# Solvability of some partial functional integrodifferential equations with finite delay and optimal controls in Banach spaces

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## Abstract

In this work, we consider the control system governed by some partial functional integrodifferential equations with finite delay in Banach spaces. We assume that the undelayed part admits a resolvent operator in the sense of Grimmer. Firstly, some suitable conditions are established to guarantee the existence and uniqueness of mild solutions for a broad class of partial functional integrodifferential infinite dimensional control systems. Secondly, it is proved that, under generally mild conditions of cost functional, the associated Lagrange problem has an optimal solution, and that for each optimal solution there is a minimizing sequence of the problem that converges to the optimal solution with respect to the trajectory, the control, and the functional in appropriate topologies. Our results extend and complement many other important results in the literature. Finally, a concrete example of application is given to illustrate the effectiveness of our main results.

## Keywords

Partial functional integrodifferential equations Finite delay Resolvent operator Solvability Mild solutions Optimal controls## Mathematics Subject Classification

93B05 45K05 47H08 47H10 34K37## Background

*C*

_{0}-semigroup \(\left( T(t)\right) _{t\ge 0}\) on a separable reflexive Banach space

*X*; for

*t*≥ 0,

*B*(

*t*) is a closed linear operator with domain \({\mathcal {D}}(B(t))\supset {\mathcal {D}}(A)\). The control

*u*(

*t*) takes values from another separable reflexive Banach space

*U*. The operator

*C*(

*t*) belongs to \({\mathcal {L}}(U,X)\), the Banach space of bounded linear operators from

*U*into

*X*, and \({\mathcal {C}}([-r,0],X)\) denotes the Banach space of continuous functions \(\varphi{:}\, [-r,0]\rightarrow X\) with supremum norm \(\Vert \varphi \Vert =\sup \nolimits _{\theta \in [-r,0]}\Vert \varphi (\theta )\Vert\),

*x*

_{ t }denotes the history function of \({\mathcal {C}}\) defined by

*C*

_{0}-semigroup. They used the techniques of a priori estimation.

Zhou (2014) considered a controlled stochastic delay partial differential equation with Neumann boundary conditions and studied the optimal control problem by means of the associated backward stochastic differential equations. In Motta and Rampazzo (2013), the authors discussed the assymptotic controllability and the optimal control of some control system where the state approaches asymptotically a target, while paying an integral cost with a nonnegative Lagrangian. Wang and Zhou (2011) discussed the optimal controls of a Lagrange problem for fractional evolution equations. In Wei et al. (2006), the authors studied the optimal controls for nonlinear impulsive integrodifferential equations of mixed type on Banach spaces. In Li and Liu (2015), the authors studied the existence of mild solutions and the optimal controls of a Lagrange problem for some impulsive fractional semilinear differential equations, using the techniques of a priori estimation.

Motivated by these works, we investigate the solvability and the existence of optimal controls of a Lagrange problem for Eq. (1), which is a more general class than those studied by the authors mentioned above. Using the techniques of a priori estimation of mild solutions and without any compactness assumptions made, the existence and uniqueness of mild solutions is obtained using the theory of resolvent operator for integral equations. Furthermore, to the best of our knowledge, the optimal controls for partial functional integrodifferential Eq. (1) with finite delay are untreated in the literature, and this fact motivates us to extend the existing ones and make new development of the present work on this issue.

## A model in heat conduction in materials with memory

*t*and position \(\xi\). The balance law for the heat transfer is given by:

*e*and

*q*on

*w*and \(\nabla w\), respectively. For instance assuming the Fourier Law i.e.,

*c*

_{1},

*c*

_{2}are positive constants, one deduces from (2) the classical heat equation

*w*of the body \(\Omega\) is known up to

*t*= 0, and the temperature of the boundary \(\partial \Omega\) of \(\Omega\) is constant (=0) for all

*t*, we are led to the following system:

*b*> 0 is arbitrarily fixed. If we prescribe

*h*(in addition to

*f*) then (8) is a Cauchy-Dirichlet problem for an integrodifferential equation in the unknown

*w*, which has been studied by several authors in the last decades, see e.g., Grimmer and Pritchard (1983), Grimmer and Kappelf (1984), Lunardi and Sinestrari (1986) and references therein.

*r*> 0) up to the present time

*t*, the temperature of the boundary \(\partial \Omega\) of \(\Omega\) is constant (=0) for all

*t*, and the external heat supply depends on the this thermal history of the body, then, system (8) becomes the following integrodifferential equation with finite delay:

*r*is a positive number. Let \(\lambda (t)\beta (t,\xi )\) denote the heating intensity, added to the system to control and regulate the heat supply. Then system (9) becomes

*X*is a Banach space.

An example of a material with memory is *Shape-memory polymers* (SMPs), which are polymeric smart materials that have the ability to return from a deformed state (temporary shape) to their original (permanent) shape induced by an external stimulus (trigger), such as temperature change. That is they act adaptively to their environment, they can easily be shaped into different forms at a low temperature, but return to their original shape on heating.

*L*is a scalar function.

Equation (11) has been studied by many authors [see e.g., Ezzinbi et al. (2009) and the references contained in it]. But to the best of our knowledge, this equation has never been considered for optimal control.

The rest of the paper is organized as follows: In second section, we present some basic definitions and preliminaries results, which will be used in the subsequent sections. In third section, we obtain an a priori estimation of mild solutions of Eq. (1). In fourth section, sufficient conditions are established for the existence and uniqueness of mild solutions of Eq. (1), by applying a well known fixed point theorem, and extension by continuity techniques. In fifth section, we investigate the existence of optimal controls of a Lagrange optimal control problem for Eq. (1). Finally, the last section, an example is given to illustrate the main results of this work.

## Resolvent operators and Balder’s theorem

In this section we introduce some definitions and Lemmas that will be used throughout the paper.

*A*and

*B*(

*t*) are closed linear operators on a Banach space

*X*.

In the sequel, we assume *A* and \(\left( B(t)\right) _{t\ge 0 }\) satisfy the following conditions:

### \((\mathbf{H_1})\)

*A* is a densely defined closed linear operator in *X*. Hence \({\mathcal {D}}(A)\) is a Banach space equipped with the graph norm defined by, \(|y|=\Vert Ay\Vert +\Vert y\Vert\) which will be denoted by \((X_1,|\cdot |)\).

### \((\mathbf{H_2})\)

*X*such that

*B*(

*t*) is continuous when regarded as a linear map from \((X_1,|\cdot |)\) into \((X,\Vert \cdot \Vert )\) for almost all \(t\ge 0\) and the map \(t\mapsto B(t)y\) is measurable for all \(y\in X_1\) and \(t\ge 0\), and belongs to \(W^{1,1}({\mathbb {R}}^{+},X)\). Moreover there is a locally integrable function \(b:{\mathbb {R}}^{+}\rightarrow {\mathbb {R}}^{+}\) such that

### Remark 1

Note that \((\mathbf{H_2})\) is satisfied in the modelling of Heat Conduction in materials with memory and viscosity. More details can be found in Liang et al. (2008).

Let \({\mathcal {L}}(X)\) be the Banach space of bounded linear operators on *X*,

### **Definition 1**

- (i)
\(R(0)=Id_X\) and \(\Vert R(t)\Vert \le Ne^{\beta t}\) for some constants

*N*and \(\beta\). - (ii)
For all \(x\in X\), the map \(t\mapsto R(t)x\) is continuous for \(t\ge 0\).

- (iii)Moreover for \(x\in X_1\), \(R(\cdot )x\,\in \,{\mathcal {C}}^1({\mathbb {R}}^{+};X)\cap {\mathcal {C}}({\mathbb {R}}^{+};X_1)\) and$$\begin{aligned} R'(t)x&= AR(t)x+\int _0^tB(t-s)R(s)xds\\&= R(t)Ax+\int _0^tR(t-s)B(s)xds. \end{aligned}$$

Observe that the map defined on \({\mathbb {R}}^{+}\) by \(t\mapsto R(t)x_0\) solves Eq. (12) for \(x_0\in {\mathcal {D}}(A)\).

### **Theorem 2**

(Grimmer 1983) *Assume that* \((H_1)\) *and* \((H_2)\) *hold. Then, the linear Eq.* (12) *has a unique resolvent operator* \(\left( R(t)\right) _{t\ge 0 }\).

### Remark 2

The following Theorem is needed in the proof of the existence of optimal controls.

### **Theorem 3**

*Let*\((\varSigma ,{\mathcal {F}},\mu )\)

*be a finite nonatomic measure space*, \((Y,\Vert \cdot \Vert )\)

*a separable Banach space*,

*and*\((V,|\cdot |)\)

*a separable reflexive Banach space*,

*and*\(V'\)

*its dual. Let*\(\theta : \varSigma \times Y\times V\rightarrow (-\infty ,+\infty ]\)

*be a given*\({\mathcal {F}}\times {\mathcal {L}}(Y\times V)\)-

*measurable function*.

*The associated integral functional*\(I_{\theta }:L_Y^1\times L_V^1\rightarrow [-\infty ,+\infty ]\)

*is defined by:*

*where*\(L_Y^1\)

*denotes the space of all absolutely summable functions from*\(\varSigma\)

*to*

*Y*.

*The following three conditions*

- (i)
\(\theta (t,\cdot ,\cdot )\)

*is sequencially lower semicontinuous on*\(X\times V,\ \mu\)-*a.e.*, - (ii)
\(\theta (t,x,\cdot )\)

*is convex on**V**for*\(x\in Y\), \(\mu\)-*a.e.*, - (iii)
*There exist*\(\sigma >0\)*and*\(\psi \in L_{{\mathbb {R}}}^1\)*such that*$$\begin{aligned} \theta (t,x,v)\ge \psi (t)-\sigma (\Vert x\Vert +|v|)\quad for\; all\ \ x\in Y,\quad v\in V,\quad \mu {\text {-a.e}}., \end{aligned}$$

*are sufficient for sequential strong–weak lower semicontinuity*\(I_{\theta }\)

*on*\(L_Y^1\times L_V^1\).

*Moreover, they are also necessary, provided that*\(I_{\theta }(\overline{x},\overline{v})<+\infty\)

*for some*\(\overline{x}\in L_Y^1,\overline{v}\in L_V^1\).

### **Theorem 4**

(Mazur’s theorem) *Let* *Z* *be a Banach space and* *G* *be a convex and closed set in* *Z*. *Then* *G* *is weakly closed in* *Z*.

## Existence of mild solutions for Eq. (1)

We make the following assumptions.

### \(\mathbf{(H_3)}\)

- (i)
\(f(\cdot ,\psi )\) is measurable for \(\psi \in {\mathcal {C}}\),

- (ii)
For any \(\rho >0\), there exists \(L_f(\rho )>0\) such that

$$\begin{aligned} \Vert f(t,\psi _1)-f(t,\psi _2)\Vert \le L_f(\rho )\Vert \psi _1-\psi _2\Vert \quad {\text {for}}\ \Vert \psi _1\Vert \le \rho ,\ \Vert \psi _2\Vert \le \rho \ {\text {and}}\ t\in [0,b], \end{aligned}$$ - (iii)There exists \(a_f>0\) such that$$\begin{aligned} \Vert f(t,\psi )\Vert \le a_f(1+\Vert \psi \Vert )\quad {\text {for all}}\ \ \psi \in {\mathcal {C}}\ \ {\text {and}}\ \ \ \ t\in [0,b]. \end{aligned}$$

### \(\mathbf{(H_4)}\)

Let *U* be the separable reflexive Banach space from which the control *u* takes values and assume \(C\in L^{\infty }(I;{\mathcal {L}}(U,X))\).

### \(\mathbf{(H_5)}\)

The multivalued map \(\varGamma :I\rightarrow 2^U{\setminus }\{\emptyset \}\) has closed, convex, and bounded values, \(\varGamma\) is graph measurable, and \(\varGamma (\cdot )\subseteq \Omega\) where \(\Omega\) is a bounded set in *U*.

### **Theorem 5**

Wang et al. (2012) \({\mathcal {U}}_{ad}\ne \emptyset\) *and* \({\mathcal {U}}_{ad}\subset L^2(I,U)\) *is bounded, closed and convex. Also*, \(Cu\in L^2(I,U)\ \ {\text {for all}}\ \ u\in {\mathcal {U}}_{ad}.\)

### **Definition 6**

We have the following Theorem on existence of mild solutions to Eq. (1) with respect to a given control \(u\in {\mathcal {U}}_{ad}\).

### **Theorem 7**

*Assume that* H_{1}–H_{5} *hold. Then for each* \(u\in {\mathcal {U}}_{ad}\), *Eq.* (1) *has a unique mild solution on* \([-r,b]\).

### Proof

*f*that

Let \(\varphi \in {\mathcal {C}},\ \rho =\Vert \varphi \Vert +1\) and \(\rho ^*=L_f(\rho )\rho +\sup _{s\in [0,b_1]}\Vert f(s,0)\Vert\).

Then, \(E_{\varphi }\) is a closed subset of \({\mathcal {C}}([-r,b_1];X)\) which is endowed with the uniform norm topology.

*K*is a strict contraction on \(E_{\varphi }\). It follows from the contraction mapping principle that

*K*has a unique fixed point \(x\in E_{\varphi }\), which is the unique mild solution of Eq. (1) with respect to

*u*on \([-r,b_1]\). \(\square\)

Using the same arguments, we can show that *x* can be extended to a maximal interval of existence \([0,t_{\max }[\).

### **Lemma 8**

(Ezzinbi et al. 2009) *If* \(t_{\max }<b\), *then*, \(\limsup _{t\rightarrow t_{\max }}\Vert x(t)\Vert =\infty\).

## Continuous dependence and existence of the optimal control solving Eq. (1)

In this section, we discuss the continuous dependence of the mild solutions of Eq. (1) on the controls and initial states, and the existence of solutions of the Lagrange problem associated to Eq. (1).

We have the following a priori estimation.

### **Lemma 9**

*Suppose (H* _{ 1 } *)–(H* _{ 3 } *) holds and assume that Eq.* (1) *has a mild solution* \(x_u\) *on* \([-r,b]\) *with respect to* \(u\in {\mathcal {U}}_{ad}\). *Then, there exists a constant* \(\rho >0\) *independent of* *u* *such that* \(\Vert x_u(t)\Vert \le \rho \quad {\text {for}}\ t\in [0,b]\), (\(\rho\) *depends only on* \({\mathcal {U}}_{ad}\) *and* \(\varphi\)).

### Proof

*z*on \([-r,0]\). Observe that

*x*satifies (13) if and only if \(z(0)=0\) and

We have the following theorem on continuous dependence of the mild solutions of Eq. (1) on the controls and initial states.

### **Theorem 10**

*For all*\(\lambda >0\),

*there exists*\(\gamma ^*(\lambda )>0\)

*such that for all*\(\varphi ^1,\ \varphi ^2\in B(0,\lambda )\),

*where*

*and*\(u^i\in {\mathcal {U}}_{ad}\), for

*i*= 1, 2.

### Proof

*i*= 1, 2, be two mild solutions of Eq. (1), corresponding to the controls \(u^i\in {\mathcal {U}}_{ad}\) and \(\lambda >0\) such that \(\varphi ^1,\ \varphi ^2\in B(0,\lambda )\).

For the existence of solutions to problem \(({\mathcal {LP}})\), we make the following assumptions.\(\mathbf{(H_L)}\)

- (i)
The functional \({\mathcal {L}}\,:\,I\times {\mathcal {C}}\times X\times U\rightarrow \mathbb {R}\cup \{\infty \}\) is Borel measurable.

- (ii)
\({\mathcal {L}}(t,\,\cdot ,\,\cdot ,\,\cdot )\) is sequencially lower semicontinuous on \({\mathcal {C}}\times X\times U\) for almost all \(t\in I\).

- (iii)
\({\mathcal {L}}(t,\,\psi ,\,y,\,\cdot )\) is convex on

*U*for each \(\psi \in {\mathcal {C}},\ \ y\in X\) and almost all \(t\in I\). - (iv)
There exist constants \(\nu ,\ \beta \ge 0,\ \gamma >0\), and \(\mu \in L^1(I)\) nonnegative such that

$$\begin{aligned} {\mathcal {L}}(t,\,\psi ,\,y,\,u)\ge \mu (t)+\nu \Vert \psi \Vert +\beta \Vert y\Vert +\gamma \Vert u\Vert . \end{aligned}$$

### **Theorem 11**

*Assume that hypotheses*

*(H*

_{ 1 }

*)–(H*

_{ 5 }

*)*

*and (H*

_{ L }

*) hold.*

*Then the Lagrange problem*\(({\mathcal {LP}})\)

*admits at least one optimal pair, that is there exists an admissible control pair*\((x^0,u^0)\in {\mathcal {C}}([-r,b],X)\times {\mathcal {U}}_{ad}\)

*such that*

### Proof

If \(\inf \left\{ {\mathcal {J}}(u)\,:\,u\in {\mathcal {U}}_{ad}\right\} =\infty\), we are done. \(\square\)

Without loss of generality, assume that \(\inf \left\{ {\mathcal {J}}(u)\,:\,u\in {\mathcal {U}}_{ad}\right\} =\delta <\infty\).

_{L}), \({\mathcal {L}}(t,\,\psi ,\,y,\,\cdot )\) is weakly lower semicontinuous, so we have that

### **Lemma 12**

*Let*\((u^n)_{n\ge 1}\subset {\mathcal {U}}_{ad}\)

*and*\(u^0\in {\mathcal {U}}_{ad}\)

*such that*\((u^n)_{n\ge 1}\)

*converges weakly to*\(u^0\).

*Then,*

_{L}) implies the assumptions of Balder’s Theorem. Hence by using Balder’s Theorem, we can conclude that \((x_t,\,x,\,u)\mapsto {\int _0^b{\mathcal {L}}\left( t,\,x_t,\,x(t),\,u(t)\right) \,dt}\) is sequencially lower semicontinuous in the strong topology of \({\mathcal {C}}([-r,0],X)\times L^1(I,X)\times L^1(I,U)\).

Now, since \({\mathcal {C}}([-r,0],X)\times L^2(I,X)\times L^2(I,U)\subset {\mathcal {C}}([-r,0],X)\times L^1(I,X)\times L^1(I,U)\), \({\mathcal {J}}\) is also sequencially lower semicontinuous on \({\mathcal {C}}([-r,0],X)\times L^2(I,X)\times L^2(I,U)\), and in the strong topology of \(L^1(I,E_{\varphi }\times X\times U)\).

_{L})-(iv), \({\mathcal {J}}>-\infty\), \({\mathcal {J}}\) attains its infimum at \(u^0\in {\mathcal {U}}_{ad}\), that is

We now illustrate our main result by the following example. We observe that in Wang et al. (2012), the Langrangian function \({\mathcal {L}}\) defined by the authors in the example does not satisfy condition (H_{L})–(iv), as they claimed. We correct that here.

## Example

*t*and \(\zeta \in W^{1,1}({\mathbb {R}}^{+},\mathbb {R}^{+})\) Let \(X=U=L^2(\Omega )\).

### **Theorem 13**

[Theorem 4.1.2, p. 79 of Vrabie (2003)] *The linear operator* *A* *defined above, is the infinitesimal generator of a* *C* _{0}-*semigroup on* \(L^2(\Omega )\).

*A* generates a *C* _{0}-semigroup \(\left( T(t)\right) _{t\ge 0}\) on \(L^2(\Omega )\).

*f*satisfies (H

_{3}). Now we consider the following cost function:

_{L}). Then,

## Conclusions

In this work, we have considered a broader class of partial functional integrodifferential equations with finite delay in Banach spaces. Under some suitable conditions, we have shown the existence and uniqueness of mild solutions using contraction principle. Moreover, we showed the existence of optimal controls of the associated Lagrange problem using convex optimization techniques and Balder’s theorem. We also provided an example to illustrate our results which extend and complement many other important results in the literature.

## Notes

### Authors' contributions

PN and KE are contributed equally in solving the problem. Moreover, PN typed the manuscript and KE made some corrections. Both authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

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