Bang-Bang Property and Time-Optimal Control for Caputo Fractional Differential Systems

Abstract

In this paper, by using the controllability method, a bang-bang property and a time optimal control problem for time fractional differential systems (FDS) are considered. First, we formulate our problem and prove the existence theorem. We then state and prove the bang-bang theorem. Finally, we state the optimality conditions that characterize the optimal control. Some application examples are given to illustrate our results.

1. Introduction

Fractional calculus is one of the most novel types of calculus used in many different scientific and technical disciplines. The order of derivatives in fractional calculus can be any real number, which distinguishes fractional calculus from ordinary calculus. Therefore, fractional calculus can be considered a generalization of ordinary calculus. Fractional order differential equations have been successfully used in modeling many different problems for various applications of fractional order derivatives and integrals in classical mechanics, quantum mechanics, image processing, earthquake engineering, biomedical engineering, physics, nuclear physics, hadron spectroscopy, viscoelasticity, bioengineering, and many other fields ([1,2,3,4,5,6,7]).

In recent decades, fractional optimal control theory for partial differential equations have found a wide range of applications in science, engineering, economics, and some other fields. For example in [8], Agrawal provided a general formulation of FOCP in the Riemann–Liouville (RL) sense and described a solution scheme for FOCP for the classical optimal control problem that was based on variational virtual work coupled with the Lagrange multiplier technique. In ([9,10]), Agrawal formulated FOCP in Caputo’s sense (CFDs) instead of the RL sense and an iterative numerical scheme was applied to find the solution of that problem numerically where the time domain was divided into small segments. CFDs permit one to include the usual initial conditions in a simple manner, and consequently they are favored choices by most researchers. The fractional derivatives of the system were approximated in [11,12] , using the Grunwald–Letnikov definition that yielded a set of algebraic equations that can be solved by numerical techniques. In [13], Gastao and Torres studied the fractional optimal control in the sense of Caputo and the fractional Noether’s Theorem. In [14], Jajarmi and Baleanu dicussed the suboptimal control of fractional-order dynamic systems with delay argument.

The time-optimal control problem for non-fractional parabolic equations with control constraints and an infinite number of variables is considered in Bahaa [15]. The fractional optimal control problem for variational inequalities with control constraints and for differential systems with control constraints for infinite order systems with control constraints are studied, respectively, in Bahaa ([16,17,18]). In Bahaa ([19,20]), fractional optimal control problems for differential systems with control constraints with delay argument, and with variable fractional order, respectively, are considered. In Bahaa [21], the optimality conditions for fractional differential inclusions with non-singular Mittag–Leffler Kernel are proved. The optimality conditions of fractional diffusion equations with weak Caputo derivatives and variational formulation are formulated in Bahaa and Tang [22]. The time-optimal control of second-order and infinite-order distributed parabolic non-fractional systems involving multiple time-varying lags are investigated in Bahaa and Kotarski [23]. In Baleanu [24], a different solution scheme was suggested where an improved Grunwald–Letnikov definition was utilized to deduce a central difference formula. In [25], Hasan et al. study the fractional optimal control of distributed systems in cylindrical and spherical coordinates. A numerical scheme and formulation for fractional optimal control of cylindrical structures subjected to general initial conditions are given in [26]. In [27], Tricaud et al. studied the time-optimal control of systems with fractional dynamics.

The bang-bang property is of great importance in the theory of optimal control, as pointed out in [28,29]. In particular, the bang-bang property can be given for certain time-optimal controls governed by parabolic equations using Pontryagin’s maximum principle [30]. In [31], Phung et al. studied the bang-bang property for time-optimal controls governed by a semilinear heat equation in a rich domain with local control in a subset. In [32], Chen et al. studied the time-varying bang-bang property of time-optimal controls for the heat equation and showed the applicability of this property.

In this paper, a Bang-Bang property and a time-optimal control problem for time fractional differential systems (FDS) are considered. First, we formulate our problem and prove the existence theorem. Using the Fre’chet differentiability, we then state and prove the bang-bang theorem. Finally, we state the optimality conditions that characterize the optimal control. Some application examples are given to illustrate our results.

This paper is organized as follows: In the second section, we introduce some fractional operators and basic definitions that we use in this paper. In the third section, we summarize the time-optimal control problem for fractional systems and then introduce the main results of this work. In the fourth section, we state and prove the existence theorem. In the fifth section, we explain and prove the Bang-Bang theorem. In the sixth section, we prove the optimality conditions. In the seventh section, we present some applications and illustrate examples of this problem. Finally, in the eighth section, we draw conclusions.

2. Basic Definitions

In this section, we introduce some basic definitions related to fractional derivatives (see [33,34]).

Definition 1.

The left Riemann–Liouville fractional integral and the right Riemann–Liouville fractional integral are presented, respectively, by

$${}_{a}I^{a}f\left(t\right)=\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_a^t{(t-\tau )}^{\alpha -1}f\left(\tau \right)d\tau ,$$

(1)

$$I_b^{\alpha }f\left(t\right)=\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_t^b{(\tau -t)}^{\alpha -1}f\left(\tau \right)d\tau ,$$

(2)

where $\alpha >0,n-1<\alpha <n$. From now on, $\mathsf{\Gamma }\left(\alpha \right)$ represents the Gamma function.

The left Riemann–Liouville fractional derivative is given by

$${}_{a}D^{a}f\left(t\right)=\frac{1}{\mathsf{\Gamma }(n-\alpha )}{\left(\frac{d}{dt}\right)}^n{\int }_a^t{(t-\tau )}^{n-\alpha -1}f\left(\tau \right)d\tau .$$

(3)

The right Riemann–Liouville fractional derivative is defined by

$$D_b^{\alpha }f\left(t\right)=\frac{1}{\mathsf{\Gamma }(n-\alpha )}{\left(-\frac{d}{dt}\right)}^n{\int }_t^b{(\tau -t)}^{n-\alpha -1}f\left(\tau \right)d\tau .$$

(4)

The fractional derivative of a constant takes the form

$${}_{a}D^{a}C=C\frac{{(t-a)}^{-\alpha }}{\mathsf{\Gamma }(1-\alpha )},$$

(5)

and the fractional derivative of a power of t has the following form

$${}_{a}D^a{(t-a)}^{\beta }=\frac{\mathsf{\Gamma }(\alpha +1){(t-a)}^{\beta -\alpha }}{\mathsf{\Gamma }(\beta -\alpha +1)},$$

(6)

for $\beta >-1,\alpha \ge 0$.

The Caputo’s fractional derivatives are defined as follows:

The left Caputo fractional derivative

$${}_a{}^{C}D^{a}f\left(t\right)=\frac{1}{\mathsf{\Gamma }(n-\alpha )}{\int }_a^t{(t-\tau )}^{n-\alpha -1}{\left(\frac{d}{d\tau }\right)}^{n}f\left(\tau \right)d\tau ,$$

(7)

and the right Caputo fractional derivative

$${}^{C}D_b^{a}f\left(t\right)=\frac{1}{\mathsf{\Gamma }(n-\alpha )}{\int }_t^b{(\tau -t)}^{n-\alpha -1}{\left(-\frac{d}{d\tau }\right)}^{n}f\left(\tau \right)d\tau ,$$

(8)

where α represents the order of the derivative such that $n-1<\alpha <n$. By definition, the Caputo fractional derivative of a constant is zero. For more properties and information about these derivatives, see [35,36].

Lemma 1

(see [35,36]). Let $0<\alpha <1$. Then, for any $\varphi \in C^{\infty }\left(\overline{Q}\right)$, we have

$${\int }_0^T{\int }_{\mathsf{\Omega }}{(}_a^C{D}^{\alpha }y(x,t)+\mathcal{A}y(x,t))\varphi (x,t)dxdt={\int }_{\mathsf{\Omega }}\varphi {(x,T)}_a{I}^{1-\alpha }y(x,T)dx-{\int }_{\mathsf{\Omega }}\varphi {(x,0)}_a{I}^{1-\alpha }y(x,0^{+})dx$$

$$+{\int }_0^T{\int }_{\partial \mathsf{\Omega }}y\frac{\partial \varphi }{\partial \nu }d\sigma dt-{\int }_0^T{\int }_{\partial \mathsf{\Omega }}\frac{\partial y}{\partial \nu }\varphi d\sigma dt+{\int }_0^T{\int }_{\mathsf{\Omega }}y(x,t){(}^R{D}_b^{\alpha }\varphi (x,t)+{\mathcal{A}}^{\ast }\varphi (x,t))dxdt.$$

where ${\mathcal{A}}^{\ast }$ is conjugate of the operator $\mathcal{A}$, which is given in the next section and

$$\frac{\partial y}{\partial {\nu }_{\mathcal{A}}}=\sum _{i,j=1}^n{a}_{ij}\frac{\partial y}{\partial x_j}cos(n,x_j)\mathrm{on}\partial \mathsf{\Omega },$$

$cos(n,x_j)$ is the i-th direction cosine of $n,n$ being the normal at $\partial \mathsf{\Omega }$ exterior to Ω.

3. Time-Optimal Control Problem for Caputo Fractional Differential System

Let us consider the optimization problem in the following form

$${}_0{}^{C}D^{a}y(t;v)+\mathcal{A}\left(t\right)y(t;v)=f+Bv,x\in \mathsf{\Omega },t\in (0,T),$$

(9)

$$y(0;v)=y_0,$$

(10)

where $f,y_0$ are given two elements in $L^2\left(\mathsf{\Omega }\right)$. The second-order operator $\mathcal{A}\left(t\right)$ in the state Equation (9) takes the form

$$\mathcal{A}\left(t\right)y(x,t)=-\sum _{i,j=1}^n\frac{\partial }{\partial x_i}\left(a_{ij}(x,t)\frac{\partial y(x,t)}{\partial x_j}\right)+a_0(x,t)y(x,t),$$

(11)

where $a_{ij}(x,t),i,j=1,\cdots ,n$, be given function on $\mathsf{\Omega }$ with the properties

$$a_0(x,t),a_{ij}(x,t)\in L^{\infty }\left(\mathsf{\Omega }\right)(\mathrm{with}\mathrm{real}\mathrm{values}),$$

$$a_0(x,t)\ge \delta >0,\sum _{i,j=1}^n{a}_{ij}(x,t){\xi }_i{\xi }_j\ge \delta ({\xi }_1^2+\cdots +{\xi }_n^2),\forall \xi \in R^n,$$

a.e. on $\mathsf{\Omega }$, i.e., $\mathcal{A}\left(t\right)$ is a bounded second-order self-adjoint elliptic partial differential operator, which maps $H_0^1\left(\mathsf{\Omega }\right)$ onto $H^{-1}\left(\mathsf{\Omega }\right)$.

Let us denote by $U:=L^2(0,T;L^2\left(\mathsf{\Omega }\right))=L^2\left(Q\right)$, the space of controls and by $Y:=L^2(0,T;H_0^1\left(\mathsf{\Omega }\right))$ the space of states.

We suppose that $U_{ad}$ is a closed, convex subset of U and $y_1$ is a given element in $L^2\left(\mathsf{\Omega }\right)$,

$$B\in L(L^2\left(Q\right),L^2(0,T,H^{-1}\left(\mathsf{\Omega }\right))).$$

(12)

Assume (Controllability)

$$\left.\begin{array}{c}\mathrm{there}\mathrm{exists}\mathrm{a}v\in U_{ad},\mathrm{such}\mathrm{that}\hfill \\ y(\tau ;v)=y_1\mathrm{for}\mathrm{an}\mathrm{appropriate}\tau .\hfill \end{array}\right\}$$

(13)

The optimal time is given by

$${\tau }_0=inf\tau ,\tau \mathrm{such}\mathrm{that}\left(13\right)\mathrm{holds}.$$

(14)

For the operator $\mathcal{A}\left(t\right)$, we define the bilinear form $\pi (t;y,ɸ)$ as follows:

Definition 2.

On $H_0^1\left(\mathsf{\Omega }\right)$, we define for each $t\in ]0,t[$ the following bilinear form

$$\pi (t;y,ɸ)=\left(\mathcal{A}\left(t\right)yɸ\right){,}_{L^2\left(\mathsf{\Omega }\right)},y,ɸ\in H_0^1\left(\mathsf{\Omega }\right).$$

Then

$$\begin{array}{cc}\hfill \pi (t;y,\varphi )& ={\left(\mathcal{A}y,\varphi \right)}_{L^2\left(\mathsf{\Omega }\right)}\hfill \\ \hfill & =\left(-\sum _{i,j=1}^n\frac{\partial }{\partial x_i}\right(a_{ij}\left(x\right)\frac{\partial y}{\partial x_j}{\left)+a_0\left(x\right)y,\varphi \left(x\right)\right)}_{L^2\left(\mathsf{\Omega }\right)}\hfill \\ & ={\int }_{\mathsf{\Omega }}\sum _{i,j=1}^n{a}_{ij}\frac{\partial }{\partial x_i}y\left(x\right)\frac{\partial }{\partial x_j}\varphi \left(x\right)dx+{\int }_{\mathsf{\Omega }}{a}_0\left(x\right)y\left(x\right)\varphi \left(x\right)dx.\hfill \end{array}$$

(15)

Lemma 2.

The bilinear form (15) is coercive on $H_0^1\left(\mathsf{\Omega }\right)$; that is,

$$\pi (t;y,y)\ge {\lambda \left|\right|y\left|\right|}_{H_0^1\{a_{\alpha },2\}\left(\mathsf{\Omega }\right)}^2,\lambda >0.$$

(16)

Proof. 

$$\begin{array}{cc}\hfill \pi (t;y,y)& ={\int }_{\mathsf{\Omega }}\sum _{i,j=1}^n{a}_{ij}(x,t)\frac{\partial }{\partial x_i}y(x,t)\frac{\partial }{\partial x_j}y(x,t)dx+{\int }_{\mathsf{\Omega }}{a}_0(x,t)y(x,t)y(x,t)dx\hfill \\ & \le \sum _{i,j=1}^n{a}_{ij}(x,t)\left|\right|\frac{\partial }{\partial x_i}{y(x,t)\left|\right|}_{L^2\left(\mathsf{\Omega }\right)}^2+{\left|\right|y(x,t)\left|\right|}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ & \ge {\lambda \left|\right|y\left|\right|}_{H_0^1\left(\mathsf{\Omega }\right)}^2,\lambda >0.\hfill \end{array}$$

□

Also, we can assume that

$$\left.\begin{array}{c}\forall y,ɸ\in H_0^1\left(\mathsf{\Omega }\right)\mathrm{the}\mathrm{bilinear}\mathrm{form}t\to \pi (t;y,ɸ)\mathrm{is}\mathrm{continuously}\mathrm{differentiable}\mathrm{in}[0,T],\hfill \\ \mathrm{the}\mathrm{function}t\to \pi (t;y,ɸ)\mathrm{is}\mathrm{measurable}\mathrm{on}[0,T],\left|\pi \right(t;y,ɸ\left)\right|\le c\left|\right|y\left|\right|.\left|\right|ɸ\left|\right|,\hfill \end{array}\right\}$$

(17)

the bilinear form (15) is symmetric,

$$\pi (t;y,ɸ)=\pi (t;ɸ,y)\forall y,ɸ\in H_0^1\left(\mathsf{\Omega }\right),$$

and there exists a $\lambda$ such that

$$\pi (t;ɸ,ɸ)+\lambda {\left|ɸ\right|}^2\ge \gamma {\left|\right|ɸ\left|\right|}^2,\gamma >0,\forall ɸH_0^1\left(\mathsf{\Omega }\right),t\in [0,T].$$

(18)

Equations (9)–(18) constitute a fractional Dirichlet problem.

Remark 1

([35,36]). The operator ${}_0{}^{C}D_t^a+\mathcal{A}\left(t\right)$ is a second-order parabolic operator that maps $L^2(0,T;H_0^1\left(\mathsf{\Omega }\right))$ onto $L^2(0,T;H^{-1}\left(\mathsf{\Omega }\right))$.

Remark 2

([35,36]). Equations (9)–(18) have the unique generalized solution

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

which continuously depends on the initial condition (10) and the right-hand side of (9). Furthermore, $y\in W(0,T)$ is a continuous function $[0,T]\to L^2\left(\mathsf{\Omega }\right)$ (compare with Theorems 1.1 and 1.2 Chapt. 3 [29]).

We are going to study the following problems:

(i) the existence of optimal control, i.e., $u\in U_{ad}$ such that

$$y({\tau }_0;u)=y_1;$$

(19)

(ii) properties of the optimal control, if it exists. These problems will be treated in Section 4 and Section 5.

4. Existence Theorem

Theorem 1.

Let $\alpha \in [\frac{1}{2},1]$. We assume that (12), (13) and (17) hold and that $U_{ad}$ is bounded. Then, there exists an optimal control, that is $u\in U_{ad}$, such that (19) holds.

Proof. 

Let ${\tau }_n$ be such that

$$y({\tau }_n;v_n)=y_1,v_n\in U_{ad},$$

(20)

$${\tau }_n\to {\tau }_0.$$

(21)

$$\left.\begin{array}{c}\mathrm{Set}{y}_n=y\left(v_n\right).\mathrm{Sin}\mathrm{ce}{U}_{ad}\mathrm{is}\mathrm{bounded},\mathrm{we}\mathrm{may}\mathrm{verify}\mathrm{that}\hfill \\ y_n\left(\mathrm{resp}.{}_0{}^{C}D^a{y}_n\right)\mathrm{ranges}\mathrm{in}\mathrm{a}\mathrm{bounded}\mathrm{set}\mathrm{in}{L}^2(0,T,H_0^1\left(\mathsf{\Omega }\right))(\mathrm{resp}.L^2(0,T,H^{-1}\left(\mathsf{\Omega }\right))).\hfill \end{array}\right\}$$

(22)

We may then extract a subsequence, again denoted by $\{v_n,y_n\}$, such that

$$\left.\begin{array}{c}{v}_n\to v\mathrm{weakly}\mathrm{in}U,v\in U_{ad}\hfill \\ y_n\to y(\mathrm{resp}{}_0{}^{C}D^a{y}_n\to {}_0{}^{C}D^{a}y\left)\mathrm{weakly}\mathrm{in}{L}^2(0,T,H_0^1\left(\mathsf{\Omega }\right))(\mathrm{resp}.L^2(0,T,H^{-1}\left(\mathsf{\Omega }\right))\right).\hfill \end{array}\right\}$$

(23)

We deduce from the equality

$${}_0{}^{C}D^a{y}_n+\mathcal{A}{y}_n=f+v_n,x\in \mathsf{\Omega },t\in (0,T),$$

that

$${}_0{}^{C}D^{a}y+\mathcal{A}y=f+v,x\in \mathsf{\Omega },t\in (0,T),$$

and $y\left(0\right)=y_0$ and hence

$$y=y\left(v\right).$$

(24)

But

$$y({\tau }_n;v_n)-y({\tau }_0;v)=y({\tau }_n;v_n)-y({\tau }_0;v_n)+y({\tau }_0;v_n)-y({\tau }_0;v).$$

(25)

Now, from (23), $y({\tau }_0;v_n)\to y({\tau }_0;v)$ weakly in $L^2(0,T,H_0^1\left(\mathsf{\Omega }\right))$ and

$$\begin{array}{cc}\hfill \left|\right|y({\tau }_n;v_n)-y({\tau }_0;v_n){\left|\right|}_{H^{-1}\left(\mathsf{\Omega }\right)}& =\left|\right|\left({}_{\tau 0}I^a{}_{\tau 0}{}^{C}D^a\right)y({\tau }_n;v_n){\left|\right|}_{H^{-1}\left(\mathsf{\Omega }\right)}\hfill \\ & \le \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{{\tau }_0}^{{\tau }_n}{({\tau }_n-\tau )}^{\alpha -1}\left|\right|{}_{\tau 0}{}^{C}D^{a}y(\tau ;v_n){\left|\right|}_{H^{-1}\left(\mathsf{\Omega }\right)}d\tau \hfill \\ & \le \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\left({\int }_{{\tau }_0}^{{\tau }_n}{({\tau }_n-\tau )}^{2\alpha -2}d\tau \right)}^{\frac{1}{2}}({\int }_{{\tau }_0}^{{\tau }_n}\left|\right|{}_{\tau 0}{}^{C}D^{a}y(\tau ;v_n){\left|\right|}_{H^{-1}\left(\mathsf{\Omega }\right)}^2{d\tau )}^{\frac{1}{2}}\hfill \\ & \le C{({\tau }_n-{\tau }_0)}^{\alpha -\frac{1}{2}};\hfill \end{array}$$

hence, (25) shows that

$$y({\tau }_n;v_n)-y({\tau }_0;v)\to 0\mathrm{weakly}\mathrm{in}{H}^{-1}\left(\mathsf{\Omega }\right).$$

But from (20), it follows that $y({\tau }_0;v)=y_1$. □

Example 1

(Neumann problem with boundary control). Take $U=L^2\left(\Sigma \right)$. Let the state $y\left(v\right)$ be as follows

$$\begin{array}{cc}\hfill {}_0{}^{C}D^{a}y\left(v\right)+\mathcal{A}\left(t\right)y\left(v\right)& =f,\hfill \\ \hfill \frac{\partial y}{\partial {\nu }_{\mathcal{A}}}\left(v\right)& =v,\hfill \\ \hfill y(x,0,v)& =y_0\left(x\right).\hfill \end{array}$$

We may apply Theorem 1 to this example. Thus, the theorem deals with the case of boundary control.

Remark 3.

Theorem 1 may be modified easily to deal with the case of systems with Dirichlet boundary conditions (cf ([29]), Section 9) and where the control is exercised through the boundary.

5. Bang-Bang Theorem

We consider the same data and hypotheses as in the previous section with

$$\mathcal{A}\mathrm{independent}\mathrm{of}t,$$

(26)

$$U_{ad}=\left\{v\right|\left|v\left(t\right)\right|\le 1\mathrm{a}.\mathrm{e}.\}.$$

(27)

Then $-\mathcal{A}$ is the infinitesimal generator of a semi-group $G\left(t\right)$ in $L^2\left(Q\right)$.

Theorem 2

(Bang-Bang Theorem). We assume that (12), (17), (26), (27) and (13) hold. Let u be an optimal control, that is, an element of $U_{ad}$ satisfying (19) (we know from Theorem 1 that such elements exist). Then

$$\left|u\left(t\right)\right|=1\mathrm{a}.\mathrm{e}.\mathrm{on}[0,{\tau }_0].$$

(28)

We shall establish some lemmas before giving a proof of the Theorem 2. Let us at once isolate.

Corollary 1.

Under the hypotheses of Theorem 2, there exists a unique optimal control.

Proof. 

Let $u_1,u_2\in U_{ad};y({\tau }_0;u_i)=y_i,i=1,2\cdots$ Then

$$y({\tau }_0;(1-\theta )u_1+\theta u_2)=y_1,0<\theta <1\mathrm{and}u=(1-\theta )u_1+\theta u_2$$

does not satisfy (19) unless $u_1\left(t\right)=u_2\left(t\right)a.e.$ □

Notation

$$\left.\begin{array}{c}{K}_s=\left\{h\right|h=G\ast v\left(s\right)={\int }_0^{s}G(s-t)v\left(t\right)dt,v\in L^{\infty }(0,s,L^2\left(\mathsf{\Omega }\right))\}\hfill \\ K_s\left(e\right)=\left\{h\right|h=G\ast v\left(s\right)={\int }_0^{s}G(s-t)v\left(t\right)dt,v\in L^{\infty }(0,s,L^2\left(\mathsf{\Omega }\right))\mathrm{with}\mathrm{support}\mathrm{in}e\cap [0,s],\hfill \\ e=\mathrm{measurable}\mathrm{subset}\mathrm{of}[0,T]\}.\hfill \end{array}\right\}$$

(29)

Lemma 3.

With the notation of (29), we have:

$$K_s=K_s^{\prime },\forall s,s^{\prime }.$$

(30)

We then set

$$K=K_s,\forall s.$$

(31)

We have

$$G\left(s\right)H\subset K.$$

(32)

Proof. 

• Let $\tau >s.$ Then we may easily verify

  $$\left.\begin{array}{c}G\ast v\left(s\right)=G\ast \overline{v}\left(\tau \right),\hfill \\ \mathrm{where}\hfill \\ \overline{v}\left(\tau \right)=\left\{\begin{array}{c}0\mathrm{if}0<t<\tau -s,\hfill \\ v(t-(\tau -s\left)\right)\mathrm{if}t>\tau -s.\hfill \end{array}\right.\hfill \end{array}\right\}$$

  (33)
  Consequently

  $$K_s\subset K_{\tau }.$$

  (34)
• Let $\tau <s.$ We may again verify that

  $$\left.\begin{array}{c}G\ast v\left(s\right)=G\ast w\left(s\right),\hfill \\ \mathrm{where}\hfill \\ w\left(s\right)=v(t+\tau -s))+\frac{1}{s}G\left(t\right){\int }_0^{\tau -s}G(\tau -s-t)v\left(t\right)dt,\hfill \end{array}\right\}$$

  (35)

  which proves the reverse inclusion of (34), whence (30) and (31).
• We check that

  $$G\left(s\right)h=G\times \left(\frac{1}{s}Gh\right)\left(s\right),\mathrm{whence}\left(\mathrm{32}\right).$$

  (36)

□

The following lemma is fundamental.

Lemma 4.

For almost all $t\in e$, we have

$$K_t\left(e\right)=K.$$

(37)

Proof. 

1. Clearly, $K_t\left(e\right)\subset K_t$ and, hence, it suffices to prove that for almost all $t\in e,$$K\subset K_t\left(e\right)$. Let $h\in K_t$. Then $h\in K_s$ with s arbitrary small and therefore for any $t_1<t,$ we may represent h in the form (cf. (33))

$$h={\int }_{t_1}^{t}G(t-\sigma )v\left(\sigma \right)d\sigma ,v\in L^{\infty }(0,T;H).$$

(38)

Now, suppose that we can find a sequence $t_n$ such that

$$\left.\begin{array}{c}{t}_n<t_{n+1}<\cdots <t,t_n\to t,\hfill \\ \mathrm{measure}([t_n,t_{n+1}]\cap e)\ge \rho (t_{n+1}-t_n),\rho >0,\hfill \\ \frac{t_{n+1}-t_n}{t_{n+2}-t_{n+1}}\le c.\hfill \end{array}\right\}$$

(39)

We shall see in part 2 of the proof that

$$\mathrm{for}\mathrm{almost}\mathrm{all}t\in e,\mathrm{we}\mathrm{may}\mathrm{find}\mathrm{a}\mathrm{sequence}{t}_n\mathrm{such}\mathrm{that}\left(39\right)\mathrm{holds}.$$

(40)

In (38), let us choose as $t_1$ the first element of the sequence $\left\{t_n\right\}$. From (38), we deduce that we may write

$$\begin{array}{cc}\hfill h& =\sum _{n=1}^{\infty }G(t-t_{n+1}){\int }_{t_n}^{t_{n+1}}G(t_{n+1}-\sigma )v\left(\sigma \right)d\sigma \hfill \\ \hfill & =\sum _{n=1}^{\infty }{\int }_{e\cap [t_{n+1},t_{n+2}]}\frac{1}{\mathrm{measure}(e\cap [t_{n+1},t_{n+2}])}G(t-{\sigma }_1)\hfill \\ & \times G({\sigma }_1-t_{n+1})d{\sigma }_1{\int }_{t_n}^{t_{n+1}}G(t_{n+1}-{\sigma }_2)v\left({\sigma }_2\right)d{\sigma }_2={\int }_0^{t}G(t-\sigma )w\left(\sigma \right)d\sigma ,\hfill \end{array}$$

(41)

where

$$w\left(s\right)=\left\{\begin{array}{c}\frac{1}{\mathrm{measure}(e\cap [t_{n+1},t_{n+2}])}G(\sigma -t_{n+1}){\int }_{t_n}^{t_{n+1}}G(t_{n+1}-{\sigma }_2)v\left({\sigma }_2\right)d{\sigma }_2\hfill \\ \mathrm{in}\mathrm{the}\mathrm{set}e\cap [t_{n+1},t_{n+2}],n=1,2,\cdots ;\hfill \\ 0\mathrm{otherwise}.\hfill \end{array}\right.$$

(42)

From (39), we have

$$\left|w\left(\sigma \right)\right|\le \frac{1}{\rho (t_{n+2}-t_{n+1})}C(t_{n+1}-t_n)\le \mathrm{constant}.$$

Hence,

$$h=G\ast w\left(t\right),w\in L^{\infty }(0,T,H),w\mathrm{has}\mathrm{support}\mathrm{in}e,\mathrm{and}h\in K_t\left(e\right).$$

Then the lemma is proved except (40).

2. Proof of (40). This is a result of the measure theory. We first define

$$e_m=\left\{\sigma |\sigma \in e,measure(e\cap [\sigma -\frac{1}{k},\sigma ])\ge \frac{1}{2k},k\ge m\right\}$$

and $d_m=$ set of points of density of $e_m$. It is known that $measure\left(e-{\bigcap }_{m\ge 1}{d}_m\right)=0$ and, hence, it suffices to prove (40) for $t\in d_m$.

However, then we may construct $\left\{t_n\right\},t_n\in d_m\forall n,t_{n+1}=t_n+s_n<t,s_n>0,\frac{s_n}{s_{n+1}}\le e$ (in a manner such that the last property of (39) holds) and since $t_n\in d_m\subset e_m\forall n$ , there exists a $\rho >0$ such that

$$\mathrm{measure}([t_n,t_{n+1}]\cap e)\ge \rho (t_{n+1}-t_n).$$

□

Lemma 5.

Let $u\in U_{ad}$ be optimal with respect to $\{y_0,y_1\}$, that is, $y(\tau ;u)=y_1$ with τ minimum. Then for any $s<\tau ,u$ is optimal with respect to $\{y_0,y(s;u)\}.$

In other words, if $w\in U_{ad}$ satisfies

$$y(\sigma ;w)=y(s;u)\mathrm{for}\mathrm{some}\mathrm{appropriate}\sigma ,$$

(43)

we necessarily have $\sigma \ge s$.

Proof. 

Assume that (43) holds with $\sigma <s$. Then define v as.

$$v\left(t\right)=\left\{\begin{array}{c}w\left(t\right)in[0,\sigma ],\hfill \\ u(t+s-\sigma )in[\sigma ,T-(s-\sigma \left)\right].\hfill \end{array}\right.$$

Then, from (43),

$$y(\sigma ;v)=y(\sigma ;w)=y(s;u)$$

and

$${}_0{}^{C}D^{a}y(t;v)+\mathcal{A}y(t;v)=u(t+s-\sigma ),\mathrm{for}t\ge \sigma .$$

Hence,

$$y(t-(s-\sigma );v)=y(t,u),t\ge s,$$

and hence,

$$y(\tau -(s-\sigma );v)=y(\tau ,u)=y.$$

But since $\tau -(s-\sigma )<\tau ,$ this contradicts the hypothesis that u is optimal with respect to $\{y_0,y_1\}$. □

Lemma 6.

Let $v\in U_{ad}$ be such that

$$\left|v\right(t\left)\right|\le 1-\epsilon \mathit{almost}\mathit{everywhere},\epsilon >0,$$

(44)

$$y(\tau ;v)=y_1\mathit{for}\mathit{some}\mathit{appropriate}\tau .$$

(45)

Then $\tau >{\tau }_0$.

Proof. 

To prove this, we will verify that there exists a $s<\tau$ and $w\in U_{ad}$ such that

$$y(s;w)=y(\tau ;v)$$

(46)

(which proves that $\tau$ is not optimal). For this, we note that (46) may be written as

$${\int }_0^{\tau }G(\tau -\sigma )v\left(\sigma \right)d\left(\sigma \right)+G\left(\tau \right)y_0-G\left(s\right)y_0={\int }_0^{s}G(s-\sigma )w\left(\sigma \right)d\sigma .$$

(47)

But it may be easily verified that the left-hand side of (47) may be written as

$${\int }_0^{s}G(s-\sigma )\overline{v}\left(\sigma \right)d\sigma ,$$

with

$$\overline{v}\left(\sigma \right)=v(\sigma +\tau -s)+\frac{1}{s}G\left(\sigma \right){\int }_0^{\tau -s}G(\tau -s-{\sigma }_1)v\left({\sigma }_1\right)d{\sigma }_1+\frac{1}{s}[G(\sigma +\tau -s)y_0-G\left(\sigma \right)y_0]$$

and for $\tau -s$ sufficiently small, we have, by virtue of (44), $|\overline{v}\left(\sigma \right)|\le 1$. Therefore, we may take $w=\overline{v}$. □

Proof of Bang-Bang Theorem (Theorem 2). 

Assume that (28) did not hold. Then there would exist $e\subset [0,{\tau }_0]$, measure $\left(e\right)>0$ such that

$$\left|u\right(t\left)\right|\le 1-\epsilon ,\epsilon >0,t\in e.$$

(48)

It follows from the fundamental Lemma (4) that we can find an s such that $K_s\left(e\right)=K=K_s$. In other words, there exists a $\overline{g}\in L^{\infty }(0,T,H)$, with support in e, such that

$${\int }_0^{s}G(s-\sigma )\overline{g}\left(\sigma \right)d\sigma ={\int }_0^{s}G(s-\sigma )u\left(\sigma \right)d\sigma .$$

(49)

Let us introduce the control

$$v=\left\{\begin{array}{c}(1-\delta )u+\delta \overline{g}in(0,{\tau }_0),\delta >0\mathrm{almost}\mathrm{everywhere},\hfill \\ 0\mathrm{for}t>{\tau }_0.\hfill \end{array}\right.$$

(50)

We shall choose $\delta$ in a manner such that

$$\left|v\left(t\right)\right|\le 1-{\epsilon }_1\mathrm{almost}\mathrm{everywhere},{\epsilon }_1>0\mathrm{appropriate}.$$

(51)

This is possible. To see this, we note that on e, we have from (48)

$$\left|v\left(t\right)\right|\le (1-\delta )(1-\epsilon )+\delta |\overline{g}\left(t\right)|\le 1-{\epsilon }_2$$

for $\delta$ sufficiently small, and outside e,

$$\left|v\left(t\right)\right|=(1-\delta )\left|u\left(t\right)\right|\le (1-\delta )(sinceu\in U_{ad}).$$

But

$$y(s;v)=(1-\delta ){\int }_0^{s}G(s-\sigma )u\left(\sigma \right)d\sigma +\delta {\int }_0^{s}G(s-\sigma )\overline{g}\left(\sigma \right)d\sigma +G\left(s\right)y_0$$

and using (49), we have

$$y(s;v)=y(s;u).$$

(52)

However, then, from Lemma (6), (since we have (51) u is not optimal with respect to $\{y_0,y(s;u)\}$ and, hence, from Lemma (5), u is not optimal with respect to $\{y_0,y_1\}$, contradicting the hypotheses. □

Remark 4.

It is possible to proceed further using much simpler arguments when the semi-group G is a group—in other words when we can reverse time. Indeed we have the following result.

Theorem 3.

We assume that $-A$ is the infinitesimal generator of a group $G\left(t\right)$. We assume that

$$U_{ad}=\left\{v\right|v\left(t\right)\in H_{ad}\subset H,H_{ad}=\mathit{neighborhood}\mathrm{of}0inH\}.$$

(53)

We further suppose that there exists a $v\in U_{ad}$ such that (13) holds and that there exists an optimal control u (that is, $y({\tau }_0;u)=y_1,{\tau }_0=$ optimal time defined in (12).

Then

$$u\left(t\right)\in \partial H_{ad}(\mathit{boundary}\mathrm{of}{H}_{ad}),\mathit{almost}\mathit{everywhere}.$$

(54)

Proof. 

If (54) does not hold, there exists an $e\subset [0,{\tau }_0]$ such that

$$\left.\begin{array}{c}\mathrm{measure}\left(e\right)>0,\hfill \\ \mathrm{distance}(u\left(t\right),\partial H_{ad})\ge c_1>0,\mathrm{almost}\mathrm{everywhere}\mathrm{for}t\in e.\hfill \end{array}\right\}$$

(55)

Let ${\chi }_e$ be the characteristic function of e. Then

$$y({\tau }_0;u)=y_1=G\left({\tau }_0\right)y_0+{\int }_0^{{\tau }_0}G({\tau }_0-\sigma )u\left(\sigma \right)d\sigma$$

may be written as—by virtue of the fact that G is a group—

$$y({\tau }_0;u)=G\left({\tau }_1\right)y_0+{\int }_0^{{\tau }_1}G({\tau }_1-\sigma )\overline{u}\left(\sigma \right)d\sigma ,$$

(56)

with

$$\overline{u}\left(\sigma \right)=u\left(\sigma \right)+\frac{1}{measure\left(e\right)}{\chi }_e\left(\sigma \right)G(\sigma -{\tau }_1)[y({\tau }_0;u)-y({\tau }_1;u)],{\tau }_1<{\tau }_2.$$

(57)

We may choose ${\tau }_1$ sufficiently near to ${\tau }_0$ such that

$$|u\left(\sigma \right)-\overline{u}\left(\sigma \right)|\le \frac{c_1}{2}\mathrm{for}\sigma \in e,$$

and, therefore, from (55), $u\in U_{ad}$. Then, from (56),

$$y({\tau }_1;\overline{u})=y({\tau }_0;u)=y_1,$$

which contradicts the fact that ${\tau }_0$ is optimal. □

6. Optimality Conditions

Theorem 4.

Assume that the hypotheses of Theorem 3 hold. We further assume that $H_{ad}$ is convex. Then there exists an $h_0\in H$ such that

$$(h_0,y({\tau }_0;v)-y({\tau }_0;u))\ge 0\forall v\in U_{ad}.$$

(58)

Proof. 

1. Define the set - which is clearly convex:

$$K=\left\{h\right|h\in H,h=y({\tau }_0;v)-y({\tau }_0;0)),v\in U_{ad}\}.$$

(59)

Let us verify that

$$0\mathrm{is}\mathrm{in}\mathrm{the}interiorofK,$$

(60)

$$y_1-y({\tau }_0;0)(\mathrm{which}\mathrm{belongs}\mathrm{to}K)\mathrm{is}\mathrm{not}\mathrm{in}\mathrm{the}\mathrm{interior}\mathrm{of}K.$$

(61)

To prove (60), let $h\in H$ be arbitrary. We have to prove that for sufficiently small values of $\left|\epsilon \right|$, $\epsilon h\in K$. Now,

$$h={\int }_0^{{\tau }_0}G({\tau }_0-\sigma )G(\sigma -{\tau }_0)hd\sigma ,$$

and therefore,

$$\epsilon h=y({\tau }_0;vG(\sigma -{\tau }_0)h))-y({\tau }_0;0)\in Kfor\left|\epsilon \right|\mathrm{sufficiently}\mathrm{small}.$$

We prove (61) by contradiction. If $y_1-y({\tau }_0;0)$ were in the interior of K, for an appropriate $\epsilon >0$, we would have

$$y_1-y({\tau }_0;0)+\epsilon (y_1-G\left({\tau }_0\right)y_0)\in K$$

and there would exist a $v\in U_{ad}$ such that

$$y_1-y({\tau }_0;0)+\epsilon (y_1-G\left({\tau }_0\right)y_0)=y({\tau }_0;v)-y({\tau }_0;0),$$

whence

$$y_1=y({\tau }_0;0)+{\int }_0^{{\tau }_0}G({\tau }_0-\sigma )\frac{1}{1+\epsilon }v\left(\sigma \right)d\left(\sigma \right)=y({\tau }_0;w)$$

and therefore, $w\left(\sigma \right)\in$ interior of $U_{ad}$, contradicting (54).

2. The result now follows as a consequence of (60), (61) and of Corollary 5, Section 9 of Dunford–Schwartz [37]. □

Remark 5

(58). We may interpret this as follows: let us introduce the adjoint state by

$${}^{R}D_{\tau 0}^{a}p+{\mathcal{A}}^{\ast }p=0,\mathit{in}[0,{\tau }_0],$$

(62)

$$I_{{\tau }_0}^{1-\alpha }p\left({\tau }_0\right)=h_0.$$

(63)

Then, by using the Green formula given in Lemma 1, (58) is equivalent to

$${\int }_0^{{\tau }_0}(p\left(t\right),v\left(t\right)-u\left(t\right))dt\ge 0\forall v\in U_{ad},$$

(64)

whence, we may pass to local conditions in t:

$$(p\left(t\right),h-u\left(t\right))\ge 0\mathrm{almost}\mathrm{everywhere}\mathrm{in}[0,{\tau }_0]\forall h\in H_{ad}.$$

(65)

7. Application

In this section, we state some examples to explain our abstract conclusions.

Example 2.

If $H_{ad}=$ unit ball in H, we would have

$$u\left(t\right)=-\frac{p\left(t\right)}{\left|p\right(t\left)\right|}.$$

Example 3.

Let us consider the system with the following state

$${}_0{}^{R}D^{a}y(t;v)+\mathcal{A}\left(t\right)y(t;v)=f+Bv$$

(66)

$${}_{0}I^{1-a}y(0,v)=y_0$$

(67)

and the adjoint state is given by

$${}^{C}D_{\tau 0}^{a}p+{\mathcal{A}}^{\ast }p=0,\mathrm{in}[0,{\tau }_0],$$

(68)

$$p\left({\tau }_0\right)=h_0.$$

(69)

Then (58) is equivalent to

$${\int }_0^{{\tau }_0}(p\left(t\right),v\left(t\right)-u\left(t\right))dt\ge 0\forall v\in U_{ad}.$$

(70)

Remark 6.

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

8. Open Problems

1—In a similar manner, we can also study the time-fractional optimal control of the above systems, where the time derivative is considered as the left Atangana–Baleanu fractional derivative in the Caputo sense:

$${}_0{}^{ABC}D^{a}y(t;v)+\mathcal{A}\left(t\right)y(t;v)=f+Bv,x\in \mathsf{\Omega },t\in (0,T),$$

$$y(0;v)=y_0,$$

where ${}_0{}^{ABC}D^a$ is the left Atangana–Baleanu fractional derivative in the sense of Caputo (see [1]).

2—The problem can be extended to the time-space fractional derivative as the following:

$${}_0{}^{ABC}D^{a}y(t;v)+{\left(-\Delta \right)}^{\beta }\left(t\right)y(t;v)=f+Bv,x\in \mathsf{\Omega },t\in (0,T),$$

$$y(0;v)=y_0,$$

where ${\left(-\Delta \right)}^{\beta }$ is is the fractional Laplacian operator for $\beta \in (0,1)$ (see [35,36,38]).

9. Conclusions

In this work, we stated and proved the bang-bang theorem (Theorem 2) for (FDS) with Dirichlet and Neumann boundary conditions. The fractional derivatives were defined in the weak Caputo and Riemann–Liouville sense. The analytical results were obtained in the form of Euler–Lagrange equations for the (TOCFP). The formulation presented and the resulting equations are similar to those for classical optimal control problems. The optimization problem presented in this paper represents a generalization of the time-optimal control problems of parabolic systems with Dirichlet and Neumann boundary conditions considered in Lion’s [29] to fractional time-optimal control problems. Moreover, the main result of the paper contains necessary optimality conditions for fractional systems of non-integer order, which allows for the characterization of optimality problems.