Optimal Controls for a Class of Conformable Fractional Evolution Systems

Abstract

In the scale of abstract Banach spaces, the existence as well as the controllability are considered for a class of evolution systems involving conformable fractional derivatives. Under essential conditions on nonlinearity, among other conditions, existence theorems are obtained by employing the operator semigroup theory. The existence problem of optimal state-control pairs is further studied by establishing a minimize sequence twice. Two examples are given as applications of the abstract results.

1. Introduction

Fractional derivatives have a long history. It began from L’Hospital in 1695. The most popular fractional derivatives, defined by integral form, are followed from Riemann–Liouville and Caputo; see, for example, [1,2,3,4,5]. These definitions have some disadvantages:

(1)

$D_{0^{+}}^{\alpha }\left(c\right)=0$ for $\alpha \in (n-1,n)$ is not valid in the Riemann–Liouville sense, where c is a constant.

(2)

For the Caputo derivative or the Riemann–Liouville derivative, the following formulas are not satisfied for $g,h:[0,+\infty )\to \mathbb{R}$,

(i)

$D_{0^{+}}^{\alpha }\left(gh\right)=g{D}_{0^{+}}^{\alpha }\left(h\right)+h{D}_{0^{+}}^{\alpha }\left(g\right)$.

(ii)

$D_{0^{+}}^{\alpha }\left(\frac{g}{h}\right)=\frac{h{D}_{0^{+}}^{\alpha }\left(g\right)-g{D}_{0^{+}}^{\alpha }\left(h\right)}{h^2}$.

(iii)

$D_{0^{+}}^{\alpha }(h\circ g)=D_{0^{+}}^{\alpha }h\left(g\left(t\right)\right)D_{0^{+}}^{\alpha }\left(g\right)$.

However, the “conformable fractional derivative”, introduced by Khalil et al., satisfies the above properties, which we infer from the articles [1,6,7,8,9]. Let $\alpha \in (0,1]$. For $h:[0,+\infty )\to \mathbb{R}$, the conformable fractional derivative is given by

$$\left(C_{\alpha }h\right)\left(t\right)=\underset{ϵ\to 0}{lim}\frac{h(t+ϵt^{1-\alpha })-h\left(t\right)}{ϵ},t>0,$$

where $C_{\alpha }$ represents the $\alpha$-order conformable fractional derivative operator. The higher-order conformable fractional derivative is also established in [1]. In 2015, Abdeljawad [6] further studied this fractional derivative and integral. Many properties, including the chain rule and the Laplace transform, etc., were proved. For more conclusions, we refer to [7,9,10,11]. As applications, Khalil et al. [1] gave the general solution for the initial value problem (IVP for short) of the non-homogeneous equation

$$\left\{\begin{array}{c}(C_{\frac{1}{2}}u)\left(t\right)+u\left(t\right)=t^2+2t^{\frac{3}{2}},t>0,\hfill \\ u\left(0\right)=0.\hfill \end{array}\right.$$

In [6], by applying the fractional fundamental exponential matrix, Abdeljawad studied the non-homogeneous system

$$\left(C_{\alpha }u\right)\left(t\right)=\Theta u\left(t\right)+\ell \left(t\right),t>a,0<\alpha \le 1,$$

where $\Theta$ is a matrix and $\ell :[a,b)\to {\mathbb{R}}^n$ is a vector function.

Our aim in the present article is to demonstrate the fractional partial differential equation

$$\left(C_{\frac{1}{2}}u\right)(t,\varrho )=\frac{{\partial }^2}{\partial {\varrho }^2}u(t,\varrho )+\ell (t,\varrho ),(t,\varrho )\in (0,1)\times (0,1)$$

(1)

involving the initial boundary conditions

$$u(t,0)=u(t,1)=0,t\in (0,1)$$

and

$$u(0,\varrho )={\nu }_0\left(\varrho \right),\varrho \in (0,1),$$

where ℓ is a continuous function and ${\nu }_0\in L^2(0,1)$. We will build a more general abstract framework that is useful for studying other forms of fractional partial differential equations including (1).

Let $\mathbb{B}=L^2(0,1)$. A linear operator P is defined by

$$Pu=\frac{{\partial }^2}{\partial z^2}u,u\in \mathcal{D}\left(P\right):=\{u\in \mathbb{B}:u^{\prime },u^{″}\in \mathbb{B},u\left(0\right)=u\left(1\right)=0\}.$$

Then, P is densely defined and, according to [12], it generates a $C_0$-semigroup $G\left(t\right)(t\ge 0)$ in $\mathbb{B}$. Moreover, $G\left(t\right)$ is a compact semigroup.

If we take $u\left(t\right)\left(\varrho \right)=u(t,\varrho )$ and $\ell \left(t\right)\left(\varrho \right)=\ell (t,\varrho )$, then Equation (1) with initial-boundary conditions can be rewritten as the linear evolution system

$$\left\{\begin{array}{c}(C_{\frac{1}{2}}u)\left(t\right)=Pu\left(t\right)+\ell \left(t\right),t\in (0,1),\hfill \\ u\left(0\right)={\nu }_0.\hfill \end{array}\right.$$

More generally, in the following we will establish a general frame in the Banach space $\mathbb{B}$. We investigate the abstract nonlocal evolution system involving conformable fractional derivative

$$\left\{\begin{array}{c}\left(C_{\alpha }u\right)\left(t\right)=Pu\left(t\right)+\kappa (t,u\left(t\right))+D\left(t\right)v\left(t\right),t>0,\hfill \\ u\left(0\right)={\nu }_0+\zeta \left(x\right),\hfill \end{array}\right.$$

(2)

where $P:\mathcal{D}\left(P\right)\subset \mathbb{B}\to \mathbb{B}$ generates a $C_0$-semigroup $G\left(t\right)(t\ge 0)$ in $\mathbb{B}$, $D(·):\mathbb{Y}\to \mathbb{B}$ is a linear operator and $\mathbb{Y}$ is a reflexive and separable Banach space, $v\in {\mathcal{V}}_{ad}$, ${\mathcal{V}}_{ad}$ is the admissible set that will be defined later, and $\kappa$ and $\zeta$ satisfy some essential conditions.

The optimal control problems of fractional evolution systems have been studied widely; see [2,3,12,13] and the references therein. Wang et al. [13] considered optimal relaxed controls of Caputo fractional evolution equations with impulsive effects. In [2], Kumar proved the optimal controls of delayed evolution systems involving the Caputo fractional derivative. Lian et al. [3] studied time optimal controls of evolution systems with the Riemann–Liouville derivative. Zhu et al. [12] demonstrated the optimal state-controls of Riemann–Liouville fractional evolution systems. To our knowledge, optimal controls for conformable fractional evolution systems has not been studied yet. In the present work, we first use the operator semigroup theory to establish existence theorems of the evolution system (2). Secondly, the existence of optimal state-control pairs is investigated by establishing minimize sequence twice. Examples are given in the last section to elucidate the obtained abstract results. Obviously, Equation (1) is the special case of our examples.

2. Preliminaries

Let $\mathbb{B},\mathbb{E}$ be two Banach spaces, $\Delta :=[0,b]$. Then, $C(\Delta ,\mathbb{B})$, the set of all continuous functions from $\Delta$ to $\mathbb{B}$, is a Banach space with the norm ${\parallel u\parallel }_C=\underset{t\in \Delta }{max}\parallel u\left(t\right)\parallel$ for any $u\in C(\Delta ,\mathbb{B})$. $L^p(\Delta ,\mathbb{B})$, the set of all p-order Bochner integrable functions, is a Banach space, whose norm is defined by ${\parallel f\parallel }_{L^p}=({\int }_0^b{\parallel f\left(t\right)\parallel }^p{dt)}^{\frac{1}{p}}$ for any $f\in L^p(\Delta ,\mathbb{B})$. $\mathcal{L}(\mathbb{E},\mathbb{B})$ represents the set of bounded linear operators from $\mathbb{E}$ to $\mathbb{B}$ and $\mathcal{L}\left(\mathbb{B}\right):=\mathcal{L}(\mathbb{B},\mathbb{B})$.

Let $\alpha \in (0,1]$ and $h:[0,+\infty )\to \mathbb{R}$. The following three definitions are cited from [1,6].

Definition 1.

The α-order conformable fractional derivative is given by

$$\left(C_{\alpha }h\right)\left(t\right)=\underset{ϵ\to 0}{lim}\frac{h(t+ϵt^{1-\alpha })-h\left(t\right)}{ϵ},t>0.$$

Furthermore,

$$\left(C_{\alpha }h\right)\left(0\right)=\underset{t\to 0^{+}}{lim}\left(C_{\alpha }h\right)\left(t\right).$$

Definition 2.

The α-order conformable fractional integral is given by

$$\left(I_{\alpha }h\right)\left(t\right)={\int }_0^t{s}^{\alpha -1}h\left(s\right)ds.$$

Definition 3.

The fractional Laplace transform is presented by

$$L_{\alpha }\left\{h\left(s\right)\right\}\left(\lambda \right)={\int }_0^{\infty }{s}^{\alpha -1}{e}^{-\lambda \frac{s^{\alpha }}{\alpha }}h\left(s\right)ds.$$

Lemma 1 ([1,6]).

If $h:[0,+\infty )\to \mathbb{R}$ is continuous, we infer that

$$C_{\alpha }\left(I_{\alpha }h\right)\left(t\right)=h\left(t\right),t>0.$$

Lemma 2 ([1,6]).

If $h:[0,+\infty )\to \mathbb{R}$ is differentiable, we have

$$I_{\alpha }\left(C_{\alpha }h\right)\left(t\right)=h\left(t\right)-h\left(0\right),t>0.$$

Let $h:[0,+\infty )\to \mathbb{R}$ be differentiable and $\alpha \in (0,1]$. It follows from Definition 5.1 of [6] that

$$L_{\alpha }\left\{\left(C_{\alpha }h\right)\left(t\right)\right\}\left(\lambda \right)=\lambda L_{\alpha }\left\{h\left(t\right)\right\}\left(\lambda \right)-h\left(0\right).$$

For any $h\in C^1(\Delta ,\mathbb{B})$ and ${\nu }_0\in \mathcal{D}\left(P\right)$, we first consider the linear IVP

$$\left\{\begin{array}{c}\left(C_{\alpha }u\right)\left(t\right)=Pu\left(t\right)+h\left(t\right),t>0,\hfill \\ u\left(0\right)={\nu }_0.\hfill \end{array}\right.$$

(3)

Lemma 3.

Let $P:\mathcal{D}\left(P\right)\subset \mathbb{B}\to \mathbb{B}$ generate a $C_0$-semigroup $G\left(t\right)(t\ge 0)$ in $\mathbb{B}$. Then the IVP(3) possesses a unique solution:

$$u\left(t\right)=C\left(\frac{t^{\alpha }}{\alpha }\right){\nu }_0+{\int }_0^t{s}^{\alpha -1}C\left(\frac{t^{\alpha }-s^{\alpha }}{\alpha }\right)h\left(s\right)ds,t\in \Delta .$$

Proof .

By utilizing the fractional Laplace transform to the Equation (3), we have

$$\begin{array}{ccc}\hfill L_{\alpha }\left\{u\left(t\right)\right\}\left(\lambda \right)& =& {(\lambda I-P)}^{-1}\nu \left(0\right)+{(\lambda I-P)}^{-1}{L}_{\alpha }\left\{h\left(t\right)\right\}\left(\lambda \right)\hfill \\ & =& {\int }_0^{\infty }{e}^{-\lambda t}G\left(t\right)u\left(0\right)dt+{\int }_0^{\infty }{e}^{-\lambda t}G\left(t\right)L_{\alpha }\left\{h\left(t\right)\right\}\left(\lambda \right)dt\hfill \\ & =& {\int }_0^{\infty }{e}^{-\lambda \frac{t^{\alpha }}{\alpha }}G\left(\frac{t^{\alpha }}{\alpha }\right){\nu }_0{t}^{\alpha -1}dt+{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-\lambda t}{e}^{-\lambda \frac{s^{\alpha }}{\alpha }}{s}^{\alpha -1}G\left(t\right)h\left(s\right)dsdt\hfill \\ & =& {\int }_0^{\infty }{t}^{\alpha -1}{e}^{-\lambda \frac{t^{\alpha }}{\alpha }}G\left(\frac{t^{\alpha }}{\alpha }\right){\nu }_{0}dt+{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-\lambda (t+\frac{s^{\alpha }}{\alpha })}{s}^{\alpha -1}G\left(t\right)h\left(s\right)dtds\hfill \\ & =& L_{\alpha }\left\{G\left(\frac{t^{\alpha }}{\alpha }\right){\nu }_0\right\}\left(\lambda \right)+{\int }_0^{\infty }{\int }_s^{\infty }{s}^{\alpha -1}{e}^{-\lambda \frac{t^{\alpha }}{\alpha }}G\left(\frac{t^{\alpha }-s^{\alpha }}{\alpha }\right)t^{\alpha -1}h\left(s\right)dtds\hfill \\ & =& L_{\alpha }\left\{G\left(\frac{t^{\alpha }}{\alpha }\right){\nu }_0\right\}\left(\lambda \right)+{\int }_0^{\infty }{\int }_0^t{t}^{\alpha -1}{e}^{-\lambda \frac{t^{\alpha }}{\alpha }}\left[s^{\alpha -1}G\left(\frac{t^{\alpha }-s^{\alpha }}{\alpha }\right)h\left(s\right)\right]dsdt\hfill \\ & =& L_{\alpha }\left\{G\left(\frac{t^{\alpha }}{\alpha }\right){\nu }_0\right\}\left(\lambda \right)+L_{\alpha }\left\{{\int }_0^t{s}^{\alpha -1}G\left(\frac{t^{\alpha }-s^{\alpha }}{\alpha }\right)h\left(s\right)ds\right\}\left(\lambda \right).\hfill \end{array}$$

By the inverse Laplace transform, we achieve that

$$u\left(t\right)=G\left(\frac{t^{\alpha }}{\alpha }\right){\nu }_0+{\int }_0^t{s}^{\alpha -1}G\left(\frac{t^{\alpha }-s^{\alpha }}{\alpha }\right)h\left(s\right)ds,t\in \Delta .$$

□

Remark 1.

When $\alpha =\frac{1}{2}$, the IVP(3) is mentioned in the Introduction section. By Lemma 3, if $\ell \in C^1([0,1]\times [0,1],L^2(0,1))$ and ${\nu }_0\in D\left(A\right)$, the unique solution of the Equation (1) is obtained in $L^2(0,1)$.

By Lemma 3, we present the following definition.

Definition 4.

For each $v\in {\mathcal{V}}_{ad}$, $u\in C(\Delta ,\mathbb{B})$ is named as the mild solution of (2) if

$$\begin{array}{c}\hfill u\left(t\right)=G\left(\frac{t^{\alpha }}{\alpha }\right)[{\nu }_0+\zeta \left(u\right)]+{\int }_0^t{s}^{\alpha -1}G\left(\frac{t^{\alpha }-s^{\alpha }}{\alpha }\right)[\kappa (s,u\left(s\right))+D\left(s\right)v\left(s\right)]ds,t\in \Delta .\end{array}$$

3. The Solvability of the Nonlocal System (2)

We make the essential assumptions below.

$\left(A1\right)$

The $C_0$-semigroup $G\left(t\right)(t\ge 0)$ is compact and

$$\parallel G\left(t\right)\parallel \le M,\forall t\ge 0,$$

where $M\ge 1$ is a constant.

$\left(A2\right)$

For $\kappa :\Delta \times \mathbb{B}\to \mathbb{B}$, the following conditions are held.

(i)

$\kappa (t,·)$ is continuous for every $t\in \Delta$, $\kappa (·,u)$ is strongly measurable for each $u\in \mathbb{B}$.

(ii)

There are $L_1,N_1>0$ satisfying $M{b}^{\alpha }{L}_1<\alpha$ such that

$$\parallel \kappa (t,u)\parallel \le L_1\parallel u\parallel +N_1,\forall t\in \Delta ,u\in \mathbb{B}.$$

$\left(A3\right)$

$\zeta :C(\Delta ,\mathbb{B})\to \mathbb{B}$ is completely continuous.

$\left(A4\right)$

$D\in L^{\infty }(\Delta ,\mathcal{L}(\mathbb{Y},\mathbb{B}))$.

Let $\mathcal{P}\left(\mathbb{Y}\right)$ be a group of all nonempty close and convex subsets of $\mathbb{Y}$, and $\mathbb{K}\subset \mathbb{Y}$ a bounded subset of $\mathbb{Y}$. We define

$${\mathcal{V}}_{ad}:=\{v\in L^p(\Delta ,\mathbb{K}):v\left(t\right)\in \mathcal{V}\left(t\right)a.e.t\in \Delta \},p\alpha >1,$$

where $\mathcal{V}:\Delta \to \mathcal{P}\left(\mathbb{Y}\right)$ denotes a multi-valued graph measurable mapping satisfying $\mathcal{V}(·)\subset \mathbb{K}$. By the proposition 2.1.7 of [14], ${\mathcal{V}}_{ad}\ne \varnothing$. In view of the condition $\left(A4\right)$, we infer that

$$D(·)v(·)\in L^p(\Delta ,\mathbb{B}),\forall v\in {\mathcal{V}}_{ad}.$$

Lemma 4.

Assume that $\left(A1\right)$ holds. Then $Q:L^p(\Delta ,\mathbb{B})\to C(\Delta ,\mathbb{B})$, defined by

$$\left(Q\xi \right)\left(t\right)={\int }_0^t{s}^{\alpha -1}G\left(\frac{t^{\alpha }-s^{\alpha }}{\alpha }\right)\xi \left(s\right)ds,\forall \xi \in L^p(\Delta ,\mathbb{B}),$$

is a compact operator for $p\alpha >1$.

Proof .

We show the relative compactness of $\{Q\xi :\xi \in B_{L^p}\left(r\right)\}$ in $C(\Delta ,\mathbb{B})$, where

$$B_{L^p}\left(r\right)=\{\xi \in L^p{(\Delta ,\mathbb{B}):\parallel \xi \parallel }_{L^p}\le r\}$$

for any $r>0$.

At first, the equi-continuity of $\{Q\xi :\xi \in B_{L^p}\left(r\right)\}$ in $C(\Delta ,\mathbb{B})$ is proved. Let $\xi \in B_{L^p}\left(r\right)$. For $0\le t_1<t_2\le b$, if $t_1\equiv 0$, we have

$$\begin{array}{ccc}& & \parallel \left(Q\xi \right)\left(t_2\right)-\left(Q\xi \right)\left(0\right)\parallel \hfill \\ & =& \parallel {\int }_0^{t_2}{s}^{\alpha -1}G\left(\frac{t_2^{\alpha }-s^{\alpha }}{\alpha }\right)\xi \left(s\right)ds\parallel \hfill \\ & \le & Mr\sigma t_2^{\alpha -\frac{1}{p}}\hfill \\ & \to & 0\hfill \end{array}$$

as $t_2\to 0$, where $\sigma ={\left(\frac{p-1}{p\alpha -1}\right)}^{1-\frac{1}{p}}$.

If $t_1>0$, we have

$$\begin{array}{ccc}& & \parallel \left(Q\xi \right)\left(t_2\right)-\left(Q\xi \right)\left(t_1\right)\parallel \hfill \\ & \le & \parallel {\int }_0^{t_1}{s}^{\alpha -1}[G\left(\frac{t_2^{\alpha }-s^{\alpha }}{\alpha }\right)-G\left(\frac{t_1^{\alpha }-s^{\alpha }}{\alpha }\right)]\xi \left(s\right)ds\parallel +\parallel {\int }_{t_1}^{t_2}{s}^{\alpha -1}G\left(\frac{t_2^{\alpha }-s^{\alpha }}{\alpha }\right)\xi \left(s\right)ds\parallel \hfill \\ & \le & \parallel {\int }_0^{t_1}[G\left(\frac{t_2^{\alpha }-t_1^{\alpha }}{\alpha }\right)-I]G\left(\frac{t_1^{\alpha }-s^{\alpha }}{\alpha }\right)s^{\alpha -1}\xi \left(s\right)ds\parallel +M{\int }_{t_1}^{t_2}{s}^{\alpha -1}\parallel \xi \left(s\right)\parallel ds\hfill \\ & \le & M{\int }_0^{t_1}\parallel [G\left(\frac{t_2^{\alpha }-t_1^{\alpha }}{\alpha }\right)-I]s^{\alpha -1}\xi \left(s\right)\parallel ds+Mr\sigma (t_2^{\alpha -\frac{1}{p}}-t_1^{\alpha -\frac{1}{p}})\hfill \\ & \to & 0\hfill \end{array}$$

as $t_2-t_1\to 0$.

Secondly, for any $t\in \Delta$, we show that $\{\left(Q\xi \right)\left(t\right):\xi \in B_{L^p}\left(r\right)\}$ is relatively compact in $\mathbb{B}$. The case $t=0$ is trivial. Let $t\in (0,b]$ and $ϵ\in (0,t)$. We present

$$\left(Q^{ϵ}\xi \right)\left(t\right)=G\left(\frac{{ϵ}^{\alpha }}{\alpha }\right){\int }_0^{{(t^{\alpha }-{ϵ}^{\alpha })}^{\frac{1}{\alpha }}}{s}^{\alpha -1}G\left(\frac{t^{\alpha }-{ϵ}^{\alpha }-s^{\alpha }}{\alpha }\right)\xi \left(s\right)ds.$$

Since

$$\parallel {\int }_0^{{(t^{\alpha }-{ϵ}^{\alpha })}^{\frac{1}{\alpha }}}{s}^{\alpha -1}G\left(\frac{t^{\alpha }-{ϵ}^{\alpha }-s^{\alpha }}{\alpha }\right)\xi \left(s\right)ds\parallel \le Mr\sigma b^{\alpha -\frac{1}{p}},$$

According to the compactness of $G\left(\frac{{ϵ}^{\alpha }}{\alpha }\right)$, $\{\left(Q^{ϵ}\xi \right)\left(t\right):\xi \in B_{L^p}\left(r\right)\}$ is relatively compact. We infer, for $t\in (0,b]$, that

$$\begin{array}{ccc}& & \parallel \left(Q\xi \right)\left(t\right)-\left(Q^{ϵ}\xi \right)\left(t\right)\parallel \hfill \\ & =& \parallel {\int }_{{(t^{\alpha }-{ϵ}^{\alpha })}^{\frac{1}{\alpha }}}^t{s}^{\alpha -1}G\left(\frac{t^{\alpha }-s^{\alpha }}{\alpha }\right)\xi \left(s\right)ds\parallel \hfill \\ & \le & Mr\sigma [t^{\alpha -\frac{1}{p}}-{(t^{\alpha }-{ϵ}^{\alpha })}^{1-\frac{1}{p\alpha }}]\hfill \\ & \to & 0\hfill \end{array}$$

as $ϵ\to 0$. Hence, $\{\left(Q^{ϵ}\xi \right)\left(t\right):\xi \in B_{L^p}\left(r\right)\}$ is arbitrarily close to $\{\left(Q\xi \right)\left(t\right):\xi \in B_{L^p}\left(r\right)\}$ and then $\{\left(Q\xi \right)\left(t\right):\xi \in B_{L^p}\left(r\right)\}$ is relatively compact. Therefore, the relative compactness of $\{Q\xi :\xi \in B_{L^p}\left(r\right)\}$ is proved according to the Ascoli–Arzela theorem. So, $Q:L^p(\Delta ,\mathbb{B})\to C(\Delta ,\mathbb{B})$ is compact. □

Theorem 1.

Let $\left(A1\right)$–$\left(A4\right)$ be fulfilled. Then the nonlocal system (2) possesses a mild solution in $C(\Delta ,\mathbb{B})$.

Proof .

For $v\in {\mathcal{V}}_{ad}$, we define $\Phi :C(\Delta ,\mathbb{B})\to C(\Delta ,\mathbb{B})$ by

$$(\Phi u)\left(t\right)=G\left(\frac{t^{\alpha }}{\alpha }\right)[{\nu }_0+\zeta \left(u\right)]+{\int }_0^t{s}^{\alpha -1}G\left(\frac{t^{\alpha }-s^{\alpha }}{\alpha }\right)[\kappa (s,u\left(s\right))+D\left(s\right)v\left(s\right)]ds,t\in \Delta .$$

(4)

By Definition 4, we are going to show the existence of fixed points of $\Phi$ in $C(\Delta ,\mathbb{B})$. To do so, we state the proof in three steps.

Step I. For every $r>0$, denote by $B_C\left(r\right)=\{u\in C(\Delta ,\mathbb{B}):\parallel u\left(t\right)\parallel \le r,t\in \Delta \}$. We prove that $\Phi \left(B_C\left(r\right)\right)\subset B_C\left(r\right)$ for some $r>0$. According to the complete continuity of $\zeta$, $\{\zeta \left(u\right):u\in B_C\left(r\right)\}$ is bounded, that is,

$$\underset{u\in B_C\left(r\right)}{sup}\parallel \zeta \left(u\right)\parallel \le N_2$$

for some $N_2>0$. We claim that $\Phi \left(B_C\left(r\right)\right)\subset B_C\left(r\right)$ provided that

$$r>\frac{M(\parallel {\nu }_0\parallel +\frac{b^{\alpha }}{\alpha }{N}_1+N_2+\sigma b^{\alpha -\frac{1}{p}}\parallel Dv{\parallel }_{L^p})}{1-\frac{M{b}^{\alpha }}{\alpha }{L}_1}.$$

In fact, for any $u\in B_C\left(r\right)$, by (4), we have

$$\begin{array}{ccc}\hfill \parallel (\Phi u)\left(t\right)\parallel & \le & \parallel G\left(\frac{t^{\alpha }}{\alpha }\right)[{\nu }_0+\zeta \left(u\right)]\parallel +\parallel {\int }_0^t{s}^{\alpha -1}G\left(\frac{t^{\alpha }-s^{\alpha }}{\alpha }\right)[\kappa (s,u\left(s\right))+D\left(s\right)v\left(s\right)]ds\parallel \hfill \\ & \le & M(\parallel {\nu }_0\parallel +\parallel \zeta \left(u\right)\parallel )+M{\int }_0^t{s}^{\alpha -1}\parallel \kappa (s,u\left(s\right))\parallel ds+M{\int }_0^t{s}^{\alpha -1}\parallel D\left(s\right)v\left(s\right)\parallel ds\hfill \\ & \le & M(\parallel {\nu }_0\parallel +N_2)+M{\int }_0^t{s}^{\alpha -1}(L_1\parallel u\left(s\right)\parallel +N_1)ds+M\sigma b^{\alpha -\frac{1}{p}}{\parallel Dv\parallel }_{L^p}\hfill \\ & \le & \frac{M{b}^{\alpha }}{\alpha }{L}_{1}r+M(\parallel {\nu }_0\parallel +N_2+\frac{b^{\alpha }{N}_1}{\alpha }+\sigma b^{\alpha -\frac{1}{p}}{\parallel Dv\parallel }_{L^p})\hfill \\ & \le & r,\forall t\in \Delta .\hfill \end{array}$$

That is, $\Phi \left(B_C\left(r\right)\right)\subset B_C\left(r\right)$.

Step II. We demonstrate the continuity of $\Phi :B_C\left(r\right)\to B_C\left(r\right)$.

Let ${\left\{u_n\right\}}_{n\ge 1}\subset B_C\left(r\right)$, $\underset{n\to \infty }{lim}{u}_n=\overline{u}$ in $B_C\left(r\right)$. Then,

$$\kappa (t,u_n\left(t\right))\to \kappa (t,\overline{u}\left(t\right)),\forall t\in \Delta$$

and

$$\zeta \left(u_n\right)\to \zeta \left(\overline{u}\right)$$

as $n\to \infty$. Consequently,

$$(\Phi u_n)\left(t\right)\to (\Phi \overline{u})\left(t\right),\forall t\in \Delta$$

as $n\to \infty$, which yields the continuity of $\Phi :B_C\left(r\right)\to B_C\left(r\right)$.

Step III. According to the complete continuity of $\zeta$ and Lemma 4, $\Phi$ is a compact operator in $C(\Delta ,\mathbb{B})$. Thus, $\Phi :B_C\left(r\right)\to B_C\left(r\right)$ is a completely continuous operator. Consequently, the existence of fixed points of $\Phi$ is obtained due to Schauder’s fixed point theorem. □

Theorem 2.

Let $\left(A1\right),{\left(A2\right)}^{\prime },\left(A3\right)$ and $\left(A4\right)$ hold, where

${\left(A2\right)}^{\prime }$$\kappa :\Delta \times \mathbb{B}\to \mathbb{B}$ is continuous and there is $F_1>0$ satisfying

$$\underset{(t,u)\in \Delta \times B_C\left(r\right)}{sup}\parallel \kappa (t,u\left(t\right))\parallel \le F_1.$$

Then the nonlocal system (2) possesses a mild solution.

Proof .

Suppose that $r>0$ is sufficiently large that we can easily prove $\Phi \left(B_C\left(r\right)\right)\subset B_C\left(r\right)$. The remaining proof is similar to the one of Theorem 1, so we delete the details here. □

Theorem 3.

Let $\left(A1\right),{\left(A2\right)}^{″},\left(A3\right)$ and $\left(A4\right)$ hold, where

${\left(A2\right)}^{″}$$\kappa :\Delta \times \mathbb{B}\to \mathbb{B}$ is continuous and there is $\rho \in L^p(\Delta ,\mathbb{B})$ for $p\alpha >1$ satisfying

$$\parallel \kappa (t,u)\parallel \le \rho \left(t\right)\phi (\parallel u\parallel ),\forall t\in \Delta ,u\in \mathbb{B},$$

where $\phi :[0,\infty )\to (0,\infty )$ is a nondecreasing continuous function and $\underset{r\to \infty }{lim}\frac{\phi \left(r\right)}{r}=\beta <\infty$. Then, the nonlocal system (2) admits a mild solution provided that

$$M\sigma \beta b^{\alpha -\frac{1}{p}}{\parallel \rho \parallel }_{L^p}<1.$$

(5)

Proof .

By Theorem 1, we just need to prove $\Phi \left(B_C\left(r\right)\right)\subset B_C\left(r\right)$ for some $r>0$. In fact, for any $r>0$, if there is a $u^{*}\in B_C\left(r\right)$ satisfying

$$\begin{array}{ccc}\hfill r<\parallel (\Phi u^{*})\left(t\right)\parallel & \le & M(\parallel {\nu }_0\parallel +N_2)+M{\int }_0^t{s}^{\alpha -1}\rho \left(s\right)\phi (\parallel u^{*}\left(s\right)\parallel )ds+Ma{b}^{\alpha -\frac{1}{p}}{\parallel Dv\parallel }_{L^p}\hfill \\ & \le & M(\parallel {\nu }_0\parallel +N_2)+M\sigma b^{\alpha -\frac{1}{p}}{\parallel \rho \parallel }_{L^p}\phi \left(r\right)+M\sigma b^{\alpha -\frac{1}{p}}{\parallel Dv\parallel }_{L^p},\hfill \end{array}$$

which yields

$$1\le M\sigma \beta b^{\alpha -\frac{1}{p}}{\parallel \rho \parallel }_{L^p}.$$

Hence, by (5), $\Phi \left(B_C\left(r\right)\right)\subset B_C\left(r\right)$ for some $r>0$. □

4. Existence of Optimal Controls

In our work, we remove the uniqueness of solutions of (2) without any additional conditions. Firstly, we give a boundedness result.

Lemma 5.

Let $\left(A1\right)-\left(A4\right)$ be satisfied. In addition, ζ is uniformly bounded. For fixed $v\in {\mathcal{V}}_{ad}$, if $u^v\in C(\Delta ,\mathbb{B})$ is a mild solution of (2), then

$$\parallel u^v{\parallel }_C\le R_1$$

for some $R_1>0$.

Proof .

Since $\zeta$ is uniformly bounded, there exists $N_3>0$, $\parallel \zeta \left(u\right)\parallel \le N_3$ for any $u\in C(\Delta ,\mathbb{B})$. For each $v\in {\mathcal{V}}_{ad}$, $u^v\in C(\Delta ,\mathbb{B})$ is a mild solution of (2) associated with v, so $u^v$ is a fixed point of $\Phi$ and

$$u^v\left(t\right)=G\left(\frac{t^{\alpha }}{\alpha }\right)[{\nu }_0+\zeta \left(u^v\right)]+{\int }_0^t{s}^{\alpha -1}G\left(\frac{t^{\alpha }-s^{\alpha }}{\alpha }\right)[\kappa (s,u^v\left(s\right))+D\left(s\right)v\left(s\right)]ds,t\in \Delta .$$

By $\left(A1\right)$–$\left(A4\right)$, we achieve that

$$\begin{array}{ccc}\hfill \parallel u^v\left(t\right)\parallel & \le & M(\parallel {\nu }_0\parallel +N_3)+M{\int }_0^t{s}^{\alpha -1}(L_1\parallel u^v\left(s\right)\parallel +N_1)ds+M\sigma b^{\alpha -\frac{1}{p}}{\parallel Dv\parallel }_{L^p}\hfill \\ & \le & K+M{L}_1{\int }_0^t{s}^{\alpha -1}\parallel u^v\left(s\right)\parallel ds,\forall t\in \Delta ,\hfill \end{array}$$

where $K=M(\parallel {\nu }_0\parallel +N_3)+M{N}_1\frac{b^{\alpha }}{\alpha }+M\sigma b^{\alpha -\frac{1}{p}}{\parallel Dv\parallel }_{L^p}$. The Gronwall inequality yields that

$$\parallel u^v\left(t\right)\parallel \le R_1,$$

where $R_1:=K{e}^{\frac{M{L}_1{b}^{\alpha }}{\alpha }}$. □

Let $B_C\left(R_1\right)=\{u\in C(\Delta ,\mathbb{B}){:\parallel u\parallel }_C\le R_1\}$. For any $v\in {\mathcal{V}}_{ad}$, let $S\left(v\right)$ be the set of all mild solutions of the nonlocal system (2). We then deal with the limited Lagrange problem (LLP) as follows:

Find ${\mu }_0\in {\mathcal{V}}_{ad}$ and $u^{{\mu }_0}\in S\left(v\right)$ such that the state-control pair $(u^{{\mu }_0},{\mu }_0)\in C(\Delta ,\mathbb{B})\times {\mathcal{V}}_{ad}$ satisfies

$$J(u^{{\mu }_0},{\mu }_0)\le J(u^v,v),$$

where $J(u^v,v)$ is given by

$$J(u^v,v):={\int }_0^b\omega (t,u^v\left(t\right),v\left(t\right))dt.$$

Let $\omega :\Delta \times \mathbb{B}\times \mathbb{Y}\to \mathbb{R}\cup \{+\infty \}$. We introduce the condition:

(A5)(i)

$\omega :\Delta \times \mathbb{B}\times \mathbb{Y}\to \mathbb{R}\cup \{+\infty \}$ is Borel measurable;

  (ii)

For a.e. $t\in \Delta ,\omega (t,·,·)$ is sequentially lower semicontinuous in $\mathbb{B}\times \mathbb{Y}$;

  (iii)

For a.e. $t\in \Delta$ and each $u\in \mathbb{B},\omega (t,u,·)$ is convex in $\mathbb{Y}$;

  (iv)

There are $c\ge 0$, $d>0$ and $\varphi \in L^1(\Delta ,\mathbb{R})$ satisfying

$$\omega (t,u,v)\ge \varphi \left(t\right)+c\parallel u\parallel +d{\parallel v\parallel }^p.$$

Theorem 4.

Let the assumptions $\left(A1\right)$–$\left(A5\right)$ hold. g, additionally, is uniformly bounded. Then the LLP governed by (2) has optimal state-control pairs.

Proof .

For fixed $v\in {\mathcal{V}}_{ad}$, let $J\left(v\right)=\underset{u^v\in S\left(v\right)}{inf}J(u^v,v)$. By $\left(A5\right)$, we can suppose that

$$-\infty <\underset{u^v\in S\left(v\right)}{inf}J(u^v,v)<\infty .$$

According to properties of the infimum, we can choose ${\left\{u_n^v\right\}}_{n\ge 1}\subset S\left(v\right)$ satisfying

$$\underset{n\to \infty }{lim}J(u_n^v,v)=J\left(v\right).$$

Since ${\left\{u_n^v\right\}}_{n\ge 1}\subset S\left(v\right)$, we infer that

$$u_n^v\left(t\right)=G\left(\frac{t^{\alpha }}{\alpha }\right)[{\nu }_0+\zeta \left(u_n^v\right)]+{\int }_0^t{s}^{\alpha -1}G\left(\frac{t^{\alpha }-s^{\alpha }}{\alpha }\right)[\kappa (s,u_n^v\left(s\right))+D\left(s\right)v\left(s\right)]ds,t\in \Delta .$$

(6)

By Lemma 5, there is $R_1>0$ such that $\parallel u_n^v{\parallel }_C\le R_1$. The compactness of $\zeta$ and Lemma 4 yield that ${\left\{u_n^v\right\}}_{n\ge 1}$ is relatively compact. Hence ${\left\{u_n^v\right\}}_{n\ge 1}$ possesses a subsequence, labeled by itself, converging to some ${\tilde{u}}^v\in C(\Delta ,\mathbb{B})$ and $\parallel {\tilde{u}}^v{\parallel }_C\le R_1$. By the continuity of $\zeta$ and $\kappa$, we infer that

$$\zeta \left(u_n^v\right)\to \zeta \left({\tilde{u}}^v\right),\kappa (t,u_n^v\left(t\right))\to \kappa (t,{\tilde{u}}^v\left(t\right)),\forall t\in \Delta .$$

For any $s\in (0,t)$, we have

$$\parallel s^{\alpha -1}G\left(\frac{t^{\alpha }-s^{\alpha }}{\alpha }\right)\kappa (s,u_n^v\left(s\right))\parallel \le M(L_1{R}_1+N_1)s^{\alpha -1}.$$

Taking $n\to \infty$ in (6), we can achieve that

$${\tilde{u}}^v\left(t\right)=G\left(\frac{t^{\alpha }}{\alpha }\right)[{\nu }_0+\zeta \left({\tilde{u}}^v\right)]+{\int }_0^t{s}^{\alpha -1}G\left(\frac{t^{\alpha }-s^{\alpha }}{\alpha }\right)[\kappa (s,{\tilde{u}}^v\left(s\right))+D\left(s\right)v\left(s\right)]ds,t\in \Delta .$$

This fact implies ${\tilde{u}}^v\in S\left(v\right)$.

Since the embedding from $C(\Delta ,\mathbb{B})$ to $L^1(\Delta ,\mathbb{B})$ is continuous, by $\left(A5\right)$ and Balder’s theorem (see [15]), we infer that

$$\begin{array}{ccc}\hfill J\left(v\right)& =& \underset{u^v\in S\left(v\right)}{inf}J(u^v,v)=\underset{n\to \infty }{lim}J(u_n^v,v)\hfill \\ & =& \underset{n\to \infty }{lim}{\int }_0^b\omega (t,u_n^v\left(t\right),v\left(t\right))dt\hfill \\ & \ge & {\int }_0^b\omega (t,{\tilde{u}}^v\left(t\right),v\left(t\right))dt\hfill \\ & =& J({\tilde{u}}^v,v)\hfill \\ & \ge & \underset{u^v\in S\left(v\right)}{inf}J(u^v,v)=J\left(v\right),\hfill \end{array}$$

that is, $J({\tilde{u}}^v,v)=J\left(v\right)$.

Secondly, we claim that there is a ${\mu }^0\in {\mathcal{V}}_{ad}$ satisfying $J\left({\mu }^0\right)\le J\left(v\right)$ for all $v\in {\mathcal{V}}_{ad}$. In fact, by $\left(A5\right)$, we assume that

$$-\infty <\underset{v\in {\mathcal{V}}_{ad}}{inf}J\left(v\right)<\infty .$$

Hence there is a sequence ${\left\{v_n\right\}}_{n\ge 1}\subset {\mathcal{V}}_{ad}$ satisfying

$$\underset{n\to \infty }{lim}J\left(v_n\right)=\underset{v\in {\mathcal{V}}_{ad}}{inf}J\left(v\right).$$

Because ${\left\{v_n\right\}}_{n\ge 1}\subset {\mathcal{V}}_{ad}$, the set ${\left\{v_n\right\}}_{n\ge 1}$ is bounded. Since $\mathbb{Y}$ is reflexive and separable, it follows that the space $L^p(\Delta ,\mathbb{Y})$ is reflexive for $p>\frac{1}{\alpha }$. So, there is a subsequence of ${\left\{v_n\right\}}_{n\ge 1}$, labeled by ${\left\{v_n\right\}}_{n\ge 1}$ again, converging weakly to some ${\mu }^0$ in $L^p(\Delta ,\mathbb{Y})$. Since ${\mathcal{V}}_{ad}$ is closed and convex, Mazur’s theorem guarantees that ${\mu }^0\in {\mathcal{V}}_{ad}$. Since ${\left\{v_n\right\}}_{n\ge 1}\subset {\mathcal{V}}_{ad}$, for $n\ge 1$, there exists ${\tilde{u}}^{v_n}\in S\left(v_n\right)$ satisfying $J({\tilde{u}}^{v_n},v_n)=J\left(v_n\right)$. Thus,

$${\tilde{u}}^{v_n}\left(t\right)=G\left(\frac{t^{\alpha }}{\alpha }\right)[{\nu }_0+\zeta \left({\tilde{u}}^{v_n}\right)]+{\int }_0^t{s}^{\alpha -1}G\left(\frac{t^{\alpha }-s^{\alpha }}{\alpha }\right)[\kappa (s,{\tilde{u}}^{v_n}\left(s\right))+D\left(s\right)v_n\left(s\right)]ds,t\in \Delta .$$

(7)

In addition, $\parallel {\tilde{u}}^{v_n}{\parallel }_C\le R_1$. By the compactness of $\zeta$ and Lemma 4, we infer that ${\left\{{\tilde{u}}^{v_n}\right\}}_{n\ge 1}$ is relatively compact in $C(\Delta ,\mathbb{B})$. Hence, we may suppose that $\underset{n\to \infty }{lim}{\tilde{u}}^{v_n}={\tilde{u}}^{{\mu }_0}$. On the other hand, since $v_n$ weakly tends to ${\mu }^0$, by Lemma 4 and $\left(A1\right)$, we achieve that

$${\int }_0^t{s}^{\alpha -1}G\left(\frac{t^{\alpha }-s^{\alpha }}{\alpha }\right)D\left(s\right)v_n\left(s\right)ds\to {\int }_0^t{s}^{\alpha -1}G\left(\frac{t^{\alpha }-s^{\alpha }}{\alpha }\right)D\left(s\right){\mu }^0\left(s\right)ds,t\in \Delta .$$

Thus, taking $n\to \infty$ on both sides of (7), we deduce that

$${\tilde{u}}^{{\mu }_0}\left(t\right)=G\left(\frac{t^{\alpha }}{\alpha }\right)[{\nu }_0+\zeta \left({\tilde{u}}^{{\mu }_0}\right)]+{\int }_0^t{s}^{\alpha -1}G\left(\frac{t^{\alpha }-s^{\alpha }}{\alpha }\right)[\kappa (s,{\tilde{u}}^{{\mu }_0}\left(s\right))+D\left(s\right){\mu }_0\left(s\right)]ds,t\in \Delta .$$

This fact yields ${\tilde{u}}^{{\mu }_0}\in S\left({\mu }_0\right)$. Hence, according to the continuity of $C(\Delta ,\mathbb{B})\hookrightarrow L^1(\Delta ,\mathbb{B})$ and Balder’s theorem, we infer that

$$\begin{array}{ccc}\hfill \underset{v\in {\mathcal{V}}_{ad}}{inf}J\left(v\right)& =& \underset{n\to \infty }{lim}{\int }_0^b\omega (t,{\tilde{u}}^{v_n}\left(t\right),v_n\left(t\right))dt\hfill \\ & \ge & {\int }_0^b\omega (t,{\tilde{u}}^{{\mu }_0}\left(t\right),{\mu }_0\left(t\right))dt\hfill \\ & =& J({\tilde{u}}^{{\mu }_0},{\mu }_0)\hfill \\ & \ge & \underset{v\in {\mathcal{V}}_{ad}}{inf}J\left(v\right).\hfill \end{array}$$

From above arguments, we conclude that

$$J({\tilde{u}}^{{\mu }_0},{\mu }_0)=\underset{v\in {\mathcal{V}}_{ad}}{inf}J\left(v\right)=\underset{v\in {\mathcal{V}}_{ad}}{inf}\underset{u^v\in S\left(v\right)}{inf}J(u^v,v).$$

Therefore, the pair $({\tilde{u}}^{{\mu }_0},{\mu }_0)$ is an optimal state-control pair of the LLP. □

5. Examples

Example 1.

We demonstrate the fractional partial differential system

$$\left\{\begin{array}{c}\left(C_{\frac{1}{2}}u\right)(t,\varrho )=\frac{{\partial }^2}{\partial {\varrho }^2}u(t,\varrho )+\frac{e^{-t}}{3}u(t,\varrho )+sinu(t,\varrho )+e^{-2t}v(t,\varrho ),(t,\varrho )\in (0,1)\times (0,1),\hfill \\ u(t,0)=u(t,1)=0,t\in (0,1),\hfill \\ u(0,\varrho )={\nu }_0\left(\varrho \right)+{\int }_0^{1}K(t,s)u(s,\varrho )ds,\varrho \in (0,1)\hfill \end{array}\right.$$

(8)

with a cost function $J(u,v)$ given by

$$J(u,v)={\int }_0^1{\int }_0^1{\left(\right|u(t,\varrho )|}^2+{\left|v(t,\varrho )\right|}^2)d\varrho dt,$$

(9)

where $K:[0,1]\times [0,1]\to \mathbb{R}$ is a two-variable continuous function.

Let $\mathbb{B}=\mathbb{Y}=L^2(0,1)$. Define P by

$$Pu=\frac{{\partial }^2}{\partial z^2}u,u\in \mathcal{D}\left(P\right):=\{u\in \mathbb{B}:u^{\prime },u^{″}\in \mathbb{B},u\left(0\right)=u\left(1\right)=0\}.$$

Then, by [12], P generates a compact analytic semigroup $G\left(t\right)(t\ge 0)$:

$$G\left(t\right)u=\sum _{k=1}^{\infty }{e}^{-k^2{\pi }^{2}t}\langle u,e_k\rangle e_k,\forall u\in \mathbb{B},$$

where $e_k\left(u\right)=\sqrt{2}sin\left(k\pi u\right)$. It follows that $\parallel G\left(t\right)\parallel \le 1$.

Let $u\left(t\right)\left(\varrho \right)=u(t,\varrho )$ and

$$\kappa (t,u\left(t\right))\left(\varrho \right)=\frac{e^{-t}}{3}u(t,\varrho )+sinu(t,\varrho ),$$

$$\zeta \left(u\right)\left(\varrho \right)={\int }_0^{1}K(t,s)u(s,\varrho )ds,$$

$$D\left(t\right)v\left(t\right)\left(\varrho \right)=e^{-2t}v(t,\varrho ).$$

Then the fractional partial differential system (8) can be rewritten as (2) with

$$J(u,v)={\int }_0^1{(\parallel u\left(t\right)\parallel }^2+{\parallel v\left(t\right)\parallel }^2)dt.$$

(10)

Obviously, $\left(A1\right)$–$\left(A5\right)$ are satisfied. Therefore, by Theorem 1, the system (8) admits one mild solution on $[0,1]$. By Theorem 4, the system (8) possesses optimal state-control pairs.

Remark 2.

By Remark 1, we know that Equation (1) with initial boundary conditions mentioned in the Introduction has solutions under suitable conditions of ℓ and ${\nu }_0$. Equation (8) is more general than (1) because of its nonlinearity and nonlocal conditions. Even without nonlocal conditions, Example 1 still shows that the conclusions of this article are more general.

Example 2.

Let $\Omega \subset {\mathbb{R}}^N$ be bounded and $\partial \Omega$ sufficiently smooth. Denote by

$$\Pi (\varrho ,\mathcal{G})u=\sum _{i,j=1}^N\frac{\partial }{\partial {\varrho }_i}\left(a_{ij}\left(\varrho \right)\frac{\partial x}{\partial {\varrho }_j}\right)+a_0\left(\varrho \right)u$$

the uniformly elliptic differential operator in $\overline{\Omega }$, where $a_0\in C^{\theta }(\Omega )$ with $a_0\left(\varrho \right)\ge 0$ for $\varrho \in \overline{\Omega }$ and $a_{ij}\in C^{1+\theta }\left(\overline{\Omega }\right),i,j=1,2,\cdots ,N$ for each $\theta \in (0,1)$. That is, ${\left[a_{ij}\left(\varrho \right)\right]}_{N\times N}$ is a positive definite symmetric matrix for every $\varrho \in \overline{\Omega }$ and there is $\tau >0$ satisfying

$$\sum _{i,j=1}^N{a}_{ij}\left(\varrho \right){\eta }_i{\eta }_j\ge \tau {\left|\eta \right|}^2,\forall \eta =({\eta }_1,{\eta }_2,\cdots ,{\eta }_N)\in {\mathbb{R}}^N,\varrho \in \overline{\Omega }.$$

Consider the optimal controls of the fractional system

$$\left\{\begin{array}{c}\left(C_{\frac{1}{2}}u\right)(t,\varrho )=\Pi (\varrho ,\mathcal{G})u(t,\varrho )+F(t,\varrho ,u(t,\varrho ))+Dv(t,\varrho ),(t,\varrho )\in (0,1)\times (0,1),\hfill \\ {u|}_{\partial \Omega }=0,t\in (0,1),\hfill \\ u(0,\varrho )={\nu }_0\left(\varrho \right)+{\rm Y}\left(u\right)\left(\varrho \right),\varrho \in (0,1)\hfill \end{array}\right.$$

(11)

with $J(u,v)$ given by (9), where the linear operator $D:L^2(\Omega )\to L^2(\Omega )$ is bounded and F, ${\rm Y}$ satisfy certain conditions.

Let $\mathbb{B}=\mathbb{Y}=L^2(\Omega )$. Define A by

$$Pu=\Pi (\varrho ,\mathcal{G})u,\forall u\in \mathcal{D}\left(P\right):=H^2(\Omega )\cap H_0^1(\Omega ).$$

Then, by [16], P generates a compact analytic semigroup $G\left(t\right)(t\ge 0)$ and $\parallel G\left(t\right)\parallel \le 1$.

Let F and ${\rm Y}$ satisfy conditions:

$\left(\mathcal{R}1\right)$

$F:[0,1]\times \overline{\Omega }\times L^2(\Omega )\to L^2(\Omega )$ is continuous and there are $L^{*}\in (0,\frac{1}{2M})$ and $N^{*}>0$ satisfying

$$\left|F(t,\varrho ,u(t,\varrho ))\right|\le L^{*}\left|u(t,\varrho )\right|+N^{*},\forall (t,\varrho )\in [0,1]\times \overline{\Omega }$$

$\left(\mathcal{R}2\right)$

${\rm Y}:C([0,1],L^2(\Omega ))\to L^2(\Omega )$ is completely continuous and uniformly bounded.

If we put $u\left(t\right)\left(\varrho \right)=u(t,\varrho )$ and

$$\kappa (t,u(t\left)\right)\left(\varrho \right)=F(t,\varrho ,u(t,\varrho \left)\right),$$

$$\zeta \left(u\right)\left(\varrho \right)={\rm Y}\left(u\right)\left(\varrho \right),$$

$$D\left(t\right)v\left(t\right)\left(\varrho \right)=Dv(t,\varrho ),$$

then the fractional system (11) can be rewritten as (2) with (10). By $\left(\mathcal{R}1\right)$ and $\left(\mathcal{R}2\right)$, the conditions $\left(A2\right)$ and $\left(A3\right)$ hold. Hence, by Theorem 4, the fractional system (11) possesses optimal state-control pairs provided that $\left(A5\right)$ holds.

6. Conclusions

In this work, when the $C_0$-semigroup $G\left(t\right)(t\ge 0)$ is a compact semigroup, we first establish existence theorems of the evolution system (2) by employing the operator semigroup theory and Schauder’s fixed point theorem. Secondly, the existence of optimal state-control pairs is investigated for the LLP which is governed by (2). The discussion is based on the minimizing sequence technique and the approximate method. Two examples are given to elucidate the obtained abstract results.