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Mathematical Sciences

, Volume 13, Issue 4, pp 407–413 | Cite as

On controllability for a nondensely defined fractional differential equation with a deviated argument

  • A. RaheemEmail author
  • M. Kumar
Open Access
Original Research
  • 219 Downloads

Abstract

This article deals with a fractional differential equation with a deviated argument defined on a nondense set. A fixed-point theorem and the concept of measure of noncompactness are used to prove the existence of a mild solution. Furthermore, by using the compactness of associated cosine family, we proved that system is approximately controllable and obtains an optimal control which minimizes the performance index. To illustrate the abstract result, we included an example.

Keywords

Approximate controllability Optimal control Deviated argument Fractional differential equation Cosine family 

Mathematics Subject Classification

93B05 49J15 34A08 47D09 47D62 34G20 34G45 47D06 47J35 

Introduction

Exact controllability and approximate controllability are some fundamental concepts in control problems. These concepts help to design and analyze the various kinds of control dynamics process in finite- and infinite-dimensional spaces. In the last few decades, control problems have been a research area of great interest. An extensive study on controllability for various kinds of control problems in abstract spaces has been done by many authors [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. In [1], authors proved the approximate controllability for semi-linear noninteger differential equations by using fixed-point technique and semigroup theory of bounded linear operators in a Hilbert space. Das et al. [2] used the concept of measure of noncompactness and the compactness of associated families of linear operators to investigate the sufficient conditions for approximate controllability of a noninteger-order control system with delay depending on state variable. More works on approximate controllability for state-dependent noninteger control have been done in [12, 13]. Most of the papers available in the literature deal with densely defined systems in abstract spaces. There are only few papers in which authors considered problems involving nondensely defined linear operators. Fu [5] has proved some results on controllability for a nondensely defined abstract differential equations. Other results on controllability for nondensely defined problems have been done in [5, 15, 16]. In [17], authors investigated some sufficient conditions for approximate controllability of delayed semilinear system by using sequence method and \(C_0\)-semigroup of bounded linear operators.

In the last decade, many authors extended the study of approximate controllability to higher order differential equations in abstract spaces [2, 4, 8, 11]. Shukla et al. [4, 8] have studied the second-order semilinear control system and proved that under some assumptions, system is approximately controllable. In [11], stochastic differential equation of order 2 has been studied. The study of controllability to noninteger-order differential equations has been done in [9, 10, 18]. In [10], authors discussed the approximate controllability for a noninteger-order stochastic control system of order \(1<\alpha \le 2.\) Few authors [18, 19] investigated the optimal controllability for noninteger-order systems. For more work on controllability of fractional differential equations, we refer [20], and papers cited in this paper.

The present work is motivated by the work of Fu [5], in which authors considered a semilinear control system with linear part is defined on a nondense domain and satisfies the Hille–Yosida condition and established some results on controllability. We extended the study of approximate and optimal controllability to a class of nondensely defined fractional differential equations with deviated argument in a Hilbert space by using the measure of noncompactness and Darbo–Sadovskii fixed-point theorem. Consider the following fractional differential equations with deviated argument in a Hilbert space \((Z, \Vert \cdot \Vert )\):
$$\begin{aligned} \left\{ \begin{array}{lll} _CD^{\alpha }z(t)=Az(t)+Ew(t)+g(t,z(t),z(b(z(t),t))), &{}\quad 0<t \le a_0,\\ \\ z(0)=z_0, &{}\quad z'(0)=z_1, \end{array}\right. \end{aligned}$$
(1)
where \(1<\alpha \le 2,\)\(_CD^{\alpha },\) denotes the (Caputo) fractional derivative, \(A: D(A) \subset Z \rightarrow Z\) is a nondensely defined linear operator which satisfies the Hille–Yosida condition, E denotes a bounded linear operator defined on W and  \(w \in L^2([0,a_0],W)\) denotes the control function, which takes values in a Banach space W. The maps gb are suitably defined functions satisfying some suitable conditions and \(z_0, z_1 \in \overline{D(A)}\). Let \(I_0=[0,a_0].\)

The organization of the rest of the manuscript is as follows. In Section 2, we state some basic concepts and assumptions. In Section 3, first we prove the existence of mild solution, then the approximate and optimal controllability for problem (1) by using a fixed-point theorem and the concept of measure of noncompactness. In the last section to illustrate the abstract result, we included an example.

Preliminaries and assumptions

In this section, we state some basic definitions, assumptions and notations.

It is well known that if A satisfies Hille–Yosida condition, then A generates an integrated semigroup which is nondegenerate and locally Lipschitz continuous [21, 22].

Define \(A_0\) on \( D(A_0) \subset \overline{D(A)}\) into \(\overline{D(A)}\) by
$$\begin{aligned} A_0z=Az, \end{aligned}$$
where
$$\begin{aligned} D(A_0)=\{z \in D(A)|Az \in \overline{D(A)}\}. \end{aligned}$$
Then, \(A_0\) generates (see [21]), a compact family \(\{C_{\alpha }(t)\}_{t \ge 0}\) of cosines on \(\overline{D(A)}.\) For \(t \ge 0,\) we define
$$\begin{aligned}&S_{\alpha }(t)=\int _{0}^{t}C_{\alpha }(s)\mathrm{d}s,\\&P_{\alpha }(t)=\frac{1}{\Gamma (\alpha -2)}\int _{0}^{t}(t-s)^{\alpha -2}C_{\alpha }(s)\mathrm{d}s. \end{aligned}$$
Let
$$\begin{aligned}&C(I_0,\overline{D(A)})=\{z:I_0 \rightarrow \overline{D(A)}|z\, \text{is continuous on}\,I_0\},\\&C_L(I_0,\overline{D(A)})=\{z \in C(I_0,\overline{D(A)})\|\Vert z(t)-z(s)\Vert \le L |t-s|\}. \end{aligned}$$
Obviously, \((C_L(I_0,\overline{D(A)}),\Vert z\Vert _{a_0})\) is a Banach space, where
$$\begin{aligned} \Vert z\Vert _{a_0}=\sup \limits _{t \in I_0} \Vert z(t)\Vert . \end{aligned}$$

Definition 1

A mild solution of problem (1) is a function \(z \in C_L(I_0,\overline{D(A)})\) satisfying the integral equation:
$$\begin{aligned} z(t)&= {} C_{\alpha }(t)z_0+S_{\alpha }(t)z_1 \\&\quad+\int _{0}^{t}P_{\alpha }(t-s)[g(s,z(s),z(b(z(s),s)))+Ew(s)]\mathrm{d}s, \\&\qquad \qquad 0 \le t \le a_0. \end{aligned}$$
(2)

Next, we define the approximate controllability on a nondensely defined domain.

Definition 2

System (1) is approximately controllable on \(I_0,\) if for every \(\epsilon _0>0\) and \(z_0, z_1, z_{a_0} \in \overline{D(A)} ,\) there is an admissible control \(w \in L^2(I_0,W)\) such that the solution z(t) of (1) satisfies
$$\begin{aligned} \Vert z(a_0)-z_{b_0}\Vert <\epsilon _0. \end{aligned}$$

Definition 3

The map \(Q_0: Z_1 \rightarrow Z_1\) is said to be an \(\phi \)-contraction if there exists a positive constant \(k_0<1\) such that \(\phi (Q_0C_0) \le k_0 \phi (C_0)\) for any bounded closed subset \(C_0 \subseteq Z_1,\) where \(Z_1\) is a Banach space and \(\phi \) denotes the Hausdorff measure of noncompactness defined on bounded subsets of \(Z_1.\)

It is convenient to define the operator
$$\begin{aligned} \Omega ^{a_0}_0=\int _{0}^{a_0}P_{\alpha }(t-s)EE^*P^*_{\alpha }(t-s)\mathrm{d}s, \end{aligned}$$
where \(E^*\) and \(P^*_{\alpha }\) are the adjoint operators of E and \(P_{\alpha }\), respectively.
For \(\delta >0,\) let
$$\begin{aligned} R(\delta , \Omega ^{a_0}_0)=(\delta I+\Omega ^{a_0}_0)^{-1} \end{aligned}$$
and
$$\begin{aligned} q(z(t))&= {} z_{a_0}-C_{\alpha }(a_0)z_0-S_{\alpha }(a_0)z_1\\&\quad-\int _{0}^{a_0}P_{\alpha }(a_0-s)g(s,z(s),b(z(s),s))\mathrm{d}s, \end{aligned}$$
then we have
$$\begin{aligned} w_z(t)=E^* P^*_{\alpha }(a_0-t)R(\delta , \Omega ^{a_0}_0)q(z(t)). \end{aligned}$$

Lemma 4

If the map \(Q_0:D(Q_0) \subseteq Z_1 \rightarrow Z_1\) is a Lipschitz continuous with constant \(k_0,\) then \(\phi (QC_0) \le k_0 \phi (C_0),\) for any bounded set \(C_0.\)

Lemma 5

[23] (Darbo–Sadovskii fixed-point theorem) A\(\phi \)-contraction continuous map\(Q_0: W_0 \rightarrow W_0\), defined on a convex, closed and bounded set\(W_0 \subseteq Z_1,\)has at least one fixed point in\(W_0.\)

We consider the following assumptions:
  1. (H1)
    Map g satisfies:
    1. (1)

      \(\Vert g(t_1,z_1,\tilde{z}_1)-g(t_2,z_2,\tilde{z}_2)\Vert _{a_0} \le L_g [|t_1-t_2|+\Vert z_1-z_2\Vert _{a_0}+\Vert \tilde{z}_1-\tilde{z}_2\Vert _{a_0}],\)

       
    2. (2)

      \(\Vert g(t,z,\tilde{z})\Vert \le G_0,\)

       
    for some \(L_g, G_0>0,\) and for all \(t, t_1, t_2 \in I_0\), \(z,z_1,z_2 \in C\left( I_0,\overline{D(A)}\right) \) and \(\tilde{z}, \tilde{z}_1,\tilde{z}_2 \in C_L\left( I_0,\overline{D(A)}\right) .\)
     
  2. (H2)
    There exists a constant \(L_b>0\) such that
    $$\begin{aligned} \Vert b(z(t),t)-b(\tilde{z}(t),t)\Vert \le L_b \Vert z(t)-\tilde{z}(t)\Vert . \end{aligned}$$
     
  3. (H3)
    There exist constants \(M_1>0,\)\(M_2>0\) and \(r_1>0\) such that
    $$\begin{aligned} \Vert C_{\alpha }(t)\Vert \le M_1,\quad \Vert S_{\alpha }(t)\Vert \le M_2,\quad \Vert P_{\alpha }(t)\Vert \le r_1, \quad t \in I_0. \end{aligned}$$
     
  4. (H4)

    There exists a constant \(M_3>0\) such that \(\Vert E\Vert \le M_3.\)

     

Lemma 6

Under assumptions (H1)–(H4), there exists\(K_w>0\) s.t. \(\Vert w_z(t)\Vert \le K_w.\)

Proof

Using the definition of control function \(w_z(t)\), and (H4), we have
$$\begin{aligned} \Vert w_z(t)\Vert&= {} \Vert E^*\Vert \Vert P^*_{\alpha }(a_0-t)\Vert \Vert R(\delta , \Omega ^{a_0}_0)\Vert \Vert q(z(t))\Vert \\&\quad\le {} \frac{M_3r_1}{\delta } \Vert q(z(t))\Vert . \end{aligned}$$
In view of assumptions (H1)–(H3), we can find a constant \(K_w>0,\) such that
$$\begin{aligned} \Vert w_z(t)\Vert \le K_w. \end{aligned}$$
\(\square \)

Main results

Theorem 1

If (H1)–(H4) hold, then for every\(z_0,z_1, z_{a_0} \in \overline{D(A)},\)system (1) has a mild solution on\(I_0,\)if
$$\begin{aligned} r_1a_0L_g(1+LL_b)\left[ 1+\dfrac{r^2_1a_0M^2_3}{\delta }\right] <\frac{1}{2}. \end{aligned}$$

Proof

To show that system (1) has a mild solution, we need to show that, there exists \( z \in C_L\left( I_0,\overline{D(A)}\right) ,\) such that
$$\begin{aligned} z(t)&= {} C_{\alpha }(t)z_0+S_{\alpha }(t)z_1\\&\quad+\int _{0}^{t}P_{\alpha }(t-s)[g(s,z(s),z(b(z(s),s)))+Ew_z(s)]\mathrm{d}s\\&\qquad \qquad 0 \le t \le a_0. \end{aligned}$$
We define the operator \(\tilde{G}_{\delta }\) on \(C_L(I_0,\overline{D(A)}),\) by
$$\begin{aligned} (\tilde{G}_{\delta }z)(t)&= {} C_{\alpha }(t)z_0+S_{\alpha }(t)z_1\\&\quad+\int _{0}^{t} P_{\alpha }(t-s) \left[ g(s,z(s),z(b(z(s),s)))+Ew_z(s) \right] \mathrm{d}s, \\&\qquad \qquad 0 \le t \le a_0. \end{aligned}$$
Let
$$\begin{aligned} B_{r_0}=\left\{z \in C_L(I_0,\overline{D(A)}) |z(0)=z_0, z'(0)=z_1,\Vert z\Vert _{a_0} \le r_0 \right\}. \end{aligned}$$
If z(t) satisfies (2), then z(t) can be written as \(z(t)=x_0(t)+x(t),\) where
$$\begin{aligned} x_0(t)&= {} C_{\alpha }(t)z_0+S_{\alpha }(t)z_1 \\ x(t)&= {} \int _{0}^{t}P_{\alpha }(t-s)[g(s,x_0(s)+x(s),x_0(b(x_0(s)+x(s),s))\\&\quad+x(b(x_0(s)+x(s),s)))+Ew_{x_0+x}(s)]\mathrm{d}s. \end{aligned}$$
Define \(\tilde{F_{\delta }}: C_L(I_0,\overline{D(A)}) \rightarrow C_L(I_0,\overline{D(A)}),\) by
$$\begin{aligned} (\tilde{F}_{\delta }x)(t)&= {} \int _{0}^{t}P_{\alpha }(t-s) [g(s,x_0(s)+x(s),x_0(b(x_0(s)+x(s),s))\\&\quad+x(b(x_0(s)+x(s),s)))+Ew_{x_0+x}(s) ]\mathrm{d}s. \end{aligned}$$
Obviously, map \(\tilde{G_{\delta }}\) has a fixed point if and only if \(\tilde{F_{\delta }}\) has a fixed point.
For \(r>0,\) we define
$$\begin{aligned} C_r=\left\{z \in C_L(I_0,\overline{D(A)}):z(0)=0,z'(0)=0,\Vert z\Vert _{a_0} \le r \right\} . \end{aligned}$$
Clearly, \(\tilde{F_{\delta }}\) is well defined on \(C_r\) which is a closed convex set in \(C_L(I_0;\overline{D(A)}).\)

To show that \(\tilde{F_{\delta }}\) has a fixed point, we use Darbo–Sadovskii fixed-point theorem. For this, we need to follow the two steps:

Step 1: There exists \(r>0,\) s.t. \(\tilde{F_{\delta }}(C_{r}) \subseteq C_r. \) Let \(x \in C_r.\) Then, using (H2), we have
$$\begin{aligned} \Vert (\tilde{F}_{\delta }x)(t) \Vert&\le {} r_1 \int _{0}^{t} [\Vert g(s,x_0(s)+x(s),x_0(b(x_0(s)+x(s),s))\\&\quad+x(b(x_0(s)+x(s),s)))+Ew_{x_0+x}(s)\Vert ]\mathrm{d}s. \end{aligned}$$
Using (H1), (H4) and Lemma 6, we get
$$\begin{aligned} \Vert (\tilde{F}_{\delta }x)(t) \Vert \le r_1(G_0a_0+M_3K_w). \end{aligned}$$
We can choose \(r>0,\) sufficiently large such that
$$\begin{aligned} \Vert (\tilde{F}_{\delta }x)(t)\Vert \le r. \end{aligned}$$
Thus, there exists \(r>0\) such that
$$\begin{aligned} \tilde{F}(C_r) \subseteq C_r. \end{aligned}$$
Step 2:\(\tilde{F_{\delta }}\) is a \(\phi \)-contraction map. For this, first, we need to show that \(\tilde{F}_{\delta }\) is Lipschitz continuous. Using (H3) and (H4), we have
$$\begin{aligned}&\Vert (\tilde{F}_{\delta }x_1)(t)-(\tilde{F}_{\delta }x_2)(t)\Vert \\&\quad \le r_1 \int _{0}^{t} \Vert g(s,x_0(s)+x_1(s),x_0(b(x_0(s)+x_1(s),s)) \\&\qquad +x_1(b(x_0(s)+x_1(s),s))) \\&\qquad -g(s,x_0(s)+x_2(s),x_0(b(x_0(s)+x_2(s),s)) \\&\qquad +x_2(b(x_0(s)+x_2(s),s))) \Vert \mathrm{d}s \\&\qquad +r_1M_3\int _{0}^{t} \Vert w_{x_0+x_1}(s)- w_{x_0+x_2}(s)\Vert \mathrm{d}s. \end{aligned}$$
(3)
Using (H1), (H2) and (H3), we get
$$\begin{aligned}&\Vert g(s,x_0(s)+x_1(s),x_0(b(x_0(s)+x_1(s),s))\\&\qquad +x_1(b(x_0(s)+x_1(s),s)))\\&\qquad -g(s,x_0(s)+x_2(s),x_0(b(x_0(s)+x_2(s),s))\\&\qquad +x_2(b(x_0(s)+x_2(s),s)))\Vert \\&\quad \le L_g\left[\Vert x_1(s)-x_2(s)\Vert +L\Vert b(x_0(s)+x_1(s),s)- b(x_0(s)+x_2(s),s)\Vert \right.\\&\qquad +\left.\Vert x_1(b(x_0(s)+x_1(s),s))-x_2(b(x_0(s)+x_2(s),s))\Vert \right] \\&\quad \le L_g \left[\Vert x_1(s)-x_2(s)\Vert +LL_b\Vert x_1(s)-x_2(s)\Vert \right.\\&\qquad +\Vert x_1(b(x_0(s)+x_1(s),s))-x_2(b(x_0(s)+x_1(s),s))\Vert \\&\qquad +\left.\Vert x_2(b(x_0(s)+x_1(s),s))-x_2(b(x_0(s)+x_2(s),s))\Vert \right] \\&\quad \le 2L_g[1+LL_b] \Vert x_1-x_2\Vert _{a_0}. \end{aligned}$$
and
$$\begin{aligned}&\Vert w_{x_0+x_1}(s)- w_{x_0+x_2}(s)\Vert \\&\quad \le \frac{M_3r_1}{\delta } \Vert q((x_0+x_1)(s))-q((x_0+x_2)(s))\Vert .\\&\quad \le \frac{2M_3r_1^2}{\delta }a_0L_g(1+LL_b)\Vert x_1-x_2\Vert _{a_0}. \end{aligned}$$
From (3), we have
$$\begin{aligned}&\Vert (\tilde{F}_{\delta }x_1)(t)-(\tilde{F}_{\delta }x_2)(t)\Vert \\&\quad \le 2r_1a_0L_g(1+LL_b)\left[ 1+\frac{r^2_1a_0M^2_3}{\delta }\right] \Vert x_1-x_2\Vert _{a_0}. \end{aligned}$$
Thus, \(\tilde{F}_{\delta }\) is Lipschitz continuous. By using Lemma 4, we have
$$\begin{aligned} \phi (\tilde{F}_{\delta }C_r) \le 2r_1a_0L_g(1+LL_b)\left[ 1+\frac{r^2_1a_0M^2_3}{\delta }\right] \phi (C_r). \end{aligned}$$
Since \(2r_1a_0L_g(1+LL_b)\left[ 1+\dfrac{r^2_1a_0M^2_3}{\delta }\right] <1,\)\(\tilde{F}_{\delta }\) is a \(\phi \)-contraction, by using Lemma 5, \(\tilde{F}_{\delta }\) has a fixed point and consequently \(\tilde{G}_{\delta }\) has a fixed point, say \(z_{\delta }.\)\(\square \)

Theorem 2

Under all the assumptions of Theorem 1, system (1) is approximately controllable on\(I_0.\)

Proof

The fixed point \(z_{\delta }\) obtained in Theorem 1 is given by:
$$\begin{aligned} z_{\delta }(t)\,=\, & {} C_{\alpha }(t)z_0+S_\alpha (t)z_1\\&+\int _{0}^{t} P_{\alpha }(t-s)[g(s,z_{\delta }(s),z_{\delta }(b(z_{\delta }(s),s))) \\&+Ew_{z_{\delta }}(s)]\mathrm{d}s, \end{aligned}$$
where
$$\begin{aligned} w_{z_{\delta }}(t)\,=\,D^* P_{\alpha }^*(a_0-t)R(\delta , \Omega _0^{a_0})q(z_{\delta }(t)), \end{aligned}$$
and
$$\begin{aligned} q(z_{\delta }(t))\,=\, & {} z_{a_0}-C_{\alpha }(a_0)z_0-S_{\alpha }(a_0)z_1\\&-\int _{0}^{a_0} P_{\alpha }(a_0-s)g(s,z_{\delta }(s),b(z_{\delta }(s),s))\mathrm{d}s. \end{aligned}$$
Thus, we have
$$\begin{aligned} z_{\delta }(a_0)-z_{a_0}\,=\, & {} - q(z_{\delta }(t))+\Omega _0^{a_0}R(\delta ,\Omega _0^{a_0})q(z_{\delta }(t)) \\= & {} -(\delta I+\Omega _0^{a_0})R(\delta ,\Omega _0^{a_0})q(z_{\delta }(t))+\Omega _0^{a_0}R(\delta ,\Omega _0^{a_0})q(z_{\delta }(t)) \\= & {} -\delta R(\delta ,\Omega _0^{a_0})q(z_{\delta }(t)). \end{aligned}$$
(4)
Now
$$\begin{aligned} \int _{0}^{a_0} \Vert g(s,z_{\delta }(s),b(z_{\delta }(s),s)) \Vert ^2\mathrm{d}s \le G^2_0a_0. \end{aligned}$$
Thus, the sequence \(\{g(t,z_{\delta }(t),b(z_{\delta }(t),t))\}\) is bounded in the Hilbert space \(L^2(I_0,Z).\) Therefore, there exists a weakly convergent subsequence \(\{g(t,z_{\delta }(t),b(z_{\delta }(t),t))\}\) (again denoted by same notation) converging to \(v(\cdot ) \in L^2(I_0,Z).\) Let
$$\begin{aligned} \mu \,=\,z_{a_0}-C_{\alpha }(a_0)z_0-S_{\alpha }(a_0)z_1-\int _{0}^{a_0}\tilde{P}_{\alpha }(a_0-s)v(s)\mathrm{d}s. \end{aligned}$$
Thus,
$$\begin{aligned} \Vert q(z_{\delta }(t))-\mu \Vert \le \left\| \int _{0}^{a_0} \tilde{P}_{\alpha }(a_0-s)[g(s,z_{\delta }(s),b(z_{\delta }(s),s))-v(s)]\mathrm{d}s \right\| . \end{aligned}$$
By using the compactness of \(P_{\alpha }(t)\), the Ascola–Arzela theorem, and following the idea used in [1], one can prove that the mapping
$$\begin{aligned} z(t) \rightarrow \int _{0}^{a_0} P_{\alpha }(t-s)z(s)\mathrm{d}s \end{aligned}$$
is compact. Therefore, we have
$$\begin{aligned} \Vert q(z_{\delta }(t))-\mu \Vert \rightarrow 0 \quad \text{as}\,\delta \rightarrow 0. \end{aligned}$$
From (4), we have
$$\begin{aligned} \Vert z_{\delta }(a_0)-z_0 \Vert\,=\, & {} \Vert \delta R(\delta , \Omega _0^{a_0}) q(z_{\delta }(t))\Vert \\\le & {} \Vert \delta R(\delta , \Omega _0^{a_0}) \Vert \Vert q(z_{\delta }(t))-\mu \Vert + \Vert \delta R(\delta , \Omega _0^{a_0}) \mu \Vert . \end{aligned}$$
This implies that \(\Vert z_{\delta }(a_0)-z_0 \Vert \rightarrow 0\) as \(\delta \rightarrow 0.\) Thus, system (1) is approximately controllable on \(I_0.\)\(\square \)
Next, we discuss the optimal controllability of problem (1). In order to proceed further, we define performance index
$$\begin{aligned} \tilde{J}(w)=\int _{0}^{a_0}\tilde{F}(t,z(t),z(b(z(t),t)),w(t))\mathrm{d}t, \end{aligned}$$
where \(\tilde{F}\) is a functional defined on \(I_0 \times C(I_0,Z) \times C_L(I_0,Z) \times W_{ad} \) and \( W_{ad}\) denotes the set of all admissible control and consequently is closed and convex in \(L^2(I_0,W).\)

Theorem 3

Under all the assumptions of Theorem 1, there exists an optimal control of problem (1) if\(r_1 a_0 L_g(2+LL_b) <1.\)

Proof

To complete the proof of the theorem, it is sufficient to find a control \(w^0 \in L^2(I_0,W)\) that minimizes the performance index \(\tilde{J}(w).\)

If \(\inf \{ \tilde{J}(w)|w \in W_{ad} \}\,=\, \infty ,\) then result trivially holds.

If \(\inf \{ \tilde{J}(w)|w \in W_{ad} \}\,=\,\epsilon _0 < \infty ,\) then there exists a sequence \(\{w^n\}\) in \(W_{ad}\) such that \(\tilde{J}(w^n) \rightarrow \epsilon _0.\) As \(W_{ad}\) is a closed and convex subset of \(L_2(I_0,W),\) the sequence \(\{w^n\}\) has a weakly convergent subsequence \(\{w^m\}\) converging to \(w^0 \in W_{ad}.\) Using Theorem 1, for each \(w^m \in W_{ad},\) there exists a mild solution \(z^m\) of (1) satisfying:
$$\begin{aligned} z^m(t)\,=\, & {} C_{\alpha }(t)z_0+S_{\alpha }(t)z_1\\&+\int _{0}^{t}P_{\alpha }(t-s)[g(s,z^m(s),z^m(b(z^m(s),s)))\\&+Ew^m(s)]\mathrm{d}s, \quad 0 \le t \le a_0. \end{aligned}$$
Similarly, corresponding to \(w^0,\) there exists a mild solution \(z^0\) of (1) satisfying:
$$\begin{aligned} z^0(t)\,=\, & {} C_{\alpha }(t)z_0+S_{\alpha }(t)z_1\\&+\int _{0}^{t}P_{\alpha }(t-s)[g(s,z^0(s),z^0(b(z^0(s),s)))\\&+Ew^0(s)]\mathrm{d}s, \quad 0 \le t \le a_0. \end{aligned}$$
Using (H3), for \(t \in [0,a_0],\) we have
$$\begin{aligned}&\Vert z^m(t)-z^0(t) \Vert \\&\quad \le r_1 \int _{0}^{t} [\Vert g(s,z^m,z^m(b(z^m(s),s)))-g(s,z^0,z^0(b(z^0(s),s))) \Vert \\&\qquad + \Vert Ew^n(s)-Ew^0(s)\Vert ]\mathrm{d}s. \end{aligned}$$
(5)
Using (H1), (H2), we get
$$\begin{aligned}&\Vert g(s,z^m(s),z^m(b(z^m(s),s)))-g(s,z^0(s),z^0(b(z^0(s),s))) \Vert \\&\quad \le L_g [\Vert z^m(s)-z^0(s)\Vert +\Vert z^m(b(z^m(s),s))-z^0(b(z^0(s),s))\Vert ] \\&\quad \le L_g [\Vert z^m(s)-z^0(s)\Vert +\Vert z^m(b(z^m(s),s))-z^m(b(z^0(s),s))\Vert \\&\qquad +\Vert z^m(b(z^0(s),s))-z^0(b(z^0(s),s))\Vert ] \\&\quad \le L_g(2+LL_b) \Vert z^m-z^0\Vert _{a_0}. \end{aligned}$$
From (5), we have
$$\begin{aligned}&\Vert z^m(t)-z^0(t)\Vert \\&\quad \le r_1\left[ L_g(2+LL_b)a_0 \Vert z^m-z^0\Vert _{a_0}+ \int _{0}^{t}\Vert Ew^m(s)-Ew^0(s)\Vert \mathrm{d}s\right] \\&\quad \le r_1 a_0 [L_g(2+LL_b) \Vert z^m-z^0\Vert _{a_0}+\Vert Ew^m-Ew^0\Vert _{a_0}]. \end{aligned}$$
Since \(r_1 a_0 L_g(2+LL_b) <1\) and \(\Vert Ew^m-Ew^0\Vert _{a_0} \rightarrow 0,\) we conclude that \(z^m \rightarrow z^0.\)
Applying Balder’s theorem, we get
$$\begin{aligned} \epsilon _0\,=\, & {} \lim \limits _{n \rightarrow \infty }\int _{0}^{a_0}\tilde{F} (t,z^m(t),z^m(b(z^m(t),t)),w^m(t))\mathrm{d}t \\\le & {} \int _{0}^{a_0} \tilde{F} (t,z^0(t), z^0(b(z^0(t),0)),w^0(t))\mathrm{d}t \\= & {} \tilde{J}(w^0) \ge \epsilon _0. \end{aligned}$$
This shows that \(\tilde{J}(w^0)\,=\,\epsilon _0,\) i.e., \(\tilde{J}\) attains its minimum value at \(w^0 \in L^2(I_0,W).\)\(\square \)

Application

Consider the following example:
$$\begin{aligned} \left\{ \begin{array}{lll}_CD^{\frac{3}{2}}z(y,s)-\frac{\partial ^2}{\partial y^2}z(y,s)=\nu _0(y,s)+g_1(y,z(y,s))+g_2(s,y,z(y,s)),\\ \qquad \qquad \qquad y \in (0,\pi ) \quad s \in (0,a_0],\\ z(0)=z(\pi )=0, \\ z(y,0)=z_0(y), \quad \frac{\partial z}{\partial s}(y,0)=z_1(y), \quad y \in (0,\pi ), \end{array}\right. \end{aligned}$$
(6)
where
$$\begin{aligned} g_1(y,z(y,s))=\int _{0}^{y}K(y,x)z(y,a_1h(t)|z(x,t)|)\mathrm{d}x, \end{aligned}$$
and the function \(g_2:R_+ \times [0,1] \times R \rightarrow R\) is locally Hölder continuous in t,  locally Lipschitz continuous in z,  uniformly in y and measurable in y. Let \(I_0=[0,a_0].\)
Choose \(Z=C([0,\pi ], R)\) and the operator A defined by \(Az=z''\) with domain
$$\begin{aligned} D(A)=\{z(\cdot ) \in Z|z'' \in Z,z(0)=z(\pi )=0\}. \end{aligned}$$
Obviously, \(\overline{D(A)} \ne Z.\) Thus, A is nondensely defined on Z.

Also \(\rho (A) \supseteq (0,+\infty ),\)\(\Vert (\lambda I-A)^{-1} \Vert \le \frac{1}{\lambda }\) for \(\lambda >0.\) This shows that A satisfies the Hille–Yosida on Z. Therefore, A generates a compact \(C_0\)-semigroup \(\{T(t)\}\) on \(\overline{D(A)}\) and consequently A generates a compact cosine family \(\{C_{\alpha }(t)\}_{t \ge 0}\) on \(\overline{D(A)}.\)

Let \(C\left( I_0,\overline{D(A)}\right) \) denote the set of all continuous functions on \(I_0,\) and
$$\begin{aligned} C_L\left( I_0,\overline{D(A)}\right) =\left\{ z \in C \left( I_0,\overline{D(A)}\right) |\Vert z(s_1)-z(s_2)\Vert \le L |s_1-s_2|\right\} . \end{aligned}$$
We define \(g:I_0 \times C \left( I_0,\overline{D(A)} \right) \times C_L \left( I_0,\overline{D(A)} \right) \rightarrow X,\) by
$$\begin{aligned} g(s,z, \phi )(y)=g_1(y,\phi )+g_2(s,y,z), \end{aligned}$$
and \((E\nu )(s)(y)=\nu _0(s,y),\)\(z(s)(y)=z(y,s)\); then, abstract formulation of problem (6) is
$$\begin{aligned} _CD^{\frac{3}{2}}z(s)= & {} Az(s)+E\nu (s)+g(s,z(s),b(z(s),s)) \\ z(0)= & {} z_0, \quad \frac{\partial z}{\partial s}(0)=z_1. \end{aligned}$$
We can easily prove that all assumptions of Theorem 2 are satisfied; therefore by using theorem, we conclude that system (6) is approximately controllability on \(I_0.\)
Define performance index
$$\begin{aligned} \tilde{J}(w)=\int _{0}^{a_0} (\Vert z(t)\Vert ^2+\Vert w(t)\Vert ^2)\mathrm{d}t. \end{aligned}$$
It can be easily checked that all the assumptions of Theorem 3 are satisfied; therefore, by using Theorem 3, we find an admissible control \(w_0\) that minimizes \(\tilde{J}(w)\).

Notes

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Authors and Affiliations

  1. 1.Department of MathematicsAligarh Muslim UniversityAligarhIndia

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