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

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## 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.

*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

*g*,

*b*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].

### **Definition 1**

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

### **Definition 2**

*z*(

*t*) of (1) satisfies

### **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 \(Z_1.\)

*E*and \(P_{\alpha }\), respectively.

### **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*\(W_0.\)

- (H1)Map
*g*satisfies: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 \(\tilde{z}, \tilde{z}_1,\tilde{z}_2 \in C_L\left( I_0,\overline{D(A)}\right) .\)- (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)
\(\Vert g(t,z,\tilde{z})\Vert \le G_0,\)

- (1)
- (H2)There exists a constant \(L_b>0\) such that$$\begin{aligned} \Vert b(z(t),t)-b(\tilde{z}(t),t)\Vert \le L_b \Vert z(t)-\tilde{z}(t)\Vert . \end{aligned}$$
- (H3)There exist constants \(M_1>0,\)\(M_2>0\) and \(r_1>0\) such that$$\begin{aligned} \Vert C_{\alpha }(t)\Vert \le M_1,\quad \Vert S_{\alpha }(t)\Vert \le M_2,\quad \Vert P_{\alpha }(t)\Vert \le r_1, \quad t \in I_0. \end{aligned}$$
- (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*

## 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*

### *Proof*

*z*(

*t*) satisfies (2), then

*z*(

*t*) can be written as \(z(t)=x_0(t)+x(t),\) where

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

*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

### **Theorem 2**

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

### *Proof*

### **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.

## Application

*t*, locally Lipschitz continuous in

*z*, uniformly in

*y*and measurable in

*y*. Let \(I_0=[0,a_0].\)

*A*defined by \(Az=z''\) with domain

*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 \(\overline{D(A)}.\)

## Notes

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