Fractional Optimal Control For Infinite Variables Parabolic SystemsWith Time Lags Given In Integral Form

1. Introduction

In recent years a considerable interest has been shown in the so-called fractional calculus, which allows us to consider integration and differentiation of any order, not necessarily integer. To a large extent this is due to the applications of the fractional calculus to problems in different areas of physics and engineering. The fractional calculus can be considered an old and yet novel topic. Starting from some speculations of Leibniz and Euler, followed by the works of other eminent mathematicians including Laplace, Fourier, Abel, Liouville and Riemann, it has undergone a rapid development especially during the past two decades.

One of the emerging branches of this study is the theory of fractional evolution equations, i.e. evolution equations where the integer derivative with respect to time is replaced by a derivative of fractional order. The increasing interest in this class of equations is motivated both by their application to problems from viscoelasticity, heat conduction in materials with memory, electrodynamics with memory, and also because they can be employed to approach nonlinear conservation laws.

Generally, the theory of fractional differential equations has received much attention over the past twenty years, since they are important in describing the natural models such as diffusion processes, stochastic processes, finance and hydrology. Concerning the literature of fractional equations we cite the books [Miller and Ross (1973), Oldham and Spanier (1974), Podlubny (1999)], the recent papers [Agrawal (2002), Baleanu and Avkar (2004), Baleanu and Muslih (2005), Baleanu and Agrawal (2006), Defterli et al (2015) ] and the references therein.

On the other hand, the general literature on fractional optimal controls for the differential evolution equations is extensive and different topics on optimal controls are considered. Concerning the motivations, relevant developments and the current status of the theory we refer the reader to the recent papers [Agrawal (2004), (2008), Agrawal and Baleanu (2007), Agrawal et al. (2010), Bahaa (2016a), (2016b), (2016c), (2017a), Baleanu et al. (2009), Doha et al. (2015), Fredrico Gastao et al. (2008), Jajarmi and Baleanu (2017), Jarad et al. (2010), (2012), Mophou (2011a), (2011b)] and the references therein.

During the last twenty years, integer differential equations and fractional differential equations with deviating argument have been applied not only in applied mathematics, physics and automatic control, but also in some problems of economy and biology. Currently, the theory of differential equations with deviating arguments constitutes a very important subfield of mathematical control theory. Consequently, fractional differential equations with deviating arguments are widely applied in optimal control problems of distributed parameter system with time delays we refer the reader to the recent papers [ Bahaa (2017), Jajarmi and Baleanu (2017), Jarad et al. (2010, (2012), Mophou and Fosting (2014)] and the references therein.

On the other hand various optimization problems associated with the integer optimal control of distributed parabolic systems with time delays appearing in the boundary conditions have been studied recently by [ Knowles, (1978) and Kowalewski, (1998); (1999); Kowalewski and Duda, (1992) and Wang, (1975)] and the references therein.

The necessary and sufficient conditions of optimality for system consists of only one integer differential equation and $(n\times n)$ integer differential systems governed by different types of partial differential equations defined on spaces of functions of infinitely many variables are discussed in [Gali & El-Saify, (1982); (1983)] in which the argument of (Lions, 1971 and Lions & Magenes, 1972) were used.

Making use of the Dubovitskii-Milyutin Theorem (Kotarski, El-Saify& Bahaa, 2002) and (Bahaa, 2003; 2005; 2008, 2017b). Kotarski et al. have obtained necessary and sufficient conditions of optimality for similar integer differential systems governed by second order operator with an infinite number of variables. The interest in the study of this class of operators is stimulated by problems in quantum field theory.

In [ Bahaa (2017), Jajarmi and Baleanu (2017), Jarad et al. (2010, (2012), Mophou and Fosting (2014)], they study the fractional optimal control problem for fractional differential equations contains time delay as a constant term appears only in the state equations with second order operator with finite number of variables.

In this paper, the fractional optimal control problem for distributed parabolic systems involving constant lags in the integral form both in the state equations and in the in the Neumann boundary condition is considered. The fractional time derivative is considered in a Caputo sense. The system contains model operator of second order with infinite number of variables. Actually distributed parameters systems with delays can be used to describe many phenomena in the real world. As is well known, heat conduction, properties of elastic-plastic material, fluid dynamics, diffusion-reaction processes, the transmission of the signals at a certain distance by using electric long lines, etc., all lie within this area.

We first study the existence and the uniqueness of the solution of the fractional differential system with time delay in a Hilbert space. Then we show that the considered optimal control problem has a unique solution. The performance index of a (FOCP) is considered as a function of both state and control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE). The time horizon is fixed. Finally, we impose some constraints on the boundary control. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of right fractional Caputo derivative, we obtain an optimality system for the fractional optimal control. Some specific properties of the optimal control are discussed. Some examples are analyzed in details.

This paper is organized as follows. In section 1, we introduce spaces of functions of infinitely many variables. In section 2 we introduce some definitions of fraction operators and green function. In section 3, we formulate the fractional mixed Neumann problem for parabolic operator with an infinite number of variables and time lags. In section 4, the boundary optimal control problem for this case is formulated, then we give the necessary and sufficient conditions for the control to be an optimal. In section 5, some illustrate examples are given, In section 6, we conclude the results and the future works in conclusion section.

1.1. 1-Sobolev Spaces with Infinite Number of Variables

This section covers the basic notations, definitions and properties, which are necessary to present this work see (Berezanskii, 1975) and [Gali and El-Saify, (1982); (1983), Bahaa (2003); (2005); (2008)].

Let ${\left(p_k\left(t\right)\right)}_{k=1}^{\infty }$ be a sequence of weights, fixed in all that follows, such that;

$$0<p_k\left(t\right)\in C^{\infty }\left({\mathbb{R}}^1\right),{\int }_{{\mathbb{R}}^1}{p}_k\left(t\right)dt=1,$$

with respect to it we introduce on the region ${\mathbb{R}}^{\infty }={\mathbb{R}}^1\times {\mathbb{R}}^1\times ....,$ the measure $d\rho \left(x\right)$ by setting,

$$d\rho \left(x\right)=p_1\left(x_1\right)d{x}_1\otimes p_2\left(x_2\right)d{x}_2\otimes ....,({\mathbb{R}}^{\infty }\ni x={\left(x_k\right)}_{k=1}^{\infty },x_k\in {\mathbb{R}}^1).$$

On ${\mathbb{R}}^{\infty }$ we construct the space $L^2({\mathbb{R}}^{\infty },d\rho \left(x\right))$ with respect to this measure i.e., $L^2({\mathbb{R}}^{\infty },d\rho \left(x\right))$ is the space of quadratic integrable functions on ${\mathbb{R}}^{\infty }$. We shall often set $L^2({\mathbb{R}}^{\infty },d\rho \left(x\right))=L^2\left({\mathbb{R}}^{\infty }\right)$.

It is classical result that $L^2\left({\mathbb{R}}^{\infty }\right)$ is a Hilbert space for the scalar product

$${(, )}_{L^2\left({\mathbb{R}}^{\infty }\right)}={\int }_{{\mathbb{R}}^{\infty }}\left(x\right)\left(x\right)d\rho \left(x\right).$$

We next consider a Sobolev space in the case of an unbounded region. For functions which are $\ell =1,2,...$ times continuously differentiable up to the boundary $\Gamma$ of ${\mathbb{R}}^{\infty }$ and which vanish in a neighborhood of ∞, we introduce the scalar product

$${(, )}_{W^{\ell }\left({\mathbb{R}}^{\infty }\right)}=\sum _{|\le \ell }{\left(D^{\beta }{D}^{\beta }\right)}_{L^2\left({\mathbb{R}}^{\infty }\right)},$$

where $D^{\beta }$ is defined by

$$D^{\beta }={\displaystyle \frac{\left|\beta \right|}{{\left(x_1\right)}^{{\beta }_1}{\left(x_2\right)}^{{\beta }_2}....}},\left|\beta \right|=\sum _{i=1}^{\infty }{\beta }_i,$$

and the differentiation is taken in the sense of generalized functions on ${\mathbb{R}}^{\infty }$, and after the completion, we obtain the Sobolev space $W^{\ell }\left({\mathbb{R}}^{\infty }\right)$. So in short, Sobolev space $W^1\left({\mathbb{R}}^{\infty }\right)$ is defined by :

$$W^1\left({\mathbb{R}}^{\infty }\right)=\left\{\right|,D\in L^2\left({\mathbb{R}}^{\infty }\right)\}.$$

As in the case of a bounded region, the space $W^1\left({\mathbb{R}}^{\infty }\right)$ form the space with positive norm ${\left|\right|.\left|\right|}_{W^1\left({\mathbb{R}}^{\infty }\right)}$. We can construct the space $W^{-1}\left({\mathbb{R}}^{\infty }\right)={\left(W^1\left({\mathbb{R}}^{\infty }\right)\right)}^{*}$ with negative norm ${\left|\right|.\left|\right|}_{W^{-1}\left({\mathbb{R}}^{\infty }\right)}$ with respect to the space $W^0\left({\mathbb{R}}^{\infty }\right)=L^2\left({\mathbb{R}}^{\infty }\right)$ with zero norm ${\left|\right|.\left|\right|}_{L^2\left({\mathbb{R}}^{\infty }\right)}$, then we have the following equipped,

$$W^1\left({\mathbb{R}}^{\infty }\right)\subseteq L^2\left({\mathbb{R}}^{\infty }\right)\subseteq W^{-1}\left({\mathbb{R}}^{\infty }\right),$$

$${\left|\right||}_{W^1\left({\mathbb{R}}^{\infty }\right)}\ge {\left|\right||}_{L^2\left({\mathbb{R}}^{\infty }\right)}\ge {\left|\right||}_{W^{-1}\left({\mathbb{R}}^{\infty }\right)}.$$

Let $L^2(0,T;W^1\left({\mathbb{R}}^{\infty }\right))$ be the space of square integrable measurable functions $t\to t\left)\mathrm{of}\right]0,T[\to W^1\left({\mathbb{R}}^{\infty }\right)$, where the variable t denotes the "time"; $t\in ]0,T[,T<\infty$. This space is a Hilbert space with respect to the scalar product

$${(,)}_{L^2(0,T;W^1\left({\mathbb{R}}^{\infty }\right))}={\int }_0^T{\left(\right(t),\left(t\right))}_{W^1\left({\mathbb{R}}^{\infty }\right)}dt,$$

and its dual is the space $L^2(0,T;W^{-1}\left({\mathbb{R}}^{\infty }\right))$, analogously, we can define the spaces $L^2(0,T;L^2\left({\mathbb{R}}^{\infty }\right))$ which we shall denote by $L^2\left(Q\right)$.

Let $\Omega \subset {\mathbb{R}}^{\infty }$ is a bounded, open set with boundary $\Gamma$, which is a $C^{\infty }$ manifold of dimension $(n-1$). Locally, $\Omega$ is totally on one side of $\Gamma$ and denote by $W^1(\Omega ,{\mathbb{R}}^{\infty },d\rho \left(x\right))$ (briefly $W^1(\Omega ,{\mathbb{R}}^{\infty }))$ the Sobolev space of vector function $y\left(x\right)$ defined on $\Omega$. The spaces considered in this paper are assumed to be real.

2. Fractional Operators

Many definitions have been given of a fractional derivative, which include Riemann-Liouville, Grünwald-Letnikov, Weyl, Caputo, Marchaud, and Riesz fractional derivative see Agrawal et al. (2002); (2004); (2007); (2008); (2010)]. We will formulate the problem in terms of the left and right Caputo fractional derivatives which will be given later.

Definition 2.1

Let $x:[a,b]\to R$ be a continuous function on $[a,b]$ and $\alpha >0$ be a real number, and $n=\left[\alpha \right]$, where $\left[\alpha \right]$ denotes the smallest integer greater than or equal to $\alpha$. The left (left RLFI) and the right (right RLFI) Riemann-Liouville fractional integrals of order $\alpha$ are defined by

$${}_{a}I_t^{\alpha }x\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{(t-s)}^{\alpha -1}x\left(s\right)ds(\mathrm{left}\mathrm{RLFI}),$$

$${}_{t}I_b^{\alpha }x\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_t^b{(s-t)}^{\alpha -1}x\left(s\right)ds(\mathrm{right}\mathrm{RLFI}),$$

where

$$\Gamma \left(\alpha \right)={\int }_0^{\infty }{e}^{-t}{u}^{\alpha -1}du,{}_{a}I_t^{0}x\left(t\right){=}_t{I}_b^{0}x\left(t\right)=x\left(t\right).$$

In the case of $\alpha =1$, the fractional integral reduces to the classical integral.

The left (left RLFD) and the right (right RLFD) Riemann-Liouville fractional derivatives of order $\alpha$ are defined by

$${}_{a}D_t^{\alpha }x\left(t\right)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\displaystyle \frac{d^n}{d{t}^n}}{\int }_a^t{(t-s)}^{n-\alpha -1}x\left(s\right)ds(\mathrm{left}\mathrm{RLFD})$$

$${}_{t}D_b^{\alpha }x\left(t\right)={\displaystyle \frac{{(-1)}^n}{\Gamma (n-\alpha )}}{\displaystyle \frac{d^n}{d{t}^n}}{\int }_t^b{(s-t)}^{n-\alpha -1}x\left(s\right)ds(\mathrm{right}\mathrm{RLFD})$$

where $\alpha \in (n-1,n),n\in N.$

Moreover, The left (left CFD) and the right (right CFD) Caputo fractional derivatives of order $\alpha$ are defined by

$${}_a{}^{C}D_t^{\alpha }x\left(t\right)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\int }_a^t{(t-s)}^{n-\alpha -1}{x}^{\left(n\right)}\left(s\right)ds(\mathrm{left}\mathrm{CFD})$$

provided that the integral is defined.

$${}_t{}^{C}D_b^{\alpha }x\left(t\right)={\displaystyle \frac{{(-1)}^n}{\Gamma (n-\alpha )}}{\int }_t^b{(s-t)}^{n-\alpha -1}{x}^{\left(n\right)}\left(s\right)ds(\mathrm{right}\mathrm{CFD})$$

provided that the integral is defined.

The relation between the right RLFD and the right CFD is as follows:

$${}_t{}^{C}D_b^{\alpha }x\left(t\right)={}_{t}D_b^{\alpha }x\left(t\right)-\sum _{k=0}^{n-1}{\displaystyle \frac{x^{\left(k\right)}\left(b\right)}{\Gamma (k-\alpha +1)}}{(b-t)}^{(k-\alpha )}.$$

If x and $x\left(i\right),i=1,...,n-1$, vanish at $t=a$, then ${}_{a}D_t^{\alpha }x\left(t\right)={}_a{}^{C}D_t^{\alpha }x\left(t\right)$, and if they vanish at $t=b$, then ${}_{t}D_b^{\alpha }x\left(t\right)={}_t{}^{C}D_b^{\alpha }x\left(t\right)$.

Further, it holds

$${}_0{}^{C}D_t^{\alpha }c=0,\mathrm{where}\mathrm{c}\mathrm{is}\mathrm{a}\mathrm{constant},$$

and

$${}_0{}^{C}D_t^{\alpha }{t}^n=\{0,\mathrm{for}n\in N_0\mathrm{and}n<\left[\alpha \right];{\displaystyle \frac{\Gamma (n+1)}{\Gamma (n+1-\alpha )}}{t}^{n-\alpha },\mathrm{for}n\in N_0\mathrm{and}n\ge \left[\alpha \right],$$

where $N_0=0,1,2,...$. We recall that for $\alpha \in N$ the Caputo differential operator coincides with the usual differential operator of integer order.

Lemmma 2.1

Let $T>0,u\in C^m\left([0,T]\right),p\in (m-1,m),m\in N$ and $v\in C^1\left([0,T]\right)$. Then for $t\in [0,T],$ the following properties hold

$${}_{a}D_t^{p}v\left(t\right)={\displaystyle \frac{d}{dt}}{}_{a}I_t^{1-p}v\left(t\right),m=1,$$

$${}_{a}D_t^p{}_{a}I_t^{p}v\left(t\right)=v\left(t\right);$$

$${}_{0}I_t^p{}_{0}D_t^{p}u\left(t\right)=u\left(t\right)-\sum _{k=0}^{m-1}{\displaystyle \frac{t^k}{k!}}{u}^{\left(k\right)}\left(0\right);$$

$$\underset{t\to 0^{+}}{lim}{}_0{}^{C}D_t^{p}u\left(t\right)=\underset{t\to 0^{+}}{lim}{}_{0}I_t^{p}u\left(t\right)=0.$$

Note also that when $T=+\infty ,{}_0{}^{C}D_t^{\alpha }f\left(t\right)$ is the Weyl fractional integral of order $\alpha$ of f.

An important tool is the integration by parts formula for Caputo fractional derivatives, which is stated in the following lemma.

Lemmma 2.2

[Agw1,Agw2]. Let $\alpha \in (0,1)$, and $x,y:[a,b]\to R$ be two functions of class $C^1$. Then the following integration by parts formula holds:

$${\int }_a^{b}y{\left(t\right)}_a^C{D}_{t}x\left(t\right)dt={{[}_t{I}_b^{1-\alpha }y\left(t\right)x\left(t\right)]}_a^b+{\int }_a^{b}x{\left(t\right)}_t{D}_b^{\alpha }y\left(t\right)dt.$$

Lemmma 2.3

(Fractional Green’s formula) ([?] ). Let $0<\alpha \le 1$. Then for any $\varphi \in C^{\infty }\left(\overline{Q}\right)$ we have

$${\int }_0^T{\int }_{\Omega }{(}_0^C{D}_t^{\alpha }y(x,t)+Ay(x,t))\varphi (x,t)dxdt={\int }_{\Omega }\varphi (x,T){}_0{}^{C}I_t^{1-\alpha }y(x,T)dx-{\int }_{\Omega }\varphi (x,0){}_0{}^{C}I_t^{1-\alpha }y(x,0^{+})dx$$

$$+{\int }_0^T{\int }_{\Omega }y{\displaystyle \frac{\varphi }{{\nu }_A}}d\Gamma dt-{\int }_0^T{\int }_{\Omega }{\displaystyle \frac{y}{{\nu }_A}}\varphi d\Gamma dt+{\int }_0^T{\int }_{\Omega }y(x,t)(-{}_0{}^{C}D_t^{\alpha }\varphi (x,t)+A^{*}\varphi (x,t))dxdt.$$

where A is a given operator which is defined by (3.6) below and

$${\displaystyle \frac{y}{{\nu }_A}}=\sum _{i,j=1}^n{a}_{ij}{\displaystyle \frac{y}{x_j}}cos(n,x_j)\mathrm{on}\Gamma ,$$

$cos(n,x_j)$ is the i-th direction cosine of n,n being the normal at $\Gamma$ exterior to $\Omega$.

We also introduce the space

$$W(0,T):=\{y:y\in L^2(0,T;H_0^1(\Omega )),{}_0{}^{C}D_t^{\alpha }y\left(t\right)\in L^2(0,T;H^{-1}(\Omega ))\}$$

in which a solution of a fractional differential systems is contained. The spaces considered in this paper are assumed to be real.

Lemmma 2.4

Let $0<\alpha <1$, X be a Banach space and $f\in C\left(\right[0,T],X)$. Then for all, $t_1,t_2\in [0,T]$

$${\left|\right|}_0{I}_t^{\alpha }f\left(t_1\right){-}_0{I}_t^{\alpha }f\left(t_2\right){\left|\right|}_X\le {\displaystyle \frac{{\left|\right|f\left|\right|}_{L^{\infty }((0,T);X)}}{\Gamma (\alpha +1)}}{|t_1-t_2|}^{\alpha }.$$

Lemmma 2.5

Since $C([0,T],X)\subset L^{\infty }((0,T);X)\subset L^2((0,T);X)$ because $[0,T]$ is a bounded subset of R , Lemma 1.4 holds for $f\in L^2((0,T);X)$ and we have that ₀$I_t^{\alpha }f\in C([0,T],X)\subset L^2((0,T);X).$

3- Fractional Neumann problem for parabolic system with time lags

The object of this section is to formulate the following fractional mixed initial boundary value problem for the parabolic system with time lag which defines the state of the system model.

$${}_0{}^{C}D_t^{\alpha }y+A\left(t\right)y+{\int }_a^{b}c(x,t)y(x,t-h;u)dh=u,x\in \Omega ,t\in (0,T),h\in (a,b),$$

$$y(x,t^{\prime })={\Phi }_0(x,t^{\prime }),x\in \Omega ,t^{\prime }\in [-b,0),$$

$$y(x,0)=y_p\left(x\right),x\in \Omega ,$$

$${\displaystyle \frac{y}{{\nu }_A}}={\int }_a^{b}d(x,t)y(x,t-h)dh+v,x\in \Gamma ,t\in (0,T),$$

$$y(x,t^{\prime })={\Psi }_0(x,t^{\prime }),x\in \Gamma ,t^{\prime }\in [-b,0),$$

where $\Omega$ has the same properties as in Section 1. We have

$$y\equiv y(x,t;u),u\equiv u(x,t),v\equiv v(x,t),$$

$$Q=\Omega \times (0,T),Q=\Omega \times [0,T],Q_0=\Omega \times [-b,0),\Sigma =\Gamma \times (0,T),{\Sigma }_0=\Gamma \times [-b,0),$$

•T is a specified positive number representing a time horizon,

•y is a function defined on Q such that $\Omega \times (0,T)\ni (x,t)\to y(x,t)\in R$,

•$u,v$ are functions defined on Q and $\Sigma$ such that

$\Omega \times (0,T)\ni (x,t)\to u(x,t)\in R$ and

$\Gamma \times (0,T)\ni (x,t)\to v(x,t)\in R$,

•c is a given real $C^{\infty }$ function defined on Q(Q closure of Q),

•d is a given real $C^{\infty }$ function defined on $\Sigma$,

•h is a time delay such that $h\in (a,b)$ and $a>0$,

•${\Phi }_0,{\Psi }_0$ are initial functions defined on $Q_0$ and ${\Sigma }_0$ such that

$\Omega \times [-b,0)\ni (x,t^{\prime })\to {\Phi }_0(x,t^{\prime })\in R$, and

$\Gamma \times [-b,0)\ni (x,t^{\prime })\to {\Psi }_0(x,t^{\prime })\in R$ respectively.

The operator $A\left(t\right)$ in the state Equation (1) is a second order operator with infinite number of variables given by:

$$A\left(t\right)y\left(x\right)=\left(-\sum _{k=1}^{\infty }{\displaystyle \frac{1}{\sqrt{p_k(x_k,t)}}}{\displaystyle \frac{2}{x_k^2}}\sqrt{p_k(x_k,t)}+q(x,t)\right)y\left(x\right)=-\sum _{k=1}^{\infty }{D}_k^{2}y\left(x\right)+q(x,t)y\left(x\right),$$

where

$$D_k{y}_{(}x)={\displaystyle \frac{1}{\sqrt{p_k(x_k,t)}}}{\displaystyle \underset{x_k}{¯}}\sqrt{p_k(x_k,t)}y\left(x\right),$$

and $q(x,t)$ is a real-valued function in x which is a bounded and measurable on $\Omega \subset {\mathbb{R}}^{\infty }$, such that $q(x,t)\ge c_0>1,c_0$ is a constant. i.e., $A\left(t\right)$ is a bounded second order self-adjoint elliptic partial differential operator with an infinite number of variables maps $W^1(\Omega ,{\mathbb{R}}^{\infty })$ onto $W^{-1}(\Omega ,{\mathbb{R}}^{\infty })$. For this operator we define the bilinear form as follows:

Definition 3.1:

For each $t\in (0,T)$, we define a family of bilinear forms $\pi (t;y, )$ on $W^1(\Omega ,{\mathbb{R}}^{\infty })$ by:

$$\pi (t;y, )=(A\left(t\right)y, {)}_{L^2(\Omega ,{\mathbb{R}}^{\infty })},y,W^1(\Omega ,{\mathbb{R}}^{\infty }).$$

Then

$$\begin{gathered} \pi {(t;y,=(A\left(t\right)y,)}_{L^2(\Omega ,{\mathbb{R}}^{\infty })}={\left(-\sum _{k=1}^{\infty }{D}_k^{2}y\left(x\right)+q(x,t)y\left(x\right),x)\right)}_{L^2(\Omega ,{\mathbb{R}}^{\infty })}= \\ {\int }_{\Omega }\sum _{k=1}^{\infty }{D}_{k}y(x)D_k\left(x\right)d\rho \left(x\right)+{\int }_{\Omega }q(x,t)y(x)\left(x\right)d\rho \left(x\right). \end{gathered}$$

Lemma 3.1

The bilinear form $\pi (t;y, )$ is coercive on $W^1(\Omega ,{\mathbb{R}}^{\infty })$ that is

$$\pi (t;y,y)\ge {\lambda \left|\right|y\left|\right|}_{W^1(\Omega ,{\mathbb{R}}^{\infty })}^2,\lambda >0.$$

Proof.  

It is well known that the ellipticity of $A\left(t\right)$ is sufficient for the coerciveness of $\pi (t;y, )$ on $W^1(\Omega ,{\mathbb{R}}^{\infty })$.

$$\pi (t;,)={\int }_{\Omega }\sum _{k=1}^{\infty }{D}_k(x)D_k\psi \left(x\right)d\rho \left(x\right)+{\int }_{\Omega }q(x,t)(x)\psi \left(x\right)d\rho \left(x\right).$$

Then

$$\begin{gathered} \pi (t;y,y)={\int }_{\Omega }\sum _{k=1}^{\infty }|D_k{y\left(x\right)|}^{2}d\rho \left(x\right)+{\int }_{\Omega }{q(x,t)\left|y\left(x\right)\right|}^{2}d\rho \left(x\right)\ge \sum _{k=1}^{\infty }\left|\right|D_k{y\left(x\right)\left|\right|}_{L^2(\Omega ,{\mathbb{R}}^{\infty })}^2+c_0{\left|\right|y\left(x\right)\left|\right|}_{L^2(\Omega ,{\mathbb{R}}^{\infty })}^2 \\ ={\left|\right|y\left(x\right)\left|\right|}_{W^1(\Omega ,{\mathbb{R}}^{\infty })}^2+c_0{\left|\right|y\left(x\right)\left|\right|}_{L^2(\Omega ,{\mathbb{R}}^{\infty })}^2\ge {\left|\right|y\left(x\right)\left|\right|}_{W^1(\Omega ,{\mathbb{R}}^{\infty })}^2=\lambda {\left|\right|y\left|\right|}_{W^1(\Omega ,{\mathbb{R}}^{\infty })}^2,\lambda >0. \end{gathered}$$

This completes the proof.

Also we have:

$$\forall y,W^1(\Omega ,{\mathbb{R}}^{\infty })\mathrm{the}\mathrm{function}t\to \pi (t;y, )\mathrm{is}\mathrm{continuously}\mathrm{differentiable}\mathrm{in}(0,T)\mathrm{and}\pi (t;y, )=\pi (t;y).$$

Equations (1)-(5) constitute a fractional Neumann problem with time lags given in integral form. Then the left-hand side of the boundary condition (4) is written in the form

$${\displaystyle \frac{y\left(u\right)}{{\nu }_A}}=\sum _{k=1}^{\infty }\left(D_{k}y\left(u\right)\right)cos(n,x_k)=g(x,t).$$

Then (4) can be written as:

$${\displaystyle \frac{y\left(u\right)}{{\nu }_A}}={\int }_a^{b}d(x,t)y(x,t-h)dh+v(x,t)=g(x,t),x\in \Gamma ,t\in (0,T),$$

${\text{where} }_{\overline{{\nu }_A}}$ is a normal derivative at $\Gamma$, directed towards the exterior of $\Omega$, and $cos(n,x_k)$ is the $k-th$ direction cosine of n, with n being the normal at $\Gamma$ exterior to $\Omega$.

Remark 3.1 We shall apply the indication $g(x,t)$ appearing in (9) to prove the existence of a unique solution for (1)-(5).

We shall formulate sufficient conditions for the existence of a unique solution of the fractional mixed initial-boundary value problem (1)-(5) for the cases where the boundary control v is an element of the space $L^2(\Sigma )$ i.e., $v\in L^2(0,T;W^0(\Sigma ,R^{\infty }))=L^2(\Sigma )$.

For this purpose, for any pair of real numbers $r,s\ge 0$, we introduce the Sobolev space $W^{r,s}\left(Q\right)$ (Lions & Magenes, 1972, Vol. 2, p. 6) defined by

$$W^{r,s}\left(Q\right)=L^2(0,T;W^r(\Omega ,R^{\infty }))\cap W^s(0,T;L^2(\Omega ,R^{\infty })),0$$

which is a Hilbert space normed by

$${\left({\int }_0^T{\left|\right|y\left(t\right)\left|\right|}_{W^r(\Omega ,R^{\infty })}^{2}dt+{\left|\right|y\left|\right|}_{W^s(0,T;L^2(\Omega ,R^{\infty }))}^2\right)}^{1/2},1$$

where $W^s(0,T;L^2(\Omega ,R^{\infty }))$ denotes the Sobolev space of order s of functions defined on $(0,T)$ and taking values in $L^2(\Omega ,R^{\infty })$.

The existence of a unique solution for the fractional mixed initial-boundary value problem (1)-(5) on the cylinder Q can be proved using a constructive method, i.e., solving at first Eqns. (1)-(5) on the sub-cylinder $Q_1$ and in turn on $Q_2$ etc., until the procedure covers the whole cylinder Q. In this way, the solution in the previous step determines the next one.

For simplicity, we introduce the following notation:

$$E_j=((j-1)a,ja),Q_j=\Omega \times E_j,{\Sigma }_j=\Gamma \times E_j$$

$$Q_0=\Omega \times [-b,0),{\Sigma }_0=\Gamma \times [-b,0)\mathrm{for}j=1,...,K,\mathrm{where}K={\displaystyle \frac{T}{a}}$$

Consequently, using Theorem 3.1 of (Lions and Magenes, 1972), one may prove the following lemma.

Lemma 3.2 Let

$$u\in W^{-{\displaystyle \frac{1}{2}},-{\displaystyle \frac{1}{4}}}\left(Q\right),v\in L^2(\Sigma ),2$$

$$f_j\in W^{-{\displaystyle \frac{1}{2}},-{\displaystyle \frac{1}{4}}}\left(Q\right),3$$

where

$$f_j(x,t)=u(x,t)-{\int }_a^{b}c(x,t)y_{j-1}(x,t-h)dh,$$

$$g_j\in L^2\left({\Sigma }_j\right),4$$

with

$$g_j(x,t)={\int }_a^{b}d(x,t)y_{j-1}(x,t-h)dh+v(x,t),$$

$$y_{j-1}(.,(j-1)a)\in W^{{\displaystyle \frac{1}{2}}}(\Omega ,R^{\infty }).5$$

Then, there exists a unique solution $y_j\in W^{{\displaystyle \frac{3}{2}},{\displaystyle \frac{3}{4}}}\left(Q_j\right)$ for the mixed initial-boundary value problem (1), (4), (15).

3. Proof

We observe that for $j=1$,

$$y_0{|}_{Q_0}(x,t-h)={\Phi }_0(x,t-h),$$

and

$$y_0{|}_{{\Sigma }_0}(x,t-h)={\Psi }_0(x,t-h).$$

Then the assumptions (13)-(15) are fulfilled if we assume that ${\Phi }_0\in W^{{\displaystyle \frac{3}{2}},{\displaystyle \frac{3}{4}}}\left(Q_0\right),y_p\in W^{{\displaystyle \frac{1}{2}}}(\Omega ,R^{\infty })$ and ${\Psi }_0\in L^2\left({\Sigma }_0\right)$. These assumptions are sufficient to ensure the existence of a unique solution $y_1\in W^{{\displaystyle \frac{3}{2}},{\displaystyle \frac{3}{4}}}\left(Q_1\right)$. In order to extend the result to $Q_2$, we have to prove that

$$y_1(.,a)\in W^{{\displaystyle \frac{1}{2}}}(\Omega ,R^{\infty }),$$

$$y_1{|}_{{\Sigma }_1}\in L^2\left({\Sigma }_1\right),$$

$$f_2\in W^{{\displaystyle \frac{-1}{2}},{\displaystyle \frac{-1}{4}}}\left(Q_2\right).$$

Then from Theorem 2.1 and 2.2 of (Kowalewski, 1998), $y_1\in W^{{\displaystyle \frac{3}{2}},{\displaystyle \frac{3}{4}}}\left(Q_1\right)$ implies that the mappings $t\to y_1(.,t)$ is continuous from $[0,a]\to W^{{\displaystyle \frac{3}{4}}}(\Omega ,R^{\infty })\subset W^{{\displaystyle \frac{1}{2}}}(\Omega ,R^{\infty })$. Thus $y_1(.,a)\in W^{{\displaystyle \frac{1}{2}}}(\Omega ,R^{\infty })$. Then using the trace theorem (Theorem 2.3 of (Kowalewski, 1998)) we can verify that $y_1\in W^{{\displaystyle \frac{3}{2}},{\displaystyle \frac{3}{4}}}(\Omega ,R^{\infty })\left(Q_1\right)$ implies that $y_1\to y_1{|}_{{\Sigma }_1}$ is a linear, continuous mapping of $W^{{\displaystyle \frac{3}{2}},{\displaystyle \frac{3}{4}}}\left(Q_1\right)\to W^{1,{\displaystyle \frac{1}{2}}}\left({\Sigma }_1\right)$. Thus $y_1{|}_{{\Sigma }_1}\in L^2\left({\Sigma }_1\right)$. Also it is easy to notice that the assumption (13) follows from the fact that $y_1\in W^{{\displaystyle \frac{3}{2}},{\displaystyle \frac{3}{4}}}\left(Q_1\right)$ and $u\in W^{-{\displaystyle \frac{1}{2}},-{\displaystyle \frac{1}{4}}}\left(Q\right)$. Then, there exists a unique solution $y_2\in W^{{\displaystyle \frac{3}{2}},{\displaystyle \frac{3}{4}}}\left(Q_2\right)$. The foregoing result is now summarized for any $Q_j,j=3,......,K$. ▪

Theorem 2.1 Let $y_p$, ${\Phi }_0$,${\Psi }_0$, v and u be given with $y_p\in W^{{\displaystyle \frac{1}{2}}}(\Omega ,R^{\infty })$, ${\Phi }_0\in W^{{\displaystyle \frac{3}{2}},{\displaystyle \frac{3}{4}}}\left(Q_0\right),{\Psi }_0\in L^2\left({\Sigma }_0\right)$, $v\in L^2(\Sigma )$ and $u\in W^{-{\displaystyle \frac{1}{2}},-{\displaystyle \frac{1}{4}}}\left(Q\right)$. Then, there exists a unique solution $y\in W^{{\displaystyle \frac{3}{2}},{\displaystyle \frac{3}{4}}}\left(Q\right)$ for the mixed initial-boundary value problem (1)-(5). Moreover, $y(.,ja)\in W^{{\displaystyle \frac{1}{2}}}(\Omega ,R^{\infty })\mathrm{for}j=1,2,...,K.$

4. Problem Formulation and Optimization Theorems

Now, we formulate the fractional optimal control problem for (1)-(5) in the context of the Theorem 2.1, that is $v\in L^2(\Sigma )$.

Let us denote by $U=L^2(\Sigma )$ the space of controls. The time horizon T is fixed in our problem.

The performance functional is given by

$$I\left(v\right)={\lambda }_1{\int }_Q{[y(x,t;v)-z_d]}^{2}d\rho dt+{\lambda }_2{\int }_{\Sigma }\left(Nv\right)vd\Gamma dt6$$

where _i ≥ 0, and ₁ + ₂ > 0, $z_d$ is a given element in Ł²(Q); N is a positive linear operator on $L^2(\Sigma )$ into $L^2(\Sigma )$.

Control constraints: We define the set of admissible controls $U_{ad}$ such that

$$U_{ad}\mathrm{is}\mathrm{closed},\mathrm{convex}\mathrm{subset}\mathrm{of}U=L^2(\Sigma ).7$$

Let $y(x,t;v)$ denote the solution of the mixed initial-boundary value problem (1)-(5) at $(x,t)$ corresponding to a given control $v\in U_{ad}$. We note from theorem 2.1 that for any $v\in U_{ad}$ the performance functional (16) is well-defined since $y\left(v\right)\in W^{{\displaystyle \frac{3}{2}},{\displaystyle \frac{3}{4}}}\left(Q\right)\subset L^2\left(Q\right)$.

Making use of the Loins’s scheme we shall derive the necessary and sufficient conditions of optimality for the optimization problem (1)-(5), (16),(17). The solving of the formulated optimal control problem is equivalent to seeking a $v^{*}\in U_{ad}$ such that

$$I\left(v^{*}\right)\le I\left(v\right),\forall v\in U_{ad}.$$

From the Lion’s scheme (Theorem 1.3 of Lions, 1971, p.10), it follows that for ₂ > 0 a unique optimal control $v^{*}$ exists. Moreover, $v^{*}$ is characterized by the following condition

$$I^{\prime }\left(v^{*}\right)(v-v^{*})\ge 0\forall v\in U_{ad}.8$$

For the performance functional of form (16) the relation (18) can be expressed as

$${}_1\int _Q(y\left(v^{*}\right)-z_d)[y\left(v\right)-y\left(v^{*}\right)]d\rho dt{+}_2{\int }_{\Sigma }N{v}^{*}(v-v^{*})d\Gamma dt\ge 0\forall v\in U_{ad}.9$$

To simplify (19), we introduce the adjoint equation, and for every $v\in U_{ad}$ , we define the adjoint variable $p=p\left(v\right)=p(x,t;v)$ as the solution of the equation

$${}_t{}^{C}D_{T-b}^{\alpha }p\left(v\right)+A^{*}\left(t\right)p\left(v\right)+{\int }_a^{b}c(x,t+h)p(x,t+h;v)dh{=}_1(y\left(v\right)-z_d),$$

$$x\in \Omega ,t\in (0,T-b),0$$

$${}_t{}^{C}D_{T-a}^{\alpha }p\left(v\right)+A^{*}\left(t\right)p\left(v\right)+{\int }_a^{T-t}c(x,t+h)p(x,t+h;v)dh{=}_1(y\left(v\right)-z_d),$$

$$x\in \Omega ,t\in (T-b,T-a),1$$

$${}_t{}^{C}D_T^{\alpha }p\left(v\right)+A^{*}\left(t\right)p\left(v\right)=0,x\in \Omega ,t\in (T-a,T),2$$

$$p(x,T;v)=0,x\in \Omega ,3$$

$${\displaystyle \frac{p\left(v\right)}{{\nu }_{A^{*}}}}(x,t)={\int }_a^{b}d(x,t+h)p(x,t+h;v)dh,x\in \Gamma ,t\in (0,T-b),4$$

$${\displaystyle \frac{p\left(v\right)}{{\nu }_{A^{*}}}}(x,t)={\int }_a^{T-t}d(x,t+h)p(x,t+h;v)dh,x\in \Gamma ,t\in (T-b,T-a),5$$

$${\displaystyle \frac{p\left(v\right)}{{\nu }_{A^{*}}}}(x,t)=0x\in \Gamma ,t\in (T-a,T),6$$

where

$${\displaystyle \frac{p\left(v\right)}{{\nu }_A}}(x,t)=\sum _{k=1}^{\infty }\left(D_{k}p\left(v\right)\right)cos(n,x_k)=g(x,t),$$

$$A^{*}\left(t\right)p\left(v\right)=(-\sum _{k=1}^{\infty }{D}_k^2+q(x,t))p\left(v\right).7$$

As in the above section with change of variables, i.e. with reversed sense of time. i.e., $t^{\prime }=T-t$, for given $z_d\in L^2\left(Q\right)$ and any $v\in L^2(\Sigma )$, there exists a unique solution $p\left(v\right)\in W^{{\displaystyle \frac{3}{2}},{\displaystyle \frac{3}{4}}}\left(Q\right)$ for problem (20)-(26).

Remark 4.1If $T<b$, then we consider (21) and (25) on $\Omega \times (0,T-a)$ and $\Gamma \times (0,T-a)$, respectively.

The existence of a unique solution for the problem (20)-(26) on the cylinder $\Omega \times (0,T)$ can be proved using a constructive method. It is easy to notice that for given $z_d$ and u, the problem (20)-(26) can be solved backwards in time starting from $t=T$, i.e. first solving (20)-(26) on the sub-cylinder $Q_K$ and in turn on $Q_{K-1}$ , etc. until the procedure covers the whole cylinder $\Omega \times (0,T)$. For this purpose, we may apply Theorem 2.1 (with an obvious change of variables).

Hence, using Theorem 2.1, the following result can be proved.

Lemma 4.1 Let the hypothesis of Theorem 2.1 be satisfied. Then for given $z_d\in L^2(\Omega ,R^{\infty })$ and any $v\in L^2(\Sigma )$, there exists a unique solution $p\left(v\right)\in W^{{\displaystyle \frac{3}{2}},{\displaystyle \frac{3}{4}}}\left(Q\right)$ for the adjoint problem (20)-(26).

We simplify (19) using the adjoint Equation (20)-(26). For this purpose setting $v=v^{*}$ in (20)-(26), multiplying both sides of (20),(21), (22) by $y\left(v\right)-y\left(v^{*}\right)$, then integrating over $\Omega \times (0,T-b)$, $\Omega \times (T-b,T-a)$ and $\Omega \times (T-a,T)$, respectively, and then adding both sides of (20)-(22), we get

$${}_1\int _Q(y(T;v^{*})-z_d)[y(T;v)-y(T;v^{*})]d\rho dt=$$

$$={\int }_0^T{\int }_{\Omega }\left({}_t{}^{C}D_{T-b}^{\alpha }p\left(v\right)+A^{*}\left(t\right)p\left(v^{*}\right)\right)[y\left(v\right)-y\left(v^{*}\right)]d\rho \left(x\right)dt$$

$$+{\int }_0^{T-b}{\int }_{\Omega }\left({\int }_a^{b}c(x,t+h)p(x,t+h;v^{*})dh\right)\times [y(x,t;v)-y(x,t;v^{*})]d\rho \left(x\right)dt$$

$$+{\int }_{T-b}^{T-a}{\int }_{\Omega }\left({\int }_a^{T-t}c(x,t+h)p(x,t+h;v^{*})dh\right)\times [y(x,t;v)-y(x,t;v^{*})]d\rho \left(x\right)dt$$

$$={\int }_{\Omega }p(x,T;v^{*})[y(x,T;v)-y(x,T;v^{*})]d\rho \left(x\right)$$

$$+{\int }_0^T{\int }_{\Omega }p\left(v^{*}\right){}_{T-b}{}^{C}D_t^{\alpha }[y\left(v\right)-y\left(v^{*}\right)]d\rho \left(x\right)dt+{\int }_0^T{\int }_{\Omega }{A}^{*}\left(t\right)p\left(v^{*}\right)[y\left(v\right)-y\left(v^{*}\right)]d\rho \left(x\right)dt$$

$$+{\int }_0^{T-b}{\int }_{\Omega }{\int }_a^{b}c(x,t+h)p(x,t+h;v^{*})\times [y(x,t;v)-y(x,t;v^{*})]dhd\rho \left(x\right)dt$$

$$+{\int }_{T-b}^{T-a}{\int }_{\Omega }{\int }_a^{T-t}c(x,t+h)p(x,t+h;v^{*})\times [y(x,t;v)-y(x,t;v^{*})]dhd\rho \left(x\right)dt.8$$

Then applying (23), the formula (28) can be expressed as

$${}_1\int _{\Omega }(y(T;v^{*})-z_d)[y(T;v)-y(T;v^{*})]d\rho \left(x\right)=$$

$$={\int }_0^T{\int }_{\Omega }p\left(v^{*}\right){}_{T-b}{}^{C}D_t^{\alpha }[y\left(v\right)-y\left(v^{*}\right)]d\rho \left(x\right)dt+{\int }_0^T{\int }_{\Omega }{A}^{*}\left(t\right)p\left(v^{*}\right)[y\left(v\right)-y\left(v^{*}\right)]d\rho \left(x\right)dt$$

$$+{\int }_0^{T-b}{\int }_{\Omega }{\int }_a^{b}c(x,t+h)p(x,t+h;v^{*})\times [y(x,t;v)-y(x,t;v^{*})]dhd\rho \left(x\right)dt$$

$$+{\int }_a^{T-t}{\int }_{\Omega }{\int }_{T-b}^{T-a}c(x,t+h)p(x,t+h;v^{*})\times [y(x,t;v)-y(x,t;v^{*})]dhd\rho \left(x\right)dt.9$$

Using (1), the first integral on the right-hand side of (29) can be rewritten as

$${\int }_0^T{\int }_{\Omega }p\left(v^{*}\right){}_{T-b}{}^{C}D_t^{\alpha }[y\left(v\right)-y\left(v^{*}\right)]d\rho \left(x\right)dt=-{\int }_0^T{\int }_{\Omega }p\left(v^{*}\right)A\left(t\right)[y\left(v\right)-y\left(v^{*}\right)]d\rho \left(x\right)dt$$

$$-{\int }_0^T{\int }_{\Omega }p(x,t;v^{*})\left({\int }_a^{b}c(x,t)\times [y(x,t-h;v)-y(x,t-h;v^{*})]dh\right)d\rho \left(x\right)dt$$

$$=-{\int }_0^T{\int }_{\Omega }p\left(v^{*}\right)A\left(t\right)[y\left(v\right)-y\left(v^{*}\right)]d\rho \left(x\right)dt$$

$$-{\int }_0^T{\int }_{\Omega }{\int }_a^{b}p(x,t;v^{*})c(x,t)\times [y(x,t-h;v)-y(x,t-h;v^{*})]dhd\rho \left(x\right)dt$$

$$=-{\int }_0^T{\int }_{\Omega }p\left(v^{*}\right)A\left(t\right)[y\left(v\right)-y\left(v^{*}\right)]d\rho \left(x\right)dt$$

$$-{\int }_a^b{\int }_0^T{\int }_{\Omega }p(x,t;v^{*})c(x,t)\times [y(x,t-h;v)-y(x,t-h;v^{*})]dtd\rho \left(x\right)dh$$

$$=-{\int }_0^T{\int }_{\Omega }p\left(v^{*}\right)A\left(t\right)[y\left(v\right)-y\left(v^{*}\right)]d\rho \left(x\right)dt$$

$$-{\int }_a^b{\int }_{\Omega }{\int }_{-h}^{T-h}p(x,t^{\prime }+h;v^{*})c(x,t^{\prime }+h)\times [y(x,t^{\prime };v)-y(x,t^{\prime };v^{*})]d{t}^{\prime }d\rho \left(x\right)dh$$

$$=-{\int }_0^T{\int }_{\Omega }p\left(v^{*}\right)A\left(t\right)[y\left(v\right)-y\left(v^{*}\right)]d\rho \left(x\right)dt$$

$$-{\int }_a^b{\int }_{\Omega }{\int }_{-h}^{0}p(x,t^{\prime }+h;v^{*})c(x,t^{\prime }+h)\times [y(x,t^{\prime };v)-y(x,t^{\prime };v^{*})]d{t}^{\prime }d\rho \left(x\right)dh$$

$$-{\int }_a^b{\int }_{\Omega }{\int }_0^{T-b}p(x,t^{\prime }+h;v^{*})c(x,t^{\prime }+h)\times [y(x,t^{\prime };v)-y(x,t^{\prime };v^{*})]d{t}^{\prime }d\rho \left(x\right)dh$$

$$-{\int }_a^b{\int }_{\Omega }{\int }_{T-b}^{T-h}p(x,t^{\prime }+h;v^{*})c(x,t^{\prime }+h)\times [y(x,t^{\prime };v)-y(x,t^{\prime };v^{*})]d{t}^{\prime }d\rho \left(x\right)dh$$

$$=-{\int }_0^T{\int }_{\Omega }p\left(v^{*}\right)A\left(t\right)[y\left(v\right)-y\left(v^{*}\right)]d\rho \left(x\right)dt$$

$$-{\int }_a^b{\int }_{\Omega }{\int }_{-h}^{0}p(x,t^{\prime }+h;v^{*})c(x,t^{\prime }+h)\times [y(x,t^{\prime };v)-y(x,t^{\prime };v^{*})]d{t}^{\prime }d\rho \left(x\right)dh$$

$$-{\int }_a^b{\int }_{\Omega }{\int }_0^{T-b}p(x,t^{\prime }+h;v^{*})c(x,t^{\prime }+h)\times [y(x,t^{\prime };v)-y(x,t^{\prime };v^{*})]d{t}^{\prime }d\rho \left(x\right)dh$$

$$-{\int }_a^{T-t}{\int }_{\Omega }{\int }_{T-b}^{T-a}p(x,t^{\prime }+h;v^{*})c(x,t^{\prime }+h)\times [y(x,t^{\prime };v)-y(x,t^{\prime };v^{*})]d{t}^{\prime }d\rho \left(x\right)dh.0$$

The second integral on the right-hand side of (29), in view of Greens formula, can be expressed as

$${\int }_0^T{\int }_{\Omega }{A}^{*}\left(t\right)p\left(v^{*}\right)[y\left(v\right)-y\left(v^{*}\right)]d\rho \left(x\right)dt={\int }_0^T{\int }_{\Omega }p\left(v^{*}\right)A\left(t\right)[y\left(v\right)-y\left(v^{*}\right)]d\rho \left(x\right)dt$$

$$+{\int }_0^T{\int }_{\Gamma }p\left(v^{*}\right)\left({\displaystyle \frac{y\left(v\right)}{{\nu }_A}}-{\displaystyle \frac{y\left(v^{*}\right)}{{\nu }_A}}\right)d\Gamma dt-{\int }_0^T{\int }_{\Gamma }{\displaystyle \frac{p\left(v^{*}\right)}{{\nu }_{A^{*}}}}[y\left(v\right)-y\left(v^{*}\right))]d\Gamma dt.1$$

Using the boundary condition (4), the second component on the right-hand side of (31) can be written as

$${\int }_0^T{\int }_{\Gamma }p\left(v^{*}\right)\left({\displaystyle \frac{y\left(v\right)}{{\nu }_A}}-{\displaystyle \frac{y\left(v^{*}\right)}{{\nu }_A}}\right)d\Gamma dt=$$

$$={\int }_0^T{\int }_{\Gamma }p(x,t;v^{*})\left({\int }_a^{b}d(x,t)\times [y(x,t-h;v)-y(x,t-h;v^{*})]dh\right)d\Gamma dt$$

$$+{\int }_0^T{\int }_{\Gamma }p(x,t;v^{*})(v-v^{*})d\Gamma dt$$

$$={\int }_0^T{\int }_{\Gamma }{\int }_a^{b}p(x,t;v^{*})d(x,t)\times [y(x,t-h;v)-y(x,t-h;v^{*})]dhd\Gamma dt$$

$$+{\int }_0^T{\int }_{\Gamma }p(x,t;v^{*})(v-v^{*})d\Gamma dt$$

$$={\int }_a^b{\int }_{\Gamma }{\int }_0^{T}p(x,t;v^{*})d(x,t)\times [y(x,t-h;v)-y(x,t-h;v^{*})]dtd\Gamma dh$$

$$+{\int }_0^T{\int }_{\Gamma }p(x,t;v^{*})(v-v^{*})d\Gamma dt$$

$$={\int }_a^b{\int }_{\Gamma }{\int }_{-h}^{T-h}p(x,t^{\prime }+h;v^{*})d(x,t^{\prime }+h)\times [y(x,t^{\prime };v)-y(x,t^{\prime };v^{*})]d{t}^{\prime }d\Gamma dh$$

$$+{\int }_0^T{\int }_{\Gamma }p(x,t;v^{*})(v-v^{*})d\Gamma dt$$

$$={\int }_a^b{\int }_{\Gamma }{\int }_{-h}^{0}p(x,t^{\prime }+h;v^{*})d(x,t^{\prime }+h)\times [y(x,t^{\prime };v)-y(x,t^{\prime };v^{*})]d{t}^{\prime }d\Gamma dh$$

$$+{\int }_a^b{\int }_{\Gamma }{\int }_0^{T-b}p(x,t^{\prime }+h;v^{*})d(x,t^{\prime }+h)\times [y(x,t^{\prime };v)-y(x,t^{\prime };v^{*})]d{t}^{\prime }d\Gamma dh$$

$$+{\int }_a^b{\int }_{\Gamma }{\int }_{T-b}^{T-h}p(x,t^{\prime }+h;v^{*})d(x,t^{\prime }+h)\times [y(x,t^{\prime };v)-y(x,t^{\prime };v^{*})]d{t}^{\prime }d\Gamma dh$$

$$+{\int }_0^T{\int }_{\Gamma }p(x,t;v^{*})(v-v^{*})d\Gamma dt$$

$$={\int }_a^b{\int }_{\Gamma }{\int }_{-h}^{0}p(x,t^{\prime }+h;v^{*})d(x,t^{\prime }+h)\times [y(x,t^{\prime };v)-y(x,t^{\prime };v^{*})]d{t}^{\prime }d\Gamma dh$$

$$+{\int }_a^b{\int }_{\Gamma }{\int }_0^{T-b}p(x,t^{\prime }+h;v^{*})d(x,t^{\prime }+h)\times [y(x,t^{\prime };v)-y(x,t^{\prime };v^{*})]d{t}^{\prime }d\Gamma dh$$

$$+{\int }_a^{T-t}{\int }_{\Gamma }{\int }_{T-b}^{T-h}p(x,t^{\prime }+h;v^{*})d(x,t^{\prime }+h)\times [y(x,t^{\prime };v)-y(x,t^{\prime };v^{*})]d{t}^{\prime }d\Gamma dh$$

$$+{\int }_0^T{\int }_{\Gamma }p(x,t;v^{*})(v-v^{*})d\Gamma dt.2$$

The last component in (31) can be rewritten as

$${\int }_0^T{\int }_{\Gamma }{\displaystyle \frac{p\left(v^{*}\right)}{{\nu }_{A^{*}}}}[y\left(v\right)-y\left(v^{*}\right)]d\Gamma dt$$

$$={\int }_0^{T-b}{\int }_{\Gamma }{\displaystyle \frac{p\left(v^{*}\right)}{{\nu }_{A^{*}}}}[y\left(v\right)-y\left(v^{*}\right)]d\Gamma dt+{\int }_{T-b}^{T-a}{\int }_{\Gamma }{\displaystyle \frac{p\left(v^{*}\right)}{{\nu }_{A^{*}}}}[y\left(v\right)-y\left(v^{*}\right)]d\Gamma dt$$

$$+{\int }_{T-a}^T{\int }_{\Gamma }{\displaystyle \frac{p\left(v^{*}\right)}{{\nu }_{A^{*}}}}[y\left(v\right)-y\left(v^{*}\right)]d\Gamma dt.3$$

Substituting (32), (33) into (31) and then (31) into (29), we obtain

$${}_1\int _{\Omega }(y(T;v^{*})-z_d)[y(T;v)-y(T;v^{*})]d\rho \left(x\right)$$

$$=-{\int }_0^T{\int }_{\Omega }p\left(v^{*}\right)A\left(t\right)[y\left(v\right)-y\left(v^{*}\right)]d\rho \left(x\right)dt$$

$$-{\int }_{-h}^0{\int }_{\Omega }{\int }_a^{b}c(x,t+h)p(x,t+h;v^{*})\times [y(x,t;v)-y(x,t;v^{*})]dhd\rho \left(x\right)dt$$

$$-{\int }_0^{T-b}{\int }_{\Omega }{\int }_a^{b}c(x,t+h)p(x,t+h;v^{*})\times [y(x,t;v)-y(x,t;v^{*})]dhd\rho \left(x\right)dt$$

$$-{\int }_{T-b}^{T-a}{\int }_{\Omega }{\int }_a^{T-t}c(x,t+h)p(x,t+h;v^{*})\times [y(x,t;v)-y(x,t;v^{*})]dhd\rho \left(x\right)dt$$

$$+{\int }_0^T{\int }_{\Omega }p\left(u^{*}\right)A\left(t\right)[y\left(v\right)-y\left(v^{*}\right)]d\rho \left(x\right)dt$$

$$+{\int }_{-h}^0{\int }_{\Gamma }{\int }_a^{b}d(x,t+h)p(x,t+h;v^{*})\times [y(x,t;v)-y(x,t;v^{*})]dhd\Gamma dt$$

$$+{\int }_0^{T-b}{\int }_{\Gamma }{\int }_a^{b}d(x,t+h)p(x,t+h;v^{*})\times [y(x,t;v)-y(x,t;v^{*})]dhd\Gamma dt$$

$$+{\int }_{T-b}^{T-a}{\int }_{\Gamma }{\int }_a^{T-t}d(x,t+h)p(x,t+h;v^{*})\times [y(x,t;v)-y(x,t;v^{*})]dhd\Gamma dt$$

$$+{\int }_0^T{\int }_{\Gamma }p(x,t;v^{*})(v-v^{*})d\Gamma dt-{\int }_0^{T-b}{\int }_{\Gamma }{\displaystyle \frac{p\left(v^{*}\right)}{{\nu }_A^{*}}}[y(x,t;v)-y(x,t;v^{*})]d\Gamma dt$$

$$-{\int }_{T-b}^{T-a}{\int }_{\Gamma }{\displaystyle \frac{p\left(v^{*}\right)}{{\nu }_A^{*}}}[y(x,t;v)-y(x,t;v^{*})]d\Gamma dt-{\int }_{T-a}^T{\int }_{\Gamma }{\displaystyle \frac{p\left(v^{*}\right)}{{\nu }_A^{*}}}[y(x,t;v)-y(x,t;v^{*})]d\Gamma dt$$

$$+{\int }_0^{T-b}{\int }_{\Omega }{\int }_a^{b}c(x,t+h)p(x,t+h;v^{*})\times [y(x,t;v)-y(x,t;v^{*})]dhd\rho \left(x\right)dt$$

$$+{\int }_{T-b}^{T-a}{\int }_{\Omega }{\int }_a^{T-t}c(x,t+h)p(x,t+h;v^{*})\times [y(x,t;v)-y(x,t;v^{*})]dhd\rho \left(x\right)dt.4$$

Afterwards, using the fact that $y(x,t;v)=y(x,t;v^{*})={\Phi }_0(x,t)$ for $x\in \Omega$ and $t\in [-b,0)$ and $y(x,t^{\prime };v)=y(x,t^{\prime };v^{*})={\Psi }_0(x,t^{\prime })$ for $x\in \Gamma$ and $t^{\prime }\in [-b,0)$, we obtain

$${}_1\int _{\Omega }(y(T;v^{*})-z_d)[y(T;v)-y(T;v^{*})]d\rho \left(x\right)={\int }_0^T{\int }_{\Gamma }p\left(u^{*}\right)(v-v^{*})d\Gamma dt.5$$

Substituting (35) into (19) gives

$${\int }_0^T{\int }_{\Gamma }\left(p\left(v^{*}\right){+}_{2}N{v}^{*}\right)(v-v^{*})d\Gamma dt\ge 0\forall v\in U_{ad}.6$$

The foregoing result is now summarized.

Theorem 3.1 For the problem (1)-(5), with the performance functional (16) with $z_d\in L^2\left(Q\right)$ and ₂ > 0 and with conditions (5), (17), there exists a unique optimal control $v^{*}$ which satisfies the maximum condition (36).

5. Applications

To illustrate the practical applications of the algorithms mentioned above we shall formulate the following fractional control problems as examples.

Example 5.1 Let $U_{ad}=U=L^2(\Sigma )$, the case where there are no constraints on the control. Thus the maximum condition (36) is satisfied when

$$v^{*}={-}_2^{-1}{N}^{-1}p\left(v^{*}\right).$$

If N is the identity operator on $L^2(\Sigma )$, then from lemma 3.1 it follows that $v^{*}\in W^{{\displaystyle \frac{3}{2}},{\displaystyle \frac{3}{4}}}\left(Q\right)$.

Example 5.2 We can also consider an analogous fractional optimal control problem where the performance functional is given by

$$\widehat{I}\left(v\right)={\lambda }_1{\int }_{\Sigma }{[y(x,t;v)-z_d]}^{2}d\Gamma dt+{\lambda }_2{\int }_{\Sigma }\left(Nv\right)vd\Gamma dt7$$

where $z_d\in L^2(\Sigma )$.

From theorem 2.1 and the Trace Theorem (Lions& Magenes, 1972, Vol 2, p.9), for each $v\in L^2(\Sigma )$, there exists a unique solution $y\left(v\right)\in W^{{\displaystyle \frac{3}{2}},{\displaystyle \frac{3}{4}}}\left(Q\right)$ with ${y|}_{\Sigma }\in L^2(\Sigma )$. Thus, $\widehat{I}$ is well defined. Then, the optimal control $v^{*}$ is characterized by

$${}_1\int _{\Sigma }(y\left(v^{*}\right)-z_d)[y\left(v\right)-y\left(v^{*}\right)]d\Gamma dt{+}_2{\int }_{\Sigma }N{v}^{*}(v-v^{*})d\Gamma dt\ge 0\forall v\in U_{ad}.8$$

We define the adjoint variable $p=p\left(v^{*}\right)=p(x,t;v^{*})$ as the solution of the equations

$${}_{T-b}{}^{C}D_t^{\alpha }p\left(v^{*}\right)+A^{*}\left(t\right)p\left(v^{*}\right)+{\int }_a^{b}c(x,t+h)p(x,t+h;v^{*})dh=0x\in \Omega ,t\in (0,T-b),9$$

$${}_{T-a}{}^{C}D_t^{\alpha }p\left(v^{*}\right)+A^{*}\left(t\right)p\left(v^{*}\right)+{\int }_a^{T-t}c(x,t+h)p(x,t+h;v^{*})dh=0,x\in \Omega ,t\in (T-b,T-a),0$$

$${}_T{}^{C}D_t^{\alpha }p\left(v^{*}\right)+A^{*}\left(t\right)p\left(v^{*}\right)=0,x\in \Omega ,t\in (T-a,T),1$$

$$p(x,T;v^{*})=0,x\in \Omega ,2$$

$${\displaystyle \frac{p\left(v^{*}\right)}{{\nu }_{A^{*}}}}(x,t)={\int }_a^{b}d(x,t+h)p(x,t+h;v^{*})dh{+}_1(y\left(v^{*}\right)-z_d),x\in \Gamma ,t\in (0,T-b),3$$

$${\displaystyle \frac{p\left(v^{*}\right)}{{\nu }_{A^{*}}}}(x,t)={\int }_a^{T-t}d(x,t+h)p(x,t+h;v^{*})dh{+}_1(y\left(v^{*}\right)-z_d),x\in \Gamma ,t\in (T-b,T-a),4$$

$${\displaystyle \frac{p\left(v^{*}\right)}{{\nu }_{A^{*}}}}(x,t){=}_1(y\left(v^{*}\right)-z_d)x\in \Gamma ,t\in (T-a,T).5$$

As in the above section, we have the following result.

Lemma 5.2 Let the hypothesis of Theorem 2.1 be satisfied. Then, for given $z_d\in L^2(\Sigma )$ and any $v\in L^2(\Sigma )$, there exists a unique solution $p\left(v^{*}\right)\in W^{{\displaystyle \frac{3}{2}},{\displaystyle \frac{3}{4}}}\left(Q\right)$ to the adjoint problem (39)-(45).

Using the adjoint equations (39)-(45) in this case, the condition (38) can also be written in the following form

$${\int }_0^T{\int }_{\Gamma }\left(p\left(v^{*}\right){+}_{2}N{v}^{*}\right)(v-v^{*})d\Gamma dt\ge 0\forall v\in U_{ad}.6$$

The following result is now summarized.

Theorem 5.2 For the problem (1)-(5) with the performance function (37) with $z_d\in L^2(\Sigma )$ and ₂ > 0, and with constraint (17), and with adjoint equations (39)-(45), there exists a unique optimal control $v^{*}$ which satisfies the maximum condition (46).

Example 5.2 Case: $v\in L^2\left(Q\right)$. We can also consider an analogous optimal control problem where the performance functional is given by

$$\widehat{\widehat{I}}\left(v\right)={\lambda }_1{\int }_Q{[y(x,t;v)-z_d]}^{2}d\rho dt+{\lambda }_2{\int }_Q\left(Nv\right)vd\rho dt7$$

where $z_d\in L^2\left(Q\right)$.

From theorem 2.1 and the Trace Theorem (Lions & Magenes, 1972, Vol 2, p.9), for each $v\in L^2\left(Q\right)$, there exists a unique solution $y\left(v\right)\in W^{{\displaystyle \frac{3}{2}},{\displaystyle \frac{3}{4}}}\left(Q\right)$. Thus, $\widehat{\widehat{I}}$ is well defined. Then, the optimal control $v^{*}$ is characterized by

$${}_1\int _Q(y\left(v^{*}\right)-z_d)[y\left(v\right)-y\left(v^{*}\right)]d\rho dt{+}_2{\int }_{Q}N{v}^{*}(v-v^{*})d\rho dt\ge 0\forall v\in U_{ad}.8$$

We define the adjoint variable $p=p\left(v^{*}\right)=p(x,t;v^{*})$ as the solution of the equations

$${}_{T-b}{}^{C}D_t^{\alpha }p\left(v^{*}\right)+A^{*}\left(t\right)p\left(v^{*}\right)+{\int }_a^{b}c(x,t+h)p(x,t+h;v^{*})dh{=}_1(y\left(v^{*}\right)-z_d),$$

$$x\in \Omega ,t\in (0,T-b),9$$

$${}_{T-a}{}^{C}D_t^{\alpha }p\left(v^{*}\right)+A^{*}\left(t\right)p\left(v^{*}\right)+{\int }_a^{T-t}c(x,t+h)p(x,t+h;v^{*})dh{=}_1(y\left(v^{*}\right)-z_d),$$

$$x\in \Omega ,t\in (T-b,T-a),0$$

$${}_T{}^{C}D_t^{\alpha }p\left(v^{*}\right)+A^{*}\left(t\right)p\left(v^{*}\right){=}_1(y\left(v^{*}\right)-z_d),x\in \Omega ,t\in (T-a,T),1$$

$$p(x,T;v^{*})=0,x\in \Omega ,2$$

$${\displaystyle \frac{p\left(v^{*}\right)}{{\nu }_{A^{*}}}}(x,t)={\int }_a^{b}d(x,t+h)p(x,t+h;v^{*})dh,x\in \Gamma ,t\in (0,T-b),3$$

$${\displaystyle \frac{p\left(v^{*}\right)}{{\nu }_{A^{*}}}}(x,t)={\int }_a^{T-t}d(x,t+h)p(x,t+h;v^{*})dh,x\in \Gamma ,t\in (T-b,T-a),4$$

$${\displaystyle \frac{p\left(v^{*}\right)}{{\nu }_{A^{*}}}}(x,t)=0x\in \Gamma ,t\in (T-a,T).5$$

As in the above section, we have the following result.

Lemma 5.3 Let the hypothesis of Theorem 2.1 be satisfied. Then, for given $z_d\in L^2\left(Q\right)$ and any $v\in L^2\left(Q\right)$, there exists a unique solution $p\left(v^{*}\right)\in W^{{\displaystyle \frac{3}{2}},{\displaystyle \frac{3}{4}}}\left(Q\right)$ to the adjoint problem (49)-(55).

Using the adjoint equations (49)-(55) in this case, the condition (48) can also be written in the following form

$${\int }_0^T{\int }_{\Omega }\left(p\left(v^{*}\right){+}_{2}N{v}^{*}\right)(v-v^{*})d\rho dt\ge 0\forall v\in U_{ad}.6$$

The following result is now summarized.

Theorem 5.3 For the problem (1)-(5) with the performance function (47) with $z_d\in L^2\left(Q\right)$ and ₂ > 0, and with constraint (17), and with adjoint equations (49)-(55), there exists a unique optimal control $v^{*}$ which satisfies the maximum condition (56).

Remark 5.1 If we take $\alpha =1$ in the previews sections we obtain the classical results in the optimal control with integer derivatives.

6. Conclusions

The fractional control problems presented in the paper constitutes a generalization of the optimal boundary control problem of a parabolic system with Neumann boundary condition involving constant time lag appearing both in the state equations and in the boundary conditions considered in (Kowalewski, 1998; 1999), (Kowalewski & Duda, 1992), (Kotarski & El-Saify & Bahaa, 2002).

Also the main result of the paper contains necessary and sufficient conditions of optimality for fractional control problems involving second order operator with infinite number of variables that give characterization of optimal control (Theorem 4.6). But it is easily seen that obtaining analytical formulas for fractional optimal control is very difficult. This results from the fact that state equations (62), adjoint equations (63) and minimum condition (64) are mutually connected that cause that the usage of derived conditions is difficult. Therefore we must resign from the exact determination of the optimal control and therefore we are forced to use approximation methods. Also it is evident that by modifying:

• - the boundary conditions,
• - the nature of the control (distributed, boundary),
• - the nature of the observation,
• - the initial differential system,

an infinity of variations on the above problem are possible to study with the help of (Lions, 1971) and Dubovitskii-Milyutin formalisms (Bahaa, 2003, 2005a,b, 2008). Those problems need further investigations and form tasks for future research. These ideas mentioned above will be developed in forthcoming papers.