# Optimal Control for a Class of Riemann-Liouville Fractional Evolution Inclusions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- (i)
- ${I}_{{0}^{+}}^{1-\alpha}x\left(0\right)+g\left(x\right)={x}_{0}$.
- (ii)
- there exists $f\left(t\right)\in F(t,x(t\left)\right)$ such that

**Lemma**

**1**

- $\left(i\right)$
- for every $t\ge 0$ and any $x\in X$,$$\parallel {T}_{\alpha}\left(t\right)x\parallel \le \frac{M}{\Gamma \left(\alpha \right)}\parallel x\parallel .$$
- $\left(ii\right)$
- for $t\ge 0$, the operator ${T}_{\alpha}\left(t\right)$ is strongly continuous.
- $\left(iii\right)$
- for $t>0$, if the operator $T\left(t\right)$ is compact, ${T}_{\alpha}\left(t\right)$ is a compact operator.

**Definition**

**2**

**.**Let X and Z be two topological spaces, and $F:X\to P\left(Z\right)$ be a multi-valued mapping.

- (1)
- If $F\left(x\right)$ is convex (closed) in Z for all $x\in X$, then F is said to be convex (closed)-valued.
- (2)
- If $F\left(D\right)$ is relatively compact for every bounded subset D of X, then F is said to be completely continuous.
- (3)
- If ${F}^{-1}\left(V\right)=\{x\in X\mid F\left(x\right)\subseteq V\}$ is an open subset of X for every open subset V of Z, then F is said to be upper semi-continuous(u.s.c.) on X.
- (4)
- If the graph ${G}_{F}=\{(x,y)\in X\times Z\mid y\in F\left(x\right)\}$ is a closed subset of $X\times Z$, then F is said to be closed.
- (5)
- If there is an element $x\in X$ satisfying $x\in F\left(x\right)$, then F is said to have a fixed point in X.

**Lemma**

**2**

**.**If the multi-valued mapping F is completely continuous with nonempty compact values, then F is u.s.c. if, and only if F has a closed graph.

**Lemma**

**3**

**Definition**

**3**

**.**A sequence ${\left\{{f}_{n}\right\}}_{n\ge 1}\subset {L}^{1}(J,X)$ is said to be semi-compact if

- $\left(i\right)$
- there exists a function $\omega \in {L}^{1}(J,{\mathbb{R}}^{+})$ such that$$\parallel {f}_{n}\left(t\right)\parallel \le \omega \left(t\right),\phantom{\rule{3.33333pt}{0ex}}a.e.\phantom{\rule{3.33333pt}{0ex}}t\in J;$$
- $\left(ii\right)$
- for a.e. $t\in J$, the set $\{{f}_{n}\left(t\right)\mid n\in \mathbb{N}\}$ is relatively compact in X.

**Lemma**

**4**

**.**If a sequence in ${L}^{1}(J,X)$ is semi-compact, then it is weakly compact in ${L}^{1}(J,X)$.

**Lemma**

**5**

**.**Let X be a Banach space and D be a compact subset of X. Then $\overline{conv}\left(D\right)$ is compact, where $\overline{conv}\left(D\right)$ denotes the convex closure of D.

**Lemma**

**7**

**.**Let $0<a\le b$ and $\theta \in (0,1]$, where we have

**Lemma**

**8**

**.**Let W be a nonempty closed, convex and bounded subset in the Banach space X, and $\Psi :W\to {2}^{W}\backslash \{\varnothing \}$ be a u.s.c. condensing multi-valued mapping. If for every $x\in W,\phantom{\rule{3.33333pt}{0ex}}\Psi \left(x\right)$ is convex and closed in W and $\Psi \left(W\right)\subseteq W,$ then Ψ has one fixed point in W.

**Lemma**

**9**

**.**Let W be a nonempty subset of X, which is convex, closed and bounded. Suppose that $\Psi :W\to {2}^{W}\backslash \{\varnothing \}$ is u.s.c. with convex and closed values, $\Psi \left(W\right)\subseteq W$ and $\Psi \left(W\right)$ is a compact set, then Ψ has one fixed-point in W.

## 3. Existence of Mild Solutions

- $\left(i\right)$
- for each $x\in X$, $F(t,x)$ is measurable to t and for every $t\in J$, $F(t,x)$ is u.s.c. to x. For every $x\in X$,$${S}_{F,x}=\{f\in {L}^{1}(J,X)\mid f\left(t\right)\in F(t,x),\phantom{\rule{3.33333pt}{0ex}}a.e.\phantom{\rule{3.33333pt}{0ex}}t\in J\}$$
- $\left(ii\right)$
- There exists a continuous nondecreasing function $\psi :[0,\infty )\to (0,\infty )$ satisfying $\Lambda :=\underset{r\to \infty}{lim}\frac{\psi \left(r\right)}{r}<+\infty $ and $m\in {L}^{p}(J,{\mathbb{R}}^{+})(p>\frac{1}{\alpha})$ such that$${\parallel F(t,x\left(t\right))\parallel :=sup\{\parallel f\left(t\right)\parallel \mid f\left(t\right)\in F(t,x\left(t\right)),\phantom{\rule{3.33333pt}{0ex}}t\in J\}\le m\left(t\right)\psi (\parallel x\parallel}_{{C}_{1-\alpha}}).$$

**Remark**

**1.**

**Lemma**

**10**

**.**Let the assumption $\left({H}_{1}\right)$ be fulfilled. Then for each $h\in {L}^{p}(J,X)$ with $p>\frac{1}{\alpha}$, the operator $\mathcal{B}:{L}^{p}(J,X)\to {C}_{1-\alpha}(J,X)$, given by

**Theorem**

**1.**

**Proof of**

**Theorem 1.**

**Remark**

**2.**

**Theorem**

**2.**

**Proof of**

**Theorem 2.**

**Corollary**

**1.**

**Corollary**

**2.**

## 4. Existence of Optimal Control

**Lemma**

**11.**

**Proof of**

**Lemma 11.**

**Lemma**

**12.**

**Theorem**

**3.**

**Proof of**

**Theorem 3.**

**Theorem**

**4.**

**Remark**

**4.**

**Remark**

**5.**

## 5. An Application

**Example**

**1.**

**Remark**

**6.**

**Remark**

**7.**

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Henderson, J.; Ouahab, A. Fractional functional differential inclusions with finite delay. Nonlinear Anal.
**2009**, 70, 2091–2105. [Google Scholar] [CrossRef] - Cernea, A. On a fractional integro-differential inclusion. Electron. J. Qual. Theory Differ. Equ.
**2014**, 25, 1–11. [Google Scholar] [CrossRef] - Ahmad, B.; Ntouyas, S.K. Fractional differential inclusions with fractional separated boundary conditions. Fract. Calc. Appl. Anal.
**2012**, 15, 362–382. [Google Scholar] [CrossRef] - Chaudhary, R.; Reich, S. Extremal mild solutions to fractional delay integro-differential equations with non-instantaneous impulses. Appl. Anal.
**2021**. [Google Scholar] [CrossRef] - Du, M.L.; Wang, Z.H. Initialized fractional differential equations with Riemann-Liouville fractional-order derivative. Eur. Phys. J. Spec. Top.
**2011**, 193, 49–60. [Google Scholar] [CrossRef] - Zhou, Y.; Zhang, L.; Shen, X.H. Existence of mild solutions for fractional evolution equations. J. Integral Equ. Appl.
**2013**, 25, 557–586. [Google Scholar] [CrossRef] - Pan, X.; Li, X.W.; Zhao, J. Solvability and optimal controls of semilinear Riemann-Liouville fractional differential equations. Abstr. Appl. Anal.
**2014**, 2014, 216919. [Google Scholar] [CrossRef] [PubMed] - Kumar, S. Mild solution and fractional optimal control of semilinear system with fixed delay. J. Optim. Theory Appl.
**2017**, 174, 108–121. [Google Scholar] [CrossRef] - Lian, T.T.; Fan, Z.B.; Li, G. time-optimal controls for fractional differential systems with Riemann-Liouville derivatives. Fract. Calc. Appl. Anal.
**2018**, 21, 1524–1541. [Google Scholar] [CrossRef] - Hu, S.; Papageorgiou, N.S. Handbook of Multivalued Analysis: Volume II: Applications; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2013. [Google Scholar]
- Wang, J.R.; Zhou, Y. Existence and controllability results for fractional semilinear differential inclusions. Nonlinear Anal.
**2011**, 12, 3642–3653. [Google Scholar] [CrossRef] - Kamenskii, M.; Obukhovskii, V.; Zecca, P. Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces; De Gruyter: Berlin, Germany, 2001. [Google Scholar]
- Alsarori, N.A.; Ghadle, K.P. On the mild solution for nonlocal impulsive fractional semilinear differential inclusion in Banach spaces. J. Math. Model.
**2018**, 6, 239–258. [Google Scholar] [CrossRef] - Chang, Y.K.; Pei, Y.T.; Ponce, R. Existence and optimal controls for fractional stochastic evolution equations of Sobolev type via fractional resolvent operators. J. Optim. Theory Appl.
**2019**, 182, 558–572. [Google Scholar] [CrossRef] - Deimling, K. Multivalued Differential Equations; De Gruyter: Berlin, Germany, 1992. [Google Scholar]
- Bohnenblust, H.F.; Karlin, S. On a Theorem of Ville, in: Contributions to the Theory of Games; Princeton University Press: Princeton, NJ, USA, 1950; pp. 155–160. [Google Scholar]
- Ye, H.P.; Gao, J.M.; Ding, Y.S. A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl.
**2007**, 328, 1075–1081. [Google Scholar] [CrossRef] [Green Version] - Balder, E.J. Necessary and sufficient conditions for L
_{1}-strong-weak lower semicontinuity of integral functional. Nonlinear Anal.**1987**, 11, 1399–1404. [Google Scholar] [CrossRef] [Green Version]

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Yang, H.; Ren, Q.
Optimal Control for a Class of Riemann-Liouville Fractional Evolution Inclusions. *Symmetry* **2022**, *14*, 248.
https://doi.org/10.3390/sym14020248

**AMA Style**

Yang H, Ren Q.
Optimal Control for a Class of Riemann-Liouville Fractional Evolution Inclusions. *Symmetry*. 2022; 14(2):248.
https://doi.org/10.3390/sym14020248

**Chicago/Turabian Style**

Yang, He, and Qian Ren.
2022. "Optimal Control for a Class of Riemann-Liouville Fractional Evolution Inclusions" *Symmetry* 14, no. 2: 248.
https://doi.org/10.3390/sym14020248