Controllability and constrained controllability for nonlocal Hilfer fractional differential systems with Clarke’s subdifferential

Sobolev-type nonlocal fractional differential systems with Clarke’s subdifferential are studied. Sufficient conditions for controllability and constrained controllability for Sobolev-type nonlocal fractional differential systems with Clarke’s subdifferential are established, where the time fractional derivative is the Hilfer derivative. An example is given to illustrate the obtained results.

A Sobolev-type equation appears in several physical problems such as flow of fluids through fissured rocks, thermodynamics and propagation of long waves of small amplitude (see [1,2,3]). Nonlinear fractional differential equations can be observed in many areas such as population dynamics, heat conduction in materials with memory, seepage flow in porous media, autonomous mobile robots, fluid dynamics, traffic models, electro magnetic, aeronautics, economics (see [4,5,6,7,8,9,10,11,12,13]). Controllability means to steer a dynamical system from an arbitrary initial state to the desired final state in a given finite interval of time by using the admissible controls, and controllability results for linear and nonlinear integer order differential systems were studied by several authors (see [14,15,16,17,18,19,20,21,22,23,24,25,26,27]). The constrained controllability is concerned with the existence of an admissible control that steers the state to a given target set from a specified initial state. Few authors studied constrained controllability; for example Son [28] studied constrained approximate controllability for the heat equations and retarded equations, Klamka [29] studied constrained controllability of nonlinear systems, Klamka [30] studied constrained controllability of semilinear systems with delays, Sikora and Klamka [31] studied constrained controllability of fractional linear systems with delays in control. Furthermore, the Clarke subdifferential has been applied in mechanics and engineering, especially in nonsmooth analysis and optimization [32, 33]. However, the controllability and the constrained controllability of nonlocal Hilfer fractional differential equations with the Clarke subdifferential have not yet been considered in the literature, and this fact motivates this work. The purpose of this paper is to study the controllability of Sobolev-type nonlocal Hilfer fractional differential equation system with the Clarke subdifferential in Banach spaces and to study the constrained local controllability of Sobolev-type nonlocal Hilfer fractional differential system with the Clarke subdifferential in Banach spaces.

In order to study the controllability and constrained controllability for Clarke subdifferential Hilfer fractional differential equations with nonlocal condition, we need the following basic definitions and lemmas.

Definition 2.1

(see [34])

The fractional integral operator of order \(\mu > 0 \) for a function f can be defined as

where \(\varGamma (\cdot )\) is the Gamma function.

Definition 2.2

(see [35, 36])

The Hilfer fractional derivative of order \(0\leq \nu \leq 1 \) and \(0 < \mu < 1\) is defined as

Next we recall some definitions from multi-valued analysis (see [37])

1. (i)

   For a given Banach space X, a multi-valued map \(F: X \rightarrow 2^{X} \setminus \{ \emptyset \} := P(X) \) is convex (closed) valued, if \(F(x)\) is convex (closed) for all \(x \in X \).

2. (ii)

   F is called upper semi-continuous (u.s.c) on X, if for each \(x \in X\), the set \(F(x)\) is a non-empty, closed subset of X, and if for each open set V of X containing \(F(x)\), there exists an open neighborhood N of x such that \(F(N)\subseteq V\).

3. (iii)

   F is said to be completely continuous if \(F(V)\) is relatively compact, for every bounded subset \(V \subseteq X \).

4. (iv)

   Let \((\varOmega , \varSigma )\) be a measurable space and \((X,d)\) a separable metric space. A multi-valued map \(F: J \rightarrow P(X) \) is said to be measurable, if for every closed set \(C\subseteq X \), we have \(F^{-1} = \{ t \in J : F(t) \cap C \neq \emptyset \} \in \varSigma \).

Throughout this paper, let X is a Banach spaces with \(\|\cdot \|\) and let \(C(J,X)\) be the Banach space of all continuous maps from \(J=(0,a]\) into X.

Define \(Y= \{x: \cdot ^{(1-\nu )(1-\mu )}x(\cdot ) \in C(J,X)\}\), with norm \(\|\cdot \|_{Y} \) defined by

Obviously, Y is a Banach space.

Introduce the set \(B_{r} = \{ x \in Y : \| x \|_{Y} \leq r \}\), where \(r> 0\).

For \(x \in X\), we define two families of operators \(\{ S_{\nu , \mu }(t): t > 0 \}\) and \(\{ P_{\mu }(t): t> 0 \}\) by

where

is a function of Wright-type which satisfies

Lemma 2.1

(see [38])

The operators \(S_{\nu , \mu }\) and \(P_{\mu }\) have the following properties.

1. (i)

   \(\{ P_{\mu }(t): t > 0\}\) is continuous in the uniform operator topology.

2. (ii)

   For any fixed \(t >0\), \(S_{\nu , \mu }(t)\) and \(P_{\mu }(t)\) are linear and bounded operators, and

   $$ \bigl\Vert P_{\mu }(t) x \bigr\Vert \leq \frac{M t^{\mu -1}}{\varGamma (\mu )} \Vert x \Vert ,\qquad \bigl\Vert S_{\nu ,\mu }(t) x \bigr\Vert \leq \frac{M t^{(\nu -1)(1-\mu )}}{\varGamma ( \nu (1-\mu )+ \mu )} \Vert x \Vert . $$

   (2.3)
3. (iii)

   \(\{ P_{\mu }(t): t > 0\}\) and \(\{ S_{\nu , \mu } (t): t > 0\}\) are strongly continuous.

4. (iv)

   For every \(t > 0\), \(\{ P_{\mu }(t)\}\) and \(\{ S_{\nu , \mu } (t) \}\) are also compact operators if \(T(t)\), \(t>0\) is compact.

The operators \(A: D(A)\subset X \rightarrow Y\) and \(E: D(E) \subset X \rightarrow Y\) satisfy the following conditions:

1. (H1)

   A and E are closed linear operators.

2. (H2)

   \(D(E) \subset D(A)\) and E is bijective.

3. (H3)

   \(E^{-1}: Y \rightarrow D(E)\) is continuous.

Here, (H1) and (H2) together with the closed graph theorem imply the boundedness of the linear operator \(AE^{-1}:Y\rightarrow Y\).

1. (H4)

   For each \(t\in J\) and for \(\lambda \in \rho (-AE^{-1})\), the resolvent of \(-AE^{-1}\), the resolvent of \(R(\lambda , -AE^{-1})\) is the compact operator.

Lemma 2.2

(see [39])

Let \(T(t)\) be a uniformly continuous semigroup. If the resolvent set \(R(\lambda , A)\) of A is compact for every \(\lambda \in \rho (A)\), then \(T(t)\) is a compact semigroup.

From the above fact, \(-AE^{-1}\) generates a compact semigroup \(\{S(t),t>0\}\) in Y, which means that there exists \(M >1\) such that \(\sup_{t\in J}\|S(t)\|\leq M\).

Definition 2.3

(see [33, 37])

Let X be a Banach space with the dual space \(X^{*}\) and \(Z: X \rightarrow R\), be a locally Lipschitz functional on X. The Clarke generalized directional derivative of Z at the point \(x \in X \) in the direction \(v\in X\), denoted by \(Z^{0}{(x;v)}\), is defined by

The Clarke generalized gradient of Z at \(x\in X\), denoted by \(\partial Z(x)\), is a subset of \(X^{*}\) given by

1. (H5)

   The functional \(Z:J\times X\rightarrow R\) satisfies the following conditions:

   1. (i)

      \(Z(\cdot,x):J \to R\) is measurable for all \(x\in X\);

   2. (ii)

      \(Z(t,\cdot): X \to R\) is locally Lipschitz continuous for a.e. \(t \in J\);

   3. (iii)

      there exist a function \(\zeta \in L^{p} (J,R^{+})\) (\(0< \frac{1}{p}<\mu < 1\)) and constant \(k>0\) satisfying

      $$ \bigl\Vert {\partial Z(t,x)} \bigr\Vert _{X} = \sup \bigl\{ { \Vert z \Vert _{X}}:z \in \partial Z(t,x) \bigr\} \leq \zeta (t)+ k \Vert x \Vert _{X},\quad \forall x\in X, \mbox{a.e. } t\in J. $$

Now we define an operator \(N: L^{2}(J,X)\rightarrow 2^{L^{2}(J,X)} \) as follows:

Lemma 2.3

If (H5) holds, then for \(x\in L^{2}(J,X)\) the set \(N(x)\) has non-empty, convex and weakly compact values.

Lemma 2.4

If (H5) holds, then the operator N satisfies: if \(x_{n}\rightarrow x\) in \(L^{2}(J,X)\), \(w_{n}\rightarrow w\) weakly in \(L^{2}(J,X)\) and \(w_{n} \in N(x_{n})\), then we have \(w \in N(x)\).

Theorem 2.1

Let X be a Banach space and \(F: X \rightarrow 2^{X}\) be a compact convex valued, u.s.c. multi-valued maps such that there exists a closed neighborhood V of 0 for which \(F(V)\) is a relatively compact set. If the set \(\varOmega = \{ x \in X : \lambda x \in F(x) \textit{ for some } \lambda > 1 \} \) is bounded, then F has a fixed point.

In this section, we present and prove main results of controllability for a Sobolev-type nonlocal Hilfer fractional differential system with the Clarke subdifferential in Banach spaces in the following form:

where \(D^{\nu ,\mu }_{0+} \) is the Hilfer fractional derivative, \(0 \leq \nu \leq 1\), \(0 < \mu < 1\), A and E are closed, linear and densely defined operators with domain contained in the Banach space X and ranges contained in the Banach space Y. The state \(x(\cdot )\) takes values in the Banach space X and the control function \(u(\cdot )\) is given in \(L^{2}(J, U)\). The Banach space of admissible control functions with U a Banach space. The symbol B stands for a bounded linear from U into Y. The nonlinear operators \(f : J \times X \rightarrow Y\), \(H : J \times J \times X \rightarrow X\), \(g : J \times J \times X \times X \rightarrow Y\) and \(\partial Z(t, \cdot )\) is the Clarke subdifferential of \(Z(t,\cdot )\).

To establish the result, we need the following additional hypotheses:

1. (H6)

   \(f: J \times X \rightarrow Y \) is a continuous function and there exist constants \(N_{1} > 0\) and \(N_{2} > 0\) such that, for all \(t \in J\), \(v_{1}\), \(v_{2}\in X\) we have

   $$ \bigl\Vert f(t,v_{1}) - f(t,v_{2}) \bigr\Vert \leq N_{1} \Vert v_{1} - v_{2} \Vert , \qquad N_{2} = \bigl\Vert f(t,0) \bigr\Vert . $$
2. (H7)

   \(g: J \times J \times X \times X \rightarrow Y \) is a continuous function and there exist constants \(L_{1} > 0\) and \(L_{2} > 0\) such that, for all \(t,s \in J\), \(v_{1}, v_{2}, w_{1}, w_{2} \in X\) we have

   $$\begin{aligned}& \bigl\Vert g(t,s,v_{1},w_{1}) - g(t,s,v_{2},w_{2}) \bigr\Vert \leq L_{1} \bigl[ \Vert v_{1} - v _{2} \Vert + \Vert w_{1} - w_{2} \Vert \bigr] , \\& L_{2} = \bigl\Vert g(t,s,0,0) \bigr\Vert . \end{aligned}$$
3. (H8)

   \(H : J \times J \times X \rightarrow X\) is continuous and there exist constants \(L_{3} > 0\), \(L_{4} > 0\), such that for all \(t,s \in J\), \(v_{1}, v_{2} \in X\) we have

   $$ \bigl\Vert H(t,s,v_{1}) - H(t,s,v_{2}) \bigr\Vert \leq L_{3} \Vert v_{1} - v_{2} \Vert , \qquad L_{4} = \bigl\Vert H(t,s,0) \bigr\Vert . $$
4. (H9)

   The linear operator W from U into E defined by

   $$ Wu = \int _{0}^{a} E^{ - 1}P_{\mu }(a-s)Bu(s) \,ds, $$

   has an inverse operator \(W^{-1}\) which takes values in \(L^{2}(J, U) \setminus \ker W\), where the kernel space of W is defined by \(\ker W = \{x \in L^{2}(J, U): Wx = 0 \} \) and B is a bounded operator.

Definition 3.1

We say \(x \in C(J,X)\) is a mild solution of the system (3.1) if it satisfies the integral equation

where

The proof of mild solution of Eq. (3.1) is similar to the proof of mild solution of Eq. (1.1) in [38].

Definition 3.2

The system (3.1) is said to be controllable on J, if for every \(x_{0},x_{1} \in X\), there exists a control \(u \in L^{2}(J,U)\) such that the mild solution \(x(t)\) of the system (3.1) satisfies \(x(a)= x_{1}\), where \(x_{1}\) and a are the preassigned terminal state and time, respectively.

Theorem 3.1

If the hypotheses (H1)–(H9) are satisfied, then the system (3.1) is controllable on J provided that there exists a constant \(r>0\) such that

Proof

For any \(x \in C(J,X)\subset L^{2}(J,X)\) from Lemma 2.3 we consider the map \(V_{r}: C(J,X)\rightarrow 2^{C(J,X)}\) as follows:

We will show \(V_{r}\) has a fixed point using Theorem 2.1. Note \(V_{r}(x)\) is convex from convexity of \(N(x)\). We divide the proof into five steps.

Step 1: \(V_{r}\) maps bounded sets into bounded sets in \(C(J,X)\).

For any \(x \in B_{r}\) and \(\varPhi \in V_{r}(x)\), we choose a \(z \in N(x)\) with

Using the assumption (H9) for any arbitrary function \(x(\cdot)\), define the control

then the operator Φ takes the form

From (H7), (H8) and the Beta function, we have

From (H5)–(H9), Lemma 2.1 and Hölder’s inequality, we have

Thus \(V_{r}(B_{r}) \) is bounded in \(C(J,X)\).

Step 2: \(\{ V_{r}(x):x \in B_{r} \}\) is equicontinuous (for all \(r>0\)).

For any \(x \in B_{r}\) and \(\varPhi \in V_{r}(x)\) and \(z \in N(x)\) and from Lemma 2.1(ii) and Hölder’s inequality, we have

Thus, for all \(\varepsilon >0\) and for sufficiently small \(\delta _{1}>0\), with \(0< t\leq \delta _{1}\), we have \(\| \varPhi (t)-\varPhi (0)\|_{Y}<\frac{\varepsilon }{2} \). Hence, for all \(\varepsilon >0\), \(\forall \tau _{1} , \tau _{2} \in [0,\delta _{1}]\) and \(\forall \varPhi \in V_{r} (B_{r})\), we have \(\| \varPhi (\tau _{2})- \varPhi (\tau _{1})\|_{Y}<\varepsilon \). For any \(x \in B_{r}\), and \(\frac{\delta _{1}}{2}\leq \tau _{1}<\tau _{2}\leq a\), we obtain

From the compactness of \(T(t), t>0\), Lemma 2.1(ii), we see that the right hand side of inequality (3.4) tends to zero as \(\tau _{2} \rightarrow \tau _{1}\). Thus we see that \(\| { ( {\varPhi } )( \tau _{2} ) - ( {\varPhi } )( \tau _{1} )} \| _{Y}\) tends to zero.

For \(\forall \varepsilon >0\), \(\forall \tau _{1}, \tau _{2} \in (0,a]\), \(| \tau _{1}-\tau _{2} |<\delta _{1}\), \(\forall \varPhi \in V_{r}(B_{r}) \) we see that \(\| { ( {\varPhi } )( \tau _{2} ) - ( {\varPhi } )( \tau _{1} )} \|_{Y}<\varepsilon \) independently of \(x \in B_{r}\). Therefore, we deduce that \(\{V_{r}(x): x \in B_{r} \}\) is an equicontinuous family of functions in \(C(J,X)\).

Step 3: \(V_{r}\) is completely continuous.

We prove that, for all \(t \in J\), \(r>0\), the set \(\prod (t)= \{ \varPhi (t): \varPhi \in V_{r}(B_{r}) \}\) is relatively compact in X. Obviously, \(\prod (0)=x_{0} - q(x)\) is compact, so we only need to consider \(t>0\). Let \(0< t< a\) be fixed. For any \(x \in B_{r}\), \(\varPhi \in V_{r}(x)\), we choose \(z \in N(x) \) with

For each \(\epsilon \in (0,t)\), \(t \in (0,a]\), \(x \in B_{r}\), and any \(\delta >0\), we define

From the compactness of \(S(\epsilon ^{\mu }\delta )\), \(\epsilon ^{\mu } \delta >0\) and the bounded of \(u(s)\) we see that the set \(\prod_{\epsilon ,\delta }(t)= \{ \varPhi ^{\epsilon ,\delta }(t): \varPhi \in V_{r}(B_{r})\} \) is relatively compact in X for each \(\epsilon \in (0,t)\) and \(\delta >0\). Moreover, we have

Now we see that \(\| \varPhi (t)-\varPhi ^{\epsilon ,\delta }(t)\|_{Y}\rightarrow 0\) as \(\epsilon \rightarrow 0\), \(\delta \rightarrow 0\). Therefore, the set \(\prod (t)\), \(t>0\) is totally bounded, i.e., relatively compact in X. From the above (and step 2) and the Ascoli–Arzela theorem, we see that \(V_{r}\) is completely continuous.

Step 4: \(V_{r}\) has a closed graph.

Let \(x_{n} \rightarrow x_{*}\) as \(n \rightarrow \infty \) in \(C(J,X)\), \(\varPhi _{n} \in V_{r}(x_{n})\) and \(\varPhi _{n} \rightarrow \varPhi _{*}\) as \(n \rightarrow \infty \) in \(C(J,X)\). We prove that \(\varPhi _{*} \in V_{r}(x_{*})\). Now \(\varPhi _{n} \in V_{r}(x_{n})\), so there exist \(z_{n} \in N(x_{n})\), \(f_{n}=f(t,x_{n}(t))\), \(R_{n}(\tau )= \int _{0}^{\tau }H(\tau ,\eta ,x_{n}(\eta ))\,d\eta \) and \(g_{n}=g(t,s,x_{n}(s),R _{n}(\tau ))\) in \(L^{2}(J,X)\) with

From (H5)–(H8), \(\{z_{n}, f_{n}, g_{n}\}_{n\geq 1}\subseteq L ^{2}(J,X)\) are bounded. Hence we assume that

From (3.5), (3.6) and compactness of \(P_{\mu }(t)\), we have

Note that \(\varPhi _{n} \rightarrow \varPhi _{*}\) in \(C(J,X)\) and \(z_{n} \in N(x_{n}) \). Hence, from Lemma 2.4 we obtain \(z_{*} \in N(x_{*})\) and \(\varPhi _{*} \in V_{r}(x_{*})\), which implies \(V_{r}\) has a closed graph and \(V_{r}\) is u.s.c.

Step 5: A priori estimate.

From steps 1–4, we see that \(V_{r}\) is u.s.c. and is compact convex valued and \(V_{r}(B_{r})\) is a relatively compact set (here \(r>0\)). We now prove that the set \(\varOmega = \{ x \in C(J,X):\lambda x \in V_{r}(x), \lambda >0 \}\) is bounded. For all \(x \in \omega \), there exist \(z \in N(x)\) and f, g in \(L^{2}(J,X)\) with

Then from assumptions (H5)–(H8), we derive

It follows from (3.7) and \(\lambda ^{-1}<1\) that \(\| x(t) \| _{Y} \leq r\). Hence, \(\| x \|_{C} = \sup_{t\in J} \| x(t) \|_{Y} \leq r \), which implies the set Ω is bounded.

From Theorem 2.1, \(V_{r}\) has a fixed point, i.e., the system (3.1) is controllable and the proof is complete. □

In this section, we present the constrained local controllability of Sobolev-type nonlocal Hilfer fractional differential system with the Clarke subdifferential in Banach spaces in the following form:

where the nonlinear operators \(f_{1} : J \times X \times U \rightarrow Y\), \(H : J \times J \times X \rightarrow X\), \(g_{1} : J \times J \times X \times X \times U \rightarrow Y\) and \(\partial Z(t,\cdot )\) is the Clarke subdifferential of \(Z(t,\cdot )\).

In this section, we need the following hypotheses:

1. (H10)

   Let \(\|Bu(t)\|\leq M_{B} \|u(t)\|_{U}\) for all \(u(t)\in U\) on J where \(M_{B}>0\).

2. (H11)

   \(f_{1}: J \times X \times U \rightarrow Y \) is a uniformly continuous function in t and there exist constants \(L_{5} > 0\) such that for all \(t \in J\), \(v_{1}\), \(v_{2}\in X\), \(u_{1}, u_{2} \in U\) we have

   $$ \bigl\Vert f_{1}(t,v_{1},u_{1}) - f_{1}(t,v_{2},u_{2}) \bigr\Vert \leq L_{5} \bigl( \Vert v_{1} - v_{2} \Vert + \Vert u_{1} - u_{2} \Vert _{U} \bigr). $$
3. (H12)

   \(g_{1}: J \times J \times X \times X \times U \rightarrow Y \) is a uniformly continuous function in t and there exist constants \(L_{6} > 0\) such that for all t, \(s \in J\), \(v_{1}, v_{2} \in X\), \(u_{1}, u_{2}\in U\) we have

   $$ \bigl\Vert g_{1}(t,s,v_{1},u_{1}) - g_{1}(t,s,v_{2},u_{2}) \bigr\Vert \leq L_{6} \bigl( \Vert v _{1} - v_{2} \Vert + \Vert u_{1} - u_{2} \Vert _{U}\bigr). $$

The mild solution of the system (4.1) takes the form

where

The constrained set of controls is considered to be a closed convex cone with empty interior and vertex at origin. Let \(U_{0} \subset U\) be the constrained set of controls and let the set of admissible controls be

Definition 4.1

The attainable set at time \(a>0\), denoted by \(K_{T}(U_{0})\), is defined as

where \(x(t,u)\) is a solution of (4.1).

Let us consider the Sobolev-type linear Hilfer fractional differential system

The mild solution of (4.3) is

Let us define the following operators:

\(\mathcal{B}: U \rightarrow C(J,X)\) by

\(\mathcal{F} : X \times U \rightarrow C(J,X)\) by

and

\(\mathcal{G} : X \times X \times U \rightarrow C(J,X)\) by

Let us put the following hypotheses:

1. (H13)

   The nonlinear function \(f_{1}\), \(g_{1}\) satisfies:

   $$\begin{aligned}& f_{1}\bigl(t,x(t),u(t)\bigr)|_{u=0}=0 , \qquad D_{x} f_{1}\bigl(t,x(t),u(t)\bigr)|_{u=0}=0 , \\& D_{u} f_{1}\bigl(t,x(t),u(t) \bigr)|_{u=0}=0 ,\qquad g_{1}\bigl(t,x(t),u(t)\bigr)|_{u=0}=0 , \\& D_{x} g_{1}\bigl(t,x(t),u(t)\bigr)|_{u=0}=0 \quad \mbox{and} \quad D_{u} g_{1}\bigl(t,x(t),u(t) \bigr)|_{u=0}=0 , \end{aligned}$$

   where \(D_{x}\) and \(D_{u}\) denote the Frechet derivative on space U.

2. (H14)

   \(\mathcal{B}\), \(\mathcal{F}\) and \(\mathcal{G}\) are continuously differentiable in U.

3. (H15)

   The linear control system (4.3) is \(U_{0}\)-exactly globally controllable on J.

Definition 4.2

The system (4.1) is said to be \(U_{0}\)-exactly locally controllable on J if the attainable set \(K_{T}(U_{0})\) contains a neighborhood of \(x(0) \in X\) in the space X.

Definition 4.3

The system (4.1) is said to be \(U_{0}\)-exactly globally controllable on J if \(K_{T}(U_{0})=X\).

The main result observes the application of the generalized open mapping theorem, so we recall it in the following lemma.

Lemma 4.1

([40])

Let X, Y be Banach spaces and \(F: B_{r} (x_{0}) \subset X \rightarrow Y\) such that

for some \(k>0\) and \(T \in \mathcal{L}(X,Y)\) with \(\operatorname{rank}(T)=Y\). Then \(B_{\rho }(Fx_{0})\subset FB_{r}(x _{0})\) for some \(\rho > 0\) provided that k is sufficiently small.

Theorem 4.1

Under the assumptions (H1)–(H4), (H8) and (H10)–(H15) the nonlinear control system (4.1) is \(U_{0}\)-exactly locally controllable on J.

Proof

Let us define an operator \(\mathcal{H} : U_{\mathrm{ad}} \rightarrow X \) by \(\mathcal{H} (u)=x(a,u)\), which maps control to the final state of the trajectory. Then the integral equation (4.2) implies

By hypothesis (H14), \(\mathcal{H}\) is differentiable in \(U_{\mathrm{ad}}\). Thus

We have

and

Then, by using hypothesis (H13) in (4.5), we get

By hypothesis (H15), the linear control system (4.3) is \(U_{0}\)-exactly globally controllable, therefore the map \(D_{u} \mathcal{H} (u)|_{u={{0}}}\), mapping \(v \mapsto y(a,v)\), is a surjective map with \(D_{u} \mathcal{H} ({{0}})(U_{\mathrm{ad}})=X\). Now, let \(u_{1}\), \(u_{2} \in U_{\mathrm{ad}}\) corresponding to \(x_{1}(t)=x(t,u_{1})\) and \(x_{2}(t)=x(t,u _{2})\), respectively. Then, for all \(t \in J\),

By Gronwall’s inequality,

Therefore

and

Now

Thus, by Lemma 4.1, the operator \(\mathcal{H}\) transforms a neighborhood of zero in \(U_{\mathrm{ad}}\) onto a neighborhood of \(\mathcal{H}(0)\) in the Banach space X. This proves the theorem. □

Remark 1

Controllability for a nonlinear fractional system was studied by many authors. However, to the best of our knowledge, there are no results on the controllability of nonlocal Hilfer fractional differential equations with the Clarke subdifferential.

Remark 2

Constrained controllability of for nonlinear fractional system was studied by few authors. However, to the best of our knowledge, there are no results on the constrained local controllability of nonlocal Hilfer fractional differential equations with the Clarke subdifferential.

Remark 3

The study may be improved by finding the sufficient conditions for controllability and constrained local controllability of a Sobolev-type nonlocal Hilfer fractional stochastic differential equation system with the Clarke subdifferential.

Example 5.1

Consider the following Sobolev-type nonlocal Hilfer fractional differential system with the Clarke subdifferential in Banach spaces:

where \(D^{\frac{1}{2},\frac{2}{3}}_{0+} \) is the Hilfer fractional derivative, \(\nu = \frac{1}{2}\), \(\mu =\frac{2}{3}\). Let \(X=Y=L^{2}(0, \pi )\) and define the operators \(A: D(A)\subset X \rightarrow Y\) and \(E: D(E)\subset X \rightarrow Y\) by \(Ax= -x_{yy}\), \(Ex= x- x_{yy}\) where \(D(A)\), \(D(E)\) is given by {\(x\in X: x,x_{y} \) are absolutely continuous and \(x_{yy}\in X\), \(x(0)=x(\pi )=0\)}. The functions \(x(t)(y)=x(t,y)\), \(Bu(t)(y)=Bu(t,y)\), \(f(t,x(t))(y)=\frac{1}{30} \cos (x(t,y))\), \(g(t,s,x(s),\int _{0}^{s}{H(s,\tau ,x(\tau ))\,d\tau })(y)= \frac{1}{s^{2}+9} + \frac{1}{9} \int _{0}^{s} \frac{1}{(2+\tau )^{2}}\,d\tau \), \(H(s,\tau ,x(\tau ))= \frac{1}{9} \frac{1}{(2+\tau )^{2}}\), \(\partial Z(t,x(t))(y)=\partial Z(t,x(t,y))\) and \(q(x)(y)=\sum_{i=1} ^{m}c_{i} x(t_{i},y)\).

It is easy to verify that the function f satisfies hypothesis (H6) with \(N_{1}=N_{2}=\frac{1}{30}\).

Then A and E can be written as

where \(x_{n}(y)=\sqrt{\frac{2}{\pi }}\sin ny \), \(n=1,2,3,\ldots\) , is the orthogonal set of eigenvectors of A and \((x,x_{n})\) is the \(L^{2}\) inner product. Moreover, for \(x \in X\), we get

It is well known that A generates a compact semigroup \(\{ T(t), t>0 \}\) in X and

with

Moreover, the two operators \(P_{\frac{2}{3}}(t)\) and \(S_{\frac{1}{2}, \frac{2}{3}}(t)\) satisfy

We note that \(L_{2}=\frac{1}{9}\), \(L_{4}=\frac{1}{36}\) and choose other constants such that all hypotheses (H1)–(H9) are satisfied and

Hence, all the hypotheses of Theorem 3.1 are satisfied and the system (5.1) is controllable on \(J= (0,1]\).

Example 5.2

Consider the following Sobolev-type nonlocal Hilfer fractional differential system with the Clarke subdifferential in Banach spaces:

where \(D^{\frac{1}{3},\frac{3}{4}}_{0+} \) is the Hilfer fractional derivative, \(\nu = \frac{1}{3}\), \(\mu =\frac{3}{4}\). Let \(X=Y=L^{2}(0, \pi )\) and define the operators \(A: D(A)\subset X \rightarrow Y\) and \(E: D(E)\subset X \rightarrow Y\) by \(Ax= -x_{\varsigma \varsigma }\), \(Ex= x- x_{\varsigma \varsigma }\) where \(D(A)\), \(D(E)\) is given by {\(x\in X: x,x_{\varsigma }\) are absolutely continuous and \(x_{\varsigma \varsigma }\in X\), \(x(0)=x(\pi )=0\)}. The functions \(x(t)(\varsigma )=x(t, \varsigma )\), \(Bu(t)(\varsigma )=Bu(t,\varsigma ) \partial Z(t,x(t))( \varsigma )=\partial Z(t,x(t,\varsigma ))\), \(q(x)(\varsigma )=\sum_{i=1}^{m}c_{i} x(t_{i},\varsigma )\), \(f_{1}(t,x(t),u(t))(\varsigma )= G_{1}(t,x(t,\varsigma ),u(t,\varsigma )) \), \(g_{1}(t,s,x(s),\int _{0}^{s}{H(s,\tau ,x(\tau ))\,d\tau }, u(s))(\varsigma )= G_{2}(t,s,x(s, \varsigma ),\int _{0}^{s}{G_{3}(s,\tau ,x(\tau ,\varsigma ))\,d\tau },u(t, \varsigma ))\).

Then A and E can be written as

where \(x_{n}(\varsigma )=\sqrt{\frac{2}{\pi }}\sin n\varsigma \), \(n=1,2,3,\ldots\) , is the orthogonal set of eigenvectors of A and \((x,x_{n})\) is the \(L^{2}\) inner product. Moreover, for \(x \in X\), we get

It is well known that A generates a compact semigroup \(\{ T(t), t>0 \}\) in X and

with

Moreover, the two operators \(P_{\frac{3}{4}}(t)\) and \(S_{\frac{1}{3}, \frac{3}{4}}(t)\) satisfy

Take \(L^{2}[0,\pi ]\) as the control space and \(U_{0}=\{ u(t) \in U: u(t, \varsigma ) \geq 0 \}\). The space of admissible controls is \(U_{\mathrm{ad}}= L^{2}(J;U_{0}) \subset V= L^{2} (J;U)\) and the attainable set is

The associated linear control system of the nonlocal Hilfer fractional differential system with the Clarke subdifferential (5.2) takes the form

with mild solution in the form

We can prove that all the hypotheses (H10)–(H15) are satisfied. Hence, Theorem 4.1 is satisfied and the nonlinear control system (5.2) is \(U_{0}\)-exactly locally controllable on \(J = (0,1]\).

In this paper, by using fractional calculus and the Sadovskii fixed point theorem, we studied the sufficient conditions for controllability of Sobolev-type nonlocal Hilfer fractional differential systems with Clarke’s subdifferential. In addition, we established the constrained local controllability for Sobolev-type nonlocal Hilfer fractional differential systems with Clarke’s subdifferential. Also, we provided two examples to illustrate our results. In the future we aim to study the existence of mild solution for a class of noninstantaneous and nonlocal impulsive Hilfer fractional stochastic integrodifferential equations with fractional Brownian motion and Poisson jumps.

We would like to thank the referees and the editor for their important comments and suggestions, which have significantly improved the paper.

Not applicable.

Affiliations

1. Higher Institute of Engineering, El-Shorouk Academy, El-Shorouk City, Egypt
   • Hamdy M. Ahmed
2. Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt
   • Mahmoud M. El-Borai
3. Department of Mathematics, Faculty of Science, Al-Azhar University, Cairo, Egypt
   • A. S. Okb El Bab
   •  & M. Elsaid Ramadan
4. Department of Mathematics, Faculty of Science, Islamic University in Madinah, Medina, Saudi Arabia
   • M. Elsaid Ramadan

Contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the manuscript.

Corresponding author

Correspondence to Hamdy M. Ahmed.

Competing interests

The authors declare that they have no competing interests.

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