $J(\tilde{v},u)\to J(u).$ Hence, $J(u)$ attains its minimum at $\tilde{v}\in \Psi (u)$ for every $u\in U_{ad}.$

Consequently, take $u^0\in U_{ad},$ i.e. $J(u)=\underset{u\in U_{ad}}{\inf }J(u).$ Due to the boundedness of ${\left\{u^n\right\}}_{n=0}^{\infty }$ in $L_P(I,Y)$ for $P>1,$ for a subsequence $u^0\in L^P(I,\mathcal{U})$ and by relative compactness of $v^n$ is a subsequence ${\tilde{v}}^0\in PC(I,\mathcal{V})$ as $\underset{n\to \infty }{\lim }{v}^n\to {\tilde{v}}^0.$ Since $PC(I,\mathcal{V})\to {\mathfrak{L}}^{{}^{\prime }}(I,\mathcal{R})$ is continuous with [H9], we get $$\begin{array}{cc}\underset{u\in U_{ad}}{\inf }J(u)\hfill & =\underset{n\to \infty }{\lim }{\int }_0^T\mathcal{L}(q,v^n(q),u^n(q))dq\ge {\int }_0^T\mathcal{L}(q,{\tilde{v}}^0(q),u^0(q))dt=J({\tilde{v}}^0,u^0)\hfill \\ \hspace{1em}\hfill & =J(u^0)\ge \underset{u\in U_{ad}}{\inf }J(u),\hfill \end{array}$$ $\therefore J(u)=\underset{u\in U_{ad}}{\inf }J(u),$ J attains its minimum at $u^0\in U_{ad},$ $$J({\tilde{v}}^0,u^0)=\underset{u\in U_{ad}}{\inf }J(u)=\underset{(v,u)\in A_{ad}}{\inf }J(v,u)$$ $\therefore \mathfrak{L}$ admits optimal pair $({\tilde{v}}^0,u^0)\in A_{ad}.$ □

5. Example

Let us consider the partial FDE: (5.1) $$\begin{array}{c}{}^{\mathit{ABC}}{\mathcal{D}}^p[\Xi (\nu , ϵ)-\frac{1}{4}\frac{e^{-\nu }}{1+e^{-\nu }}+{\int }_0^{\nu }\frac{1}{20}{e}^{-s}\Xi (\nu , ϵ)ds]\hfill \\ =\frac{{\partial }^2}{\partial {ϵ}^2}[\Xi (\nu , ϵ)-\frac{1}{4}\frac{e^{-\nu }}{1+e^{-\nu }}+{\int }_0^{\nu }\frac{1}{20}{e}^{-s}\Xi (\nu , ϵ)ds]\hfill \\ +Bu(\nu , ϵ)+\frac{1}{9}\frac{e^{-\nu }}{1+e^{-\nu }}+{\int }_0^{\nu }\frac{1}{36}{e}^{-s}\Xi (\nu , ϵ)ds, \nu \in (0,\frac{1}{5}]\cup (\frac{4}{5},1]\hfill \\ \alpha \Xi (\nu , 0)=\Xi (\nu , \pi )=0, \nu \in [0,1]\hfill \\ \Xi (\nu , ϵ)=\frac{e^{-(\frac{5\nu -1}{5})}}{9} \frac{|\Xi (\nu , ϵ)|}{1+|\Xi (\nu , ϵ)|}, \nu \in (0,\frac{1}{5}]\cup (\frac{4}{5},1]\hfill \\ \Xi (0, ϵ)+\sum _{i=1}^5\frac{1}{5^i}\Xi (\frac{1}{i}, ϵ)={\Xi }_0(ϵ), ϵ\in [0,1].\hfill \end{array}$$(5.1)

Let $\mathcal{V}=L^2[0, \pi ],$ $\mathbb{A}:D(\mathbb{A})\subset \mathcal{V}\to \mathcal{V}$ define as $\mathbb{A}\mathfrak{I}={\mathfrak{I}}^{{}^{″}}, \mathfrak{I}\in D(\mathbb{A})$ and $$D(\mathbb{A})=\{\mathfrak{I}\in \mathcal{V}, \mathfrak{I},{\mathfrak{I}}^{{}^{″}} \text{absolutely continuous}, {\mathfrak{I}}^{{}^{″}}\in \mathcal{V}, \mathfrak{I}(0)=\mathfrak{I}(\pi )=0\}.$$

Then, $\mathbb{A}\mathfrak{I}={\sum }_{i=1}^{\infty }{n}^2〈\mathfrak{I}, {\mathfrak{I}}_n〉,$ ${\mathfrak{I}}_n(s)=\sqrt{\frac{2}{\pi }}\hspace{0.17em}\sin \hspace{0.17em}(ns)$ and the Eigenvectors of $\mathbb{A}$ is orthogonal set and they are given by $n\in \mathbb{N}.$ Therefore, the generator of $\mathbb{A}$ is $T{(\nu )}_{\nu \ge 0}$ in $\mathcal{V},$ expressed as $$T(\nu )\mathfrak{I}=\sum _{i=1}^{\infty }{e}^{-n^2\nu }〈\mathfrak{I}, {\mathfrak{I}}_n〉, \mathfrak{I}, {\mathfrak{I}}_n\in \mathcal{V}, \nu >0,$$

This makes that $\mathbb{A}\in A^p({\sigma }_0, {\mathfrak{I}}_0).$ In addition to that, $\Vert T_p(\nu )\Vert \le M_T,$ for every $\nu \in [0,1].$

By putting, $$v(q)=\Xi (\nu ,\cdot ),$$ $$h(q,v(q),{\int }_0^q\mathcal{K}(q,s,v(s))ds)=\frac{1}{4}\frac{e^{-\nu }}{1+e^{-\nu }}+{\int }_0^{\nu }\frac{1}{20}{e}^{-s}\Xi (\nu , ϵ)ds,$$ $$g_i(q,v(q))=\frac{e^{-(\frac{5\nu -1}{5})}}{9} \frac{|\Xi (\nu , ϵ)|}{1+|\Xi (\nu , ϵ)|},$$ $$\varsigma (v)=\sum _{i=1}^5\frac{1}{5^i}\Xi (\frac{1}{i}, ϵ).$$

Equation (5.1)(5.1) $$\begin{array}{c}{}^{\mathit{ABC}}{\mathcal{D}}^p[\Xi (\nu , ϵ)-\frac{1}{4}\frac{e^{-\nu }}{1+e^{-\nu }}+{\int }_0^{\nu }\frac{1}{20}{e}^{-s}\Xi (\nu , ϵ)ds]\hfill \\ =\frac{{\partial }^2}{\partial {ϵ}^2}[\Xi (\nu , ϵ)-\frac{1}{4}\frac{e^{-\nu }}{1+e^{-\nu }}+{\int }_0^{\nu }\frac{1}{20}{e}^{-s}\Xi (\nu , ϵ)ds]\hfill \\ +Bu(\nu , ϵ)+\frac{1}{9}\frac{e^{-\nu }}{1+e^{-\nu }}+{\int }_0^{\nu }\frac{1}{36}{e}^{-s}\Xi (\nu , ϵ)ds, \nu \in (0,\frac{1}{5}]\cup (\frac{4}{5},1]\hfill \\ \alpha \Xi (\nu , 0)=\Xi (\nu , \pi )=0, \nu \in [0,1]\hfill \\ \Xi (\nu , ϵ)=\frac{e^{-(\frac{5\nu -1}{5})}}{9} \frac{|\Xi (\nu , ϵ)|}{1+|\Xi (\nu , ϵ)|}, \nu \in (0,\frac{1}{5}]\cup (\frac{4}{5},1]\hfill \\ \Xi (0, ϵ)+\sum _{i=1}^5\frac{1}{5^i}\Xi (\frac{1}{i}, ϵ)={\Xi }_0(ϵ), ϵ\in [0,1].\hfill \end{array}$$(5.1) represents (1.1)(1.1) $$\begin{array}{l}{}^{\mathit{ABC}}{\mathcal{D}}^p[v(q)-h(q,v(q),{\int }_0^q\mathcal{K}(q,s,v(s))ds)]\\ =\mathbb{A}[v(q)-h(q,v(q),{\int }_0^q\mathcal{K}(q,s,v(s))ds)]+\mathcal{B}u(q)\\ +f(q,v(q),{\int }_0^q\xi (q,s,v(s))ds), q\in \underset{i=0}{\overset{m}{\cup }}(s_i,q_{i+1}]\\ v(q)=g_i(q,v(q)), q\in \underset{i=1}{\overset{m}{\cup }}(s_i,q_i]\\ v(0)+\varsigma (v)=v_0\in V,\end{array}$$(1.1) , fulfils (HO)–(H8). Hence, (5.1)(5.1) $$\begin{array}{c}{}^{\mathit{ABC}}{\mathcal{D}}^p[\Xi (\nu , ϵ)-\frac{1}{4}\frac{e^{-\nu }}{1+e^{-\nu }}+{\int }_0^{\nu }\frac{1}{20}{e}^{-s}\Xi (\nu , ϵ)ds]\hfill \\ =\frac{{\partial }^2}{\partial {ϵ}^2}[\Xi (\nu , ϵ)-\frac{1}{4}\frac{e^{-\nu }}{1+e^{-\nu }}+{\int }_0^{\nu }\frac{1}{20}{e}^{-s}\Xi (\nu , ϵ)ds]\hfill \\ +Bu(\nu , ϵ)+\frac{1}{9}\frac{e^{-\nu }}{1+e^{-\nu }}+{\int }_0^{\nu }\frac{1}{36}{e}^{-s}\Xi (\nu , ϵ)ds, \nu \in (0,\frac{1}{5}]\cup (\frac{4}{5},1]\hfill \\ \alpha \Xi (\nu , 0)=\Xi (\nu , \pi )=0, \nu \in [0,1]\hfill \\ \Xi (\nu , ϵ)=\frac{e^{-(\frac{5\nu -1}{5})}}{9} \frac{|\Xi (\nu , ϵ)|}{1+|\Xi (\nu , ϵ)|}, \nu \in (0,\frac{1}{5}]\cup (\frac{4}{5},1]\hfill \\ \Xi (0, ϵ)+\sum _{i=1}^5\frac{1}{5^i}\Xi (\frac{1}{i}, ϵ)={\Xi }_0(ϵ), ϵ\in [0,1].\hfill \end{array}$$(5.1) is controllable.

6. Conclusions

In this paper, we explored the controllability and optimal control of NFIE with noninstantaneous impulses. The controllability findings are backed up by the set contraction principle perfectly. Fixed point and noncompact measure methods are used to achieve the results. An appropriate example is used to describe the results. We have presented some important properties of AB operators, which will be useful for further study in this direction.

Acknowledgements

This study is supported via funding from Prince Sattam Bin Abdulaziz University under project number (PSAU/2023/R/1444). The authors are thankful to the Deanship of Scientific Research at University of Bisha for supporting this work through the Fast-Track Research Support Program.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This study is supported via funding from Prince Sattam Bin Abdulaziz University under Project Number (PSAU/2023/R/1444). The authors are thankful to the Deanship of Scientific Research at University of Bisha for supporting this work through the Fast-Track Research Support Program.