Lagrange optimal controls and time optimal controls for composite fractional relaxation systems
- Tingting Lian^{1, 2},
- Zhenbin Fan^{1}Email authorView ORCID ID profile and
- Gang Li^{1}
https://doi.org/10.1186/s13662-017-1299-7
© The Author(s) 2017
Received: 12 May 2017
Accepted: 29 July 2017
Published: 11 August 2017
Abstract
By means of the theory of resolvent and Schauder’s fixed point, the existence results of semilinear composite fractional relaxation systems are acquired. Then the new approach of setting up minimizing sequences twice is used to derive the optimal pairs without Lipschitz assumptions on nonlinear functions and nonlocal items. Moreover, the reflexivity of a state space X is not required by making full use of the compact method. Our results essentially improve and generalize those on optimal controls in recent literature.
Keywords
fractional composite relaxation systems resolvent operators optimal controls time optimal mild solutionMSC
49J15 47A10 34K371 Introduction
Fractional differential equations play a critical role in many fields, such as physics, engineering, chemistry, etc., in which it is used as a tool of modeling many phenomena. So, more and more researchers pay attention to it. We refer the readers to the monographs [1–3], the recent papers [4–10] and the reference therein. Our motivation for studying the fractional relaxation system comes from the existing work (see [11]) in which system (1.1) without control item and with A being a positive constant is discussed. Especially, when \(\gamma =1/2\), the system is the classical Basset problem, which is concerned with the unsteady accelerating in a viscous fluid for gravitational force. Numerical analysis of the fractional relaxation system is carried out in [12–15].
In recent decades, in spite of many research works on the optimal control problems governed by fractional differential systems in infinite dimension Banach spaces, it has been well recognized that most of the existing results on optimal controls are obtained under the following two conditions. One is that the mild solution of the corresponding system exists uniquely, and then the Lipschitz continuity of nonlinear functions and nonlocal items is required. The other is that both of the spaces X and Y are reflexive in time optimal problems. We refer the readers to the recent papers [12, 16–24] and the references therein.
It is our intention to deal with the solvability of system (1.1) by using the properties of resolvent developed by Fan [25]. Meanwhile, Lagrange optimal control problems and time optimal control problems governed by system (1.1) are studied. Two contributions are made here. One is that we remove the Lipschitz continuity of nonlinear function f and nonlocal item m without imposing any other conditions. Inspired by Zhu and Huang [26], we derive the optimal pairs by utilizing the new technique of setting up minimizing sequences twice. The other is that we make full use of the compactness of resolvent to compensate the lack of reflexivity of X, and the conclusion about time optimal controls also holds here. Hence, our results essentially improve the related results on this topic.
The present paper is built up as follows. The basic definitions and assumptions which will be used throughout the paper are presented in Section 2. We establish the solvability of system (1.1) in Section 3. Lagrange optimal control problems subjected to system (1.1) are investigated in Section 4. Time optimal control problems governed by (1.1) are presented in Section 5. We illustrate our results with an example in Section 6.
2 Preliminaries and basic assumptions
In this paper, let \(c>0\) be fixed and \(0<\gamma <1\). \(\mathbb{R}\) and \(\mathbb{R}^{+}\) are the sets of real numbers and nonnegative real numbers, respectively. The set of all continuous functions from \([0,c]\) to the Banach space X with \(\Vert y\Vert =\sup \{\Vert y(t)\Vert , t\in [0,c]\}\) is denoted by \(C([0,c],X)\), and the set of all Bochner integrable functions from \([0,c]\) to the Banach space X with \(\Vert f\Vert _{L^{p}} =(\int_{0}^{c}\Vert f(t)\Vert ^{p}\,\mathrm{d}t)^{1/p}\) is denoted by \(L^{p} ([0,c], X)\), where \(1\leq p< \infty \). Let \(L^{\infty } ([0,c], X)\) be the set of all essentially bounded functions on \([0,c]\) with values in X and \(\Vert f\Vert _{\infty }=\operatorname{esssup}\{\Vert f(t)\Vert , t \in [0,c]\}\), and let \(\mathcal{L}(X, Y)\) be the space of all linear and continuous operators from X to Y with the operator norm \(\Vert \cdot \Vert \). \(\mathcal{L}(X)\) respecting the space \(\mathcal{L}(X, X)\).
Definition 2.1
- (1)
\(\mathcal{R}_{\gamma }(\cdot)y\in C([0, \infty), X)\) for \(y\in X\), and \(\mathcal{R}_{\gamma }(0)=I\);
- (2)
\(\mathcal{R}_{\gamma }(t)D(A)\subset D(A)\), and for all \(y\in D(A)\) and \(t\geq 0\), there holds that \(A\mathcal{R}_{\gamma }(t)y= \mathcal{R}_{\gamma }(t)Ay\);
- (3)for all \(y\in D(A)\), \(t\geq 0\), the following resolvent equation holds:$$\mathcal{R}_{\gamma }(t)y=y+ \int_{0}^{t}g_{\gamma }(t-\tau)A \mathcal{R}_{\gamma }(\tau)y\,\mathrm{d}\tau. $$
By virtue of [27], we can infer that the resolvent equation is satisfied for all \(y\in X\). Now, the mild solution of system (1.1) will be given below using the definition of resolvent and Laplace transformation, and the details can be seen in [13].
Definition 2.2
- (HA)
The resolvent \(\{\mathcal{R}_{1-\gamma }(t)\}_{t>0}\) generated by A is compact and uniformly continuous, and set \(M=\sup_{t\in [0,c]}\Vert \mathcal{R}_{1-\gamma }(t)\Vert \).
- (Hf)
- (1)
\(f(\cdot, y)\) is measurable for all \(y\in X\), and \(f(t,\cdot)\) is continuous for a.e. \(t\in [0,c]\).
- (2)there exists a function \(\eta \in L^{1}([0,c], \mathbb{R}^{+})\) with \(\Vert \eta \Vert _{L^{1}([0,c])}<\frac{1}{M}\) such thatfor all \(y\in X\).$$\begin{aligned} \bigl\Vert f(t,y)-y\bigr\Vert \leq \eta (t) \bigl(1+\Vert y \Vert \bigr) \end{aligned}$$(2.1)
- (1)
- (Hm)
m is a continuous and compact operator on \(C([0,c], X)\) with \(\Vert m(y)\Vert \leq N\) for all \(y\in C([0,c],X)\) and some \(N>0\).
- (HB)
\(B\in L^{\infty }([0,c], \mathcal{L}(Y, X))\), and \(\Vert B\Vert _{ \infty }= M_{B}\).
The admissible control set is defined bywhere \(1< p<\infty \), and the multivalued map U satisfies the condition (HU).$$U_{ad}:=S_{U}^{p}=\bigl\{ u\in L^{p} \bigl([0,c], Y\bigr): u(t)\in U(t), \mbox{a.e. }t\in [0,c] \bigr\} , $$ - (HU)\(U:[0,c]\rightarrow P_{lv}(Y)\) (the set of all nonempty closed and convex subset of Y) satisfies:
- (1)
\(U(\cdot)\) is graph measurable;
- (2)
\(U([0,c])=\{\nu \in Y: \nu \in U(t), t\in [0,c]\}\subseteq \Lambda \) for some bounded subset of Λ of Y, that is, \(\Vert U([0,c])\Vert =\sup \{\nu \in U(t): t\in [0,c]\}\leq \tilde{M}\) for some \(\tilde{M}>0\).
- (1)
In view of [28], we can infer that (HU) implies that \(U_{ad}\neq \emptyset \) and, clearly, \(U_{ad}\) is bounded, closed and convex, and \(Bu\in L^{p}([0,c], X)\).
Remark 2.3
Let \(0<\gamma <1\). In view of Fan [25], Lemma 3.8, an analytic resolvent of analyticity type \((\omega_{0},\theta_{0})\) is uniformly continuous for all \(t>0\).
The following property of resolvent plays an important role in this paper.
Lemma 2.4
[25]
- (1)
\(\lim_{h\rightarrow 0^{+}}\Vert \mathcal{R}_{1-\gamma }(t+h)-\mathcal{R}_{1-\gamma }(t)\mathcal{R}_{1-\gamma }(h)\Vert =0\);
- (2)
\(\lim_{h\rightarrow 0^{+}}\Vert \mathcal{R}_{1-\gamma }(t)-\mathcal{R}_{1-\gamma }(h)\mathcal{R}_{1-\gamma }(t-h)\Vert =0\).
3 The solvability of system (1.1)
In this section, by making use of the compactness and uniform continuity of resolvent, the mild solutions of system (1.1) are obtained without the Lipschitz continuity of f and m. More importantly, no other conditions of f and m are applied.
Theorem 3.1
Let all the assumptions listed in Section 2 be fulfilled. Then, for every \(u\in U_{ad}\), system (1.1) possesses at least one mild solution.
Proof
Now, applying Schauder’s fixed point theorem, one gets that G possesses at least one fixed point in \(W_{r}\), which is the mild solution of system (1.1). This completes the proof. □
Remark 3.2
The properties of resolvent (Lemma 2.4) are of great importance in the process of deriving the compactness of the solution operator G, through which the methods used in the sense of integer order differential equations can be successfully applied here.
Remark 3.3
Remark 3.4
4 Lagrange optimal control problems subjected to system (1.1)
- (HL)
- (1)
\((t, y, u)\rightarrow L(t, y, u)\) is measurable;
- (2)
\(L(t, \cdot, \cdot)\) is sequentially lower semi-continuous on \(X\times Y\) for a.e. \(t\in [0,c]\);
- (3)
\(L(t, y, \cdot)\) is convex on Y for each \(y\in X\) and a.e. \(t\in [0,c]\);
- (4)
\(L(t, y, u)\geq \phi (t)+a(\Vert y\Vert ^{p}+\Vert u\Vert ^{p})\) for some \(\phi \in L^{p}([0,c], \mathbb{R})\), \(a\geq 0\) and a.e. \(t\in [0,c]\).
- (1)
The following lemma will be used in the proof of our main results.
Lemma 4.1
Proof
A similar manner as that in Theorem 3.1 gives the conclusion. Also it can be seen in [13]. So we omit it. □
Theorem 4.2
Proof
In view of Theorem 3.1, there is at least one mild solution \(y\in W_{R}\) such that \((y, u)\in \mathcal{A}_{d}\) for each \(u\in U_{ad}\), that is, \(S(u)\neq \emptyset \). For clarity, we proceed in the following two steps.
Remark 4.3
The optimal pairs for the Lagrange problem (\(P_{1}\)) are obtained without the Lipschitz continuity of f and m. To compensate, the new idea of setting up minimizing sequences twice is used. Hence, our results generalize the recent existing ones in [12, 16–21], in which the Lipschitz assumptions on nonlinear function and nonlocal item are needed.
5 Time optimal control problems governed by system (1.1)
Theorem 5.1
Remark 5.2
The control \(u^{*}\), the time \(t_{(y^{*}, u^{*})}\) and the pair \((y^{*}, u^{*})\) in Theorem 5.1 are called the time optimal control, optimal time and time optimal pair, respectively.
Proof of Theorem 5.1
From Theorem 3.1, we have that there exists at least one y such that \((y,u)\in \mathcal{A}_{d}\) for each \(u\in U_{ad}\). We will proceed in two steps to check the main results.
Remark 5.3
On the basis of the solution set \(S(u)\) and the target set W, a suitable definition of optimal time is given. Then the idea of constructing time optimal sequences twice and the theory of resolvent are used to derive the existence of time optimal pairs without the Lipschitz assumptions on f and the reflexivity of X. Therefore, our results essentially improve those in [17, 22–24] and the references therein, where the Lipschitz continuity of f and the reflexivity of X are all required.
6 Applications
Now, for every \(t\in [0, 1]\), \(\theta \in [0, 1]\), let \(y(t)(\theta)=x(t, \theta)\), \(f(t, y(t))(\theta)=t^{2}(1+y(t)(\theta))+y(t)(\theta)\), \(u\in L^{2}([0,1]\times [0,1], Y)\), and \(u(t)( \theta)=u(t,\theta)\).
Declarations
Acknowledgements
The authors would like to express their gratitude to the editor and anonymous reviewers for their valuable comments and suggestions. Moreover, the work was supported by the NSF of China (11571300, 11271316), the Qing Lan Project of Jiangsu Province of China and the High-Level Personnel Support Program of Yangzhou University.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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