Study on the controllability of delayed evolution inclusions involving fractional derivatives

1.   Introduction

Fractional Cauchy problems are useful in physics to model anomalous diffusion ^[1]. Nigmatullin ^[2] introduced a generalization diffusion equation of the non-Markovian type

where $\partial^{\alpha}_t$ is the $\alpha$-order fractional derivative operator on $t$. The case $\alpha = 0$ corresponds to an elliptic equation. The case $\alpha = 1$ corresponds to a parabolic equation. The case $\alpha = 2$ corresponds to a hyperbolic equation.

Let $\Upsilon\subset \mathbb{R}^N$ be a bounded domain and $\partial\Upsilon$, the boundary of $\Upsilon$, be sufficiently smooth. We first demonstrate the linear IBVP

where $^C\partial^{\alpha}_t$ is the $\alpha$-order fractional partial derivative operator on $t$ in the Caputo sense, $\Delta$ denotes the Laplace operator, $g: (0, \infty)\times\Upsilon\rightarrow \mathbb{R}$ is a given linear function, $\omega_0(\varrho), \omega_1(\varrho)\in \mathbb{R}$ for all $\varrho\in \Upsilon$. Equation (1.1) has wide applications in viscoelastic models. When $\alpha\rightarrow1$, it is the heat equation. When $\alpha\rightarrow2$, it becomes the standard wave equation. For $\alpha \in (1, 2)$, Eq (1.1) is called the super-diffusive equation. Hence, Eq (1.1) has significant backgrounds in physical applications.

Let $A = \Delta$ and

Then the IBVP(1.1) can be rewritten as an abstract IVP

where $^CD^{\alpha}_{0^+}$ represents the Caputo fractional derivative operator. Let $\mathbb{E} = L^2(\Upsilon)$. It follows from Arendt et al. ^[3] that $A: \mathcal{D}(A)\subseteq \mathbb{E}\rightarrow \mathbb{E}$ is densely defined and closed in $\mathbb{E}$ generating a strongly continuous cosine family $\{G(t): t\geq0\}$ which is uniformly bounded. When $g: [0, \infty)\rightarrow \mathbb{E}$ is continuous and $\omega_0, \omega_1\in \mathbb{E}$, Zhou et al. ^[4] presented a new concept of mild solutions of the IVP(1.2) by employing the Laplace transform. Exact controllability results were proved on the interval $[0, c]$(where $c > 0$ is a constant) for the IVP of the semilinear fractional evolution equation (without delays) in ^[4] when the nonlinear term $f$ is global Lipschitz continuous or satisfies certain compact conditions.

It is well known that the delay appears in wide fields, such as biology, physics, economics, etc. It is meaningful to study the fractional evolution equation with time delay. Let $\mathbb{E}$ be a general Banach space. In 2023, Yang in ^[5] investigated the fractional delayed control system in $\mathbb{E}$

where $^LD^{\alpha}_t$ is the fractional derivative operator on $t$ of order $\alpha$ in the Riemann-Liouville sense, $\bar{\omega}(t) = t^{2-\alpha}\omega(t)$ for $t\in (0, c]$ with $\bar{\omega}(0) = \lim\limits_{t\rightarrow0^+}\bar{\omega}(t)$ and $\bar{\omega}_t(\theta) = \bar{\omega}(t+\theta)$ for $t\in [0, c]$ and $\theta\in [-r, 0], r > 0$, $v\in L^2([0, c], \mathbb{V})$, $\mathbb{V}$ is another Banach space, $P\in \mathcal{L}(\mathbb{V}, \mathbb{E})$ (here $\mathcal{L}(\mathbb{V}, \mathbb{E}): = \{P: \mathbb{V}\rightarrow \mathbb{E}\ \text{is a bounded operator}\}$, $\mathcal{L}(\mathbb{E}): = \mathcal{L}(\mathbb{E}, \mathbb{E})$), $\varphi\in C([-r, 0], \mathbb{E})$ with $\varphi(0)\in \mathcal{D}(A)$ and $\omega_1\in \mathbb{E}$. $\mathcal{H}_{\varrho}(t) = \frac{t^{\varrho-1}}{\Gamma(\varrho)}$ for $t, \varrho > 0$. The symbol $*$ represents the convolution. When the sine family $\{W(t): t\geq0\}$, corresponding to the cosine family $\{G(t): t\geq0\}$ generated by $A$, is compact for $t > 0$, the approximate controllability results were obtained.

In ^[6], Gou et al. studied the existence and approximate controllability of the Hilfer fractional evolution equations when the cosine family $\{G(t): t\geq0\}$, generated by the linear operator $A$, is exponentially bounded and continuous in the uniform operator topology for every $t > 0$. In ^[7], He et al. investigated the approximate controllability for a class of fractional stochastic wave equations when the sine family $\{W(t): t\geq 0\}$, corresponding to the cosine family $\{G(t): t\geq0\}$ generated by $A$, is compact for every $t > 0$. However, if we remove the global Lipschitz condition on $f$ or the compactness assumption on $\{W(t): t\geq 0\}$, how is the controllability of the fractional delayed evolution inclusions?

In the present work, we research the infinite controllability of fractional delayed evolution inclusions (FDEIs)

where $F$ is a multi-valued mapping with compact values in $\mathbb{E}$, $v\in L^2_{loc}([0, \infty), \mathbb{V})$, $\mathbb{V}$ is another Banach space, $P\in \mathcal{L}(\mathbb{V}, \mathbb{E})$, $\phi\in C([-r, 0], \mathbb{E})$ and $\omega_1\in \mathbb{E}$. For any $t\geq0$, $\omega_t(\xi) = \omega(t+\xi)$ for $\xi\in [-r, 0]$.

The infinite controllability of evolution systems is demonstrated extensively. In ^[8], Benchohra et al. proved the controllability of first-order evolution equations in a semi-infinite time horizon. The infinite controllability of differential and integrodifferential inclusions had been discussed in ^[9,10]. However, there are research articles on the infinite controllability of the fractional delayed evolution equations. Particularly, for $\alpha\in (1, 2)$, the infinite controllability of the $\alpha$-order Caputo fractional delayed evolution inclusions has not been studied.

In this paper, using a nonlinear alternative established by Frigon in ^[11], the infinite controllability of the FDEIs (1.3) is investigated in Fr$\acute{e}$chet spaces. The major results and features of our work are summarized below:

(1) The infinite controllability of the FDEIs (1.3) of order $\alpha\in (1, 2)$ in Fr$\acute{e}$chet spaces is demonstrated. It is new and novel even if $F$ is a single-valued mapping.

(2) Our result is obtained without the global Lipschitz continuity of the nonlinear term $F$. Hence, it greatly generalizes Theorem 4.1 of ^[4].

(3) We remove the assumption of compactness on the cosine family and the sine family, which are essential assumptions in ^[5,6,7]. Hence, Our work is an improvement of ^[5,6,7].

2.   Preliminaries

Let $(\mathbb{E}, \|\cdot\|)$ and $(\mathbb{V}, \|\cdot\|)$ be Banach spaces. $C([-r, 0], \mathbb{E})$ is a Banach space of $\mathbb{E}$-valued continuous functions with $\|\phi\|_C = \sup\limits_{t\in [-r, 0]}\|\phi(t)\|$.

We first introduce symbols:

$\mathcal{K}(\mathbb{E})$ is the set of nonempty subsets of $\mathbb{E}$.

$\mathcal{K}_{cl}(\mathbb{E}) = \{Y\in \mathcal{K}(\mathbb{E}): Y$ is closed $\}$,

$\mathcal{K}_{cp}(\mathbb{E}) = \{Y\in \mathcal{K}(\mathbb{E}): Y$ is compact $\}$,

$\mathcal{K}_{cl, b}(\mathbb{E}) = \{Y\in \mathcal{K}(\mathbb{E}): Y$ is closed and bounded $\}$.

For $\varrho > 0$, denote by

where $\Gamma(\varrho) = \int_0^{\infty}e^{-\theta}\theta^{\varrho-1}d\theta$. For $n\geq1$, it follows from ^[12,13] that the Caputo fractional derivative is given by

where $^LD_{0^+}^{\alpha}x(t) = \frac{d^n}{dt^n}(g_{n-\alpha}\ast x)(t)$ and $\ast$ means the convolution.

Definition 2.1. ^[14] $\{G(t): t\in \mathbb{R}\}\subset\mathcal{L}(E)$ is called a strongly continuous cosine family, if

$(i)$ $G(0) = I$;

$(ii)$ for any $\sigma_1, \sigma_2\in \mathbb{R}$, $G(\sigma_1+\sigma_2)+G(\sigma_1-\sigma_2) = 2G(\sigma_1)G(\sigma_2)$;

$(iii)$ for each $\omega\in \mathbb{E}$, $G(\cdot)\omega$ is continuous.

Define

Then, $\{W(t): t\in \mathbb{R}\}$ is called the sine family corresponding to $\{G(t): t\in \mathbb{R}\}$. Let

and, for any $\omega\in \mathcal{D}(A)$,

Then, $A$ is called the infinitesimal generator of the cosine family $\{G(t): t\in \mathbb{R}\}$. Obviously, $A$ is a linear operator and it is closed and densely defined in $\mathbb{E}$.

$(H1)$ $A$ generates a strongly continuous cosine family $\{G(t): t\geq 0\}$ in $\mathbb{E}$, and $\exists M\geq1$ such that, for any $t\geq0$,

Let

where $\mathbb{C}$ is the imaginary line. Then $\mathcal{M}_{\tau}(\theta)$ is called Mainardi's Wright-type function.

Lemma 2.1. ^[4,15,16] The function $\mathcal{M}_{\tau}(\theta)$ has the properties:

$(i)$ $\mathcal{M}_{\tau}(\theta)\geq0, \ \ \ \forall \theta > 0$;

$(ii)$ $\int_0^{\infty}\theta^{\delta}\mathcal{M}_{\tau}(\theta)d\theta = \frac{\Gamma(1+\delta)}{\Gamma(1+\tau\delta)}, \ \ \ \forall \delta\in (-1, \infty)$.

Let $\beta = \frac{\alpha}{2}$. Then $\beta\in (\frac{1}{2}, 1)$. By ^[4] we can get the following concept.

Definition 2.2. ^[4] $\omega\in C([-r, \infty], \mathbb{E})$ is called the mild solution of the FDEIs (1.3) if

$(i)$ $\omega'(0) = \omega_1\in \mathbb{E}$ and, for all $t\in [-r, 0]$, $\omega(t) = \phi(t)$;

$(ii)$ There exists $g\in L^{1}_{loc}([0, \infty), \mathbb{E})$ with $g(t)\in F(t, \omega_t), \ a.e.\ t\geq0$ satisfying

where

Lemma 2.2. ^[4] Let $(H1)$ hold. For $\forall \omega\in \mathbb{E}$, the families $\{G_{\beta}(t): t\geq0\}, \{\mathcal{S}_{\beta}(t): t\geq0\}$ and $\{\mathcal{T}_{\beta}(t): t\geq0\}$ satisfy

and

For any $B_1, B_2\in \mathcal{K}(\mathbb{E})$, the Hausdorff pseudometric $D_{\nu}, \nu\in \Lambda$ induced by $d_{\nu}$ is defined by

with $\inf\emptyset = \infty$. Particularly, when $\mathbb{E}$ is locally-convex and complete, the set $B_1\subset \mathbb{E}$ is bounded if $D_{\nu}(\{0\}, B_1) < \infty$ for every $\nu\in \Lambda$.

Definition 2.3. ^[11] Let $F: \mathbb{E}\rightarrow \mathcal{K}(\mathbb{E})$ be a multi-valued map. $F$ is named as an admissible contraction with constants $\{\kappa_{\nu}\}_{\nu\in \Lambda}$ if for $\forall \nu\in \Lambda$, $\exists \kappa_{\nu}\in (0, 1)$ satisfying

$(i)$ for $\forall x, z\in \mathbb{E}$, $D_{\nu}(F(x), F(z))\leq \kappa_{\nu}d_{\nu}(x, z)$;

$(ii)$ for $\forall \epsilon > 0, x\in \mathbb{E}$, $\exists z\in F(x)$ satisfying

Lemma 2.3. ^[11] Let $D\subset \mathbb{E}$ be an open neighborhood of the origin in the Fr$\acute{e}$chet space $\mathbb{E}$. If $Q: \overline{D}\rightarrow \mathcal{K}(\mathbb{E})$ is an admissible contraction and $Q$ is bounded, then either

$(i)$ $Q$ admits fixed points,

or

$(ii)$ there is $\mu\in [0, 1)$ satisfying $\omega\in \mu Q\omega$ for $\forall \omega\in \partial D$.

3.   Controllability of the FDEIs (1.3)

Definition 3.1. ^[10] For each $b > 0$ and any $\bar{x}\in \mathbb{E}$, if there is $v\in L^2([0, b], \mathbb{V})$ such that the mild solution $\omega$ of the FDEIs (1.3) corresponding to this $v$ satisfies $\omega(b) = \bar{x}$, then we say that the FDEIs (1.3) is infinitely controllable on $(0, \infty)$.

We first define $H_d: \mathcal{K}(\mathbb{E})\times\mathcal{K}(\mathbb{E})\rightarrow \mathbb{R}^+\cup\{\infty\}$ by

where $d(\mathbb{A}, \varsigma) = \inf\limits_{\varpi\in \mathbb{A}}d(\varpi, \varsigma)$ and $d(\varpi, \mathbb{B}) = \inf\limits_{\varsigma\in \mathbb{B}}d(\varpi, \varsigma)$. Then $(\mathcal{K}_{cl, b}(\mathbb{E}), H_d)$ is a metric space. To prove our controllability result, we further make the assumptions:

$(H2)$ $F: [0, \infty)\times C([-r, 0], \mathbb{E})\rightarrow \mathcal{K}_{cp}(\mathbb{E})$ satisfies

(ⅰ) $F(\cdot, \omega)$ is strongly measurable for each $\omega\in C([-r, 0], \mathbb{E})$;

(ⅱ) $F(t, \cdot)$ is continuous for $a.e.\ t\in [0, \infty)$;

(ⅲ) for every $\omega\in C([-r, 0], \mathbb{E})$, there is $p\in L^2_{loc}([0, \infty), \mathbb{R}^+)$ satisfying, for $a.e.\ t\geq0$,

$(H3)$ For any $R > 0$, there is $\rho\in L^1_{loc}([-r, \infty), \mathbb{R}^+)$ satisfying

for any $\omega_1, \omega_2\in C([-r, 0], \mathbb{E})$ with $\|\omega_1\|_C, \|\omega_2\|_C\leq R$.

For each $b > 0$, define $\Phi: L^2([0, b], \mathbb{V})\rightarrow \mathbb{E}$ by

$(H4)$ $\Phi$ is a linear operator and satisfies

(ⅰ) $\Phi^{-1}$ exists and takes values in $L^2([0, b], \mathbb{V})\setminus Ker(\Phi)$;

(ⅱ) $\exists M_1, M_2 > 0$ s.t.

For each $b > 0$, we introduce a semi-norm in $C([-r, \infty), \mathbb{E})$ as

where $L_b(t) = \int_0^tl_b(s)ds$ and

In the following we always choose $\sigma > 2$ large enough. Then $C([-r, \infty), \mathbb{E})$ is a Fr$\acute{e}$chet space with $\|\cdot\|_b$.

Theorem 3.1. Let $(H1)-(H4)$ be satisfied. Then the FDEIs (1.3) is infinitely controllable on $(0, \infty)$.

Proof. For each $b > 0$ and any $\omega\in C([-r, \infty), \mathbb{E})$, let $S_{F, \omega}^b: = \{g\in L^1([0, b], \mathbb{E}): g(t)\in F(t, \omega_t)\ a.e.\ t\in [0, b]\}$. By $(H4)$, we choose

where $g\in S_{F, \omega}^b$.

If $\omega\in C([-r, \infty), \mathbb{E})$ is a mild solution of the FDEIs (1.3) corresponding to the control $v_\omega^b$, by Definition 2.2, we easily check that $\omega(b) = \bar{x}$. Hence the FDEIs (1.3) is infinitely controllable on $(0, \infty)$ owing to Definition 3.1.

In the following we prove that the FDEIs (1.3) admits a mild solution in $C([-r, \infty), \mathbb{E})$. First, we define $h$ in $C([-r, \infty), \mathbb{E})$ by

where $g\in S_{F, \omega}: = \{g\in L_{loc}^1([0, \infty), \mathbb{E}): g(t)\in F(t, \omega_t)\ a.e.\ t\geq0\}$. Define $Q$ by

Then $Q: C([-r, \infty), \mathbb{E})\rightarrow \mathcal{K}(C([-r, \infty), \mathbb{E}))$. Next, we are going to prove that $Q$ has fixed points.

For each $b > 0$, denote by $\mathcal{C}: = C([-r, b], \mathbb{E})$. If $\omega\in \mathcal{C}$ is a possible mild solution of (1.3) on $[-r, b]$, that is, $\omega\in Q\omega$. Then there is $g\in S_{F, \omega}^b$ satisfying, for any $t\in [0, b]$,

By (3.1), we can infer that

By choosing $K = \max\{\|\phi\|_C, M^*\}$, we obtain that

Then $Q$ is bounded.

Let $R = K+1$ be the constant in $(H3)$. Denote by

Clearly, $D\subset \mathcal{C}$ is an open neighborhood of zero. Next, we will show that $Q: \overline{D}\rightarrow \mathcal{K}(\mathcal{C})$ is an admissible contraction.

Let $\omega, z\in \overline{D}$ and $h\in Q\omega$. There is $g\in S_{F, \omega}^b$ satisfying

It follows from $(H3)$ that

Thus, there exists $\gamma\in F(t, z_t)$ satisfying

Let

Since the Proposition Ⅲ.4 of ^[8] yields that the set $\Omega(t) = D_*(t)\cap F(t, z_t)$ is measurable, it follows that there is $\bar{g}\in \Omega(t), t\in [0, b]$. Thus, $\bar{g}\in F(t, z_t)$ for all $t\in [0, b]$ and

Define

Then $\bar{h}\in Qz$ and

Therefore,

Analogously, by interchanging $\omega$ and $z$, we have

On the other hand, for any $\omega\in \mathcal{C}$, since $F(t, \omega_t)$ has compact values, by the definition of $Q$, we get $Q\omega\subset \mathcal{K}_{cp}(\mathcal{C})$. Hence there exists $\omega^*\in \mathcal{C}$ satisfying $\omega^*\in Q\omega^*$.

Let $\hat{\omega}\in \overline{D}$ and $h\in Q\hat{\omega}$. For every $\epsilon > 0$, if $\omega^*\in Q\hat{\omega}$, we have

Since $h\in Q\hat{\omega}$ is arbitrary, let $h\in \{h\in \mathcal{C}: \|h-\omega^*\|_b\leq\epsilon\}$. We can achieve that

If $\omega^*\not\in Q\hat{\omega}$, then $\|\omega^*-Q\hat{\omega}\|_b\neq0$. By means of the compactness of $Q\hat{\omega}$, there is $\omega^{**}\in Q\hat{\omega}$ satisfying

Then,

Consequently, we have

Thus, we deduce that $Q: \overline{D}\rightarrow \mathcal{K}(\mathcal{C})$ is an admissible contraction. In view of the definition of $D$, there is no $\omega\in \partial D$ satisfying $\omega\in \mu Q\omega$ for all $\mu\in (0, 1)$. Then according to Lemma 2.3, $Q$ has at least one fixed point $\omega\in \mathcal{C}$ such that for each $b > 0$, $\omega(b) = \bar{x}$. Consequently, the FDEIs (1.3) is infinitely controllable on $(0, \infty)$. □

4.   Applications

Example 4.1. Let $\Upsilon\subset \mathbb{R}^{N}$ be bounded and $\partial \Upsilon$ sufficiently smooth. We study the fractional partial differential inclusions

where $^C\partial_{t}^{\frac{3}{2}}$ represents the $\frac{3}{2}$-order Caputo fractional partial derivative operator, $P: L^2(\Upsilon)\rightarrow L^2(\Upsilon)$ is linear and bounded, $v\in L^2_{loc}([0, \infty), L^2(\Upsilon))$.

Let $\mathbb{E} = \mathbb{V} = L^{2}(\Upsilon)$ and let

It follows from Section 7.2 of ^[3] that $A$ generates a strongly continuous cosine family $\{G(t): t\geq0\}$ of uniformly bounded linear operators in $\mathbb{E}$ and, for all $t\geq0$, $\|G(t)\|\leq 1$. Then $(H1)$ is achieved.

$A$ has eigenvalues $\lambda_m = m^2\pi^2, m\in \mathbb{N}$. $\vartheta_m(t) = \sqrt{\frac{2}{\pi}}\sin(m\pi t), m\in \mathbb{N}$ are corresponding eigenvectors. Then

and $\lambda_m\rightarrow \infty$ as $m\rightarrow \infty$. Hence

and

where $\langle\cdot, \cdot\rangle$ is the inner product in $\mathbb{E}$.

For any $v\in L^2_{loc}([0, \infty), \mathbb{V})$, we define $P$ on $\mathbb{V}$ by

where $a > 0$ and

Then $P: \mathbb{V}\rightarrow \mathbb{E}$. Since for any $v\in L^2_{loc}([0, \infty), \mathbb{V})$, we have

It follows that

Then there exists $M_1 > 0$ satisfying $\|P\|\leq M_1$.

For every $b > 0$, we define $\Phi: L^2([0, b], \mathbb{V})\rightarrow \mathbb{E}$ by

Let $\delta = E_{\frac{3}{2}, 1}(-\frac{1}{10})$. According to ^[4] and ^[17], for any $m\in \mathbb{N}$, we have

Then the operator $\Phi$ is surjective. Thus, for any $\omega\in \mathbb{E}$, $\Phi^{-1}: \mathbb{E}\rightarrow L^2([0, b], \mathbb{V})\setminus Ker(\Phi)$ exists and expresses by

which implies

Consequently, we have

Hence, the condition $(H4)$ holds.

Theorem 4.1. Let $\mathcal{G}: [0, \infty)\times\Upsilon\times C([-r, \infty)\times\Upsilon, \mathbb{R})\rightarrow \mathcal{K}_{cp}(L^2(\Upsilon))$ satisfies the following condition

$(F1)$ $(i)$ For each $\omega\in C([-r, \infty)\times\Upsilon, \mathbb{R})$ and for any $\varrho\in \Upsilon$, $\mathcal{G}(\cdot, \varrho, \omega)$ is strongly measurable;

$(ii)$ For $a.e.\ t\in [0, \infty)$ and for any $\varrho\in \Upsilon$, $\mathcal{G}(t, \varrho, \cdot)$ is continuous;

$(iii)$ There exists $p_1\in L^2_{loc}([0, \infty), \mathbb{R}^+)$ satisfying

$(iv)$ For any $R > 0$, there is $p_2\in L^1_{loc}([-r, \infty), \mathbb{R}^+)$ satisfying, for $a.e.\ t\in [0, \infty)$,

for any $\omega_1, \omega_2\in \mathbb{R}$ with $|\omega_1|, |\omega_2|\leq R$.

Then the inclusion (4.1) is infinitely controllable on $(0, \infty)$.

Proof. Let

and

Then, (4.1) can be rewritten as the abstract FDEIs (1.3). From the above arguments, we know that $(H1)-(H4)$ are fulfilled. Hence, by Theorem 3.1, the differential inclusion (4.1) is infinitely controllable on $(0, \infty)$. □

5.   Conclusions

In the present work, by applying a nonlinear alternative of multivalued mapping which is established by Frigon ^[11], the infinite controllability theorem of the FDEIs (1.3) is established in Fr$\acute{e}$chet spaces. All compactness conditions are removed in our theorem. The obtained result is new even if the nonlinear function $F$ is a single-valued mapping. The scheme established in this article is also useful for investigating the infinite controllability of the Riemann-Liouville (or the Hilfer) fractional evolution equations. In the future, we will further study the optimal control problems of the FDEIs (1.3).

Use of AI tools declaration

The author declares that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The research is supported by the NSF of Gansu (No. 22JR5RA875).

Conflict of interest

The author declares that they have no conflicts of interest.