近年来,无穷维空间中的分数阶微分方程有了越来越多的应用,原因在于它们对经济、力学及物理等领域诸多问题的抽象刻画^[1-2]. El-Borai^[3]较早对这类方程进行了深入研究. Zhou等人^[4]给出了Caputo分数阶微分方程温和解的定义,并给出了不同条件下温和解的存在性.在该工作的基础上, Zhou等人^[5]给出了Riemman-Liouville分数阶微分方程温和解的定义,并利用非紧性测度理论讨论了温和解的存在性. Hilfer ^[6]对Riemman-Liouville分数阶导数进行了推广,提出了Hilfer分数阶导数.该导数既包括Caputo分数阶导数,又包括Riemman-Liouville分数阶导数.随后, Gu等人^[7]研究了Hilfer分数阶微分方程温和解的存在性.更多分数阶微分方程温和解的存在性工作,详见文献[8-15].

能控性是控制理论中的一个基本但重要的概念,它包括精确能控性和逼近能控性.在无穷维空间中,由于前者的假设条件太强,对控制系统的逼近能控性研究更为普遍^[16].许多学者对分数阶微分方程的逼近能控性做了深入研究,参见文献[17-21].然而,这些工作只考虑了Caputo分数阶微分方程或Riemman-Liouville分数阶微分方程的逼近能控性.作为该领域内一个同样重要的工作, Hilfer分数阶微分方程的逼近能控性目前尚缺少深入研究.在此背景下,本文考虑如下形式的Hilfer分数阶积分微分方程的逼近能控性

(1.1) $ \begin{equation} \left\{\begin{array}{ll} D^{\nu, \mu}_{0^{+}}x(t) = Ax(t)+f\left(t, x(t), (\phi x)(t), (\varphi x)(t)\right)+(Bu)(t), \;t\in J': = (0, b], \\ I^{(1-\nu)(1-\mu)}_{0^{+}}x(t)|_{t = 0} = x_{0}, \end{array}\right. \end{equation} $

其中$ D^{\nu, \mu}_{0^{+}} $表示Hilfer分数阶导数, $ \nu\in[0, 1], \;\mu\in (0, 1) $, $ A $是Banach空间$ X $上强连续半群$ \{S(t)\}_{t\geq0} $的无穷小生成元.状态$ x(\cdot) $取值于$ X $,控制函数$ u(\cdot)\in L^{p}(J, U)(p>\frac{1}{\mu}) $, $ U $是一个Banach空间.令$ J = [0, b] $, $ B:L^{p}(J, U)\rightarrow L^{p}(J, X) $是一个有界线性算子. $ f $的定义将在后文中给出. $ \phi $和$ \psi $是两个线性算子,定义为

$ (\phi x)(t) = \int_{0}^{t}K(t, s)x(s){\rm d}s, \;(\varphi x)(t) = \int_{0}^{t}H(t, s)x(s){\rm d}s, $

其中$ K, H\in C(D, {{\Bbb R}} ^{+}) $, $ D = \{(t, s)\in {{\Bbb R}} ^{2}|0\leq s \leq t\leq b\}, \;{{\Bbb R}} ^{+} = [0, +\infty) $.

本文的目的是研究系统(1.1)的逼近能控性.值得注意的是,在文献[17-21]中,作者在非线性项是一致有界的,并且相应的分数阶线性系统是逼近能控的假设条件下讨论了分数阶微分方程的逼近能控性.与上述方法不同的是,本文提出的方法不需要这些假设条件,利用类似于文献[22-23]中的方法,并做了必要修改,使之能够适用于系统(1.1).

本文的组织结构如下:第2节给出本文相关的预备知识.第3节获得了系统(1.1)温和解的存在唯一性.在第4节中,建立了系统(1.1)逼近能控的充分条件.最后,第5节给出了一个例子.

本节给出一些预备知识.对于更多细节,详见文献[1-2, 6-7].

令$ \alpha = \nu+\mu-\nu\mu $,则$ 1-\alpha = (1-\nu)(1-\mu)\geq0 $.定义

$ C(J, X) = \{f|f\;\mbox{是一个从}\;J\; \mbox{到}\; X\;\mbox{的连续函数}\}, $

$ C(J', X) = \{f|f\;\mbox{是一个从}\;J'\; \mbox{到}\; X\;\mbox{的连续函数}\}, $

$ C_{1-\alpha}(J, X) = \{x\in C(J', X)|t^{1-\alpha}x(t)\in C(J, X)\}. $

空间$ C_{1-\alpha}(J, X) $赋予范数$ \|x\|_{C_{1-\alpha}} = \sup\limits_{t\in J}{t^{1-\alpha}\|x(t)\|} $后是一个Banach空间.令

$ k_{0} = \max\left\{|K(t, s)|\big|(t, s)\in D\right\}, {\quad} h_{0} = \max\left\{|H(t, s)|\big|(t, s)\in D\right\}. $

定义2.1^[1]  函数$ f\in AC[0, \infty) $的$ q $阶分数阶积分定义为

$ \begin{eqnarray*} I^{q}_{0^{+}}f(t) = \frac{1}{\Gamma(q)}\int^{t}_{0}\frac{f(s)}{(t-s)^{1-q}}{\rm d}s, \;t>0, \;0<q<1, \end{eqnarray*} $

假设等式右部在区间$ [0, \infty) $上有定义,其中$ \Gamma(\cdot) $是Gamma函数, $ AC[0, \infty) $是由定义在$ [0, \infty) $上的所有绝对连续函数组成的空间.

定义2.2^[1]  函数$ f\in AC[0, \infty) $的Riemann-Liouville分数阶导数$ ^{L}D^{q}_{0^{+}}f(t) $定义为

$ \begin{eqnarray*} ^{L}D^{q}_{0^{+}}f(t) = \frac{1}{\Gamma(1-q)}\frac{\rm d} {{\rm d}t}\int^{t}_{0}\frac{f(s)}{(t-s)^{q}}{\rm d}s, \;t>0, \;0<q<1, \end{eqnarray*} $

假设等式右部在区间$ [0, \infty) $上有定义.

定义2.3^[1]  函数$ f\in AC[0, \infty) $的Caputo分数阶导数$ ^{C}D^{q}_{0^{+}}f(t) $定义为

$ \begin{eqnarray*} ^{C}D^{q}_{0^{+}}f(t) = ^{L}D^{q}_{0^{+}}\left[f(t)-f(0)\right], \;t>0, \;0<q<1. \end{eqnarray*} $

此外,如果$ f\in C^{1}[0, \infty) $,那么

$ \begin{eqnarray*} ^{C}D^{q}_{0^{+}}f(t) = \frac{1}{\Gamma(1-q)}\int^{t}_{0}(t-s)^{-q}f'(s){\rm d}s, \;t>0, \;0<q<1. \end{eqnarray*} $

定义2.4^[6]  Hilfer分数阶导数$ D^{\nu, \mu}_{0^{+}}f(t), \nu\in[0, 1] $, $ \mu\in (0, 1) $定义为

$ D^{\nu, \mu}_{0^{+}}f(t) = I^{\nu(1-\mu)}_{0^{+}}\frac{\rm d}{{\rm d}t}I^{(1-\nu)(1-\mu)}_{0^{+}}f(t), $

假设等式右部在区间$ [0, \infty) $上有定义.

注2.1^[6]  (ⅰ)当$ \nu = 0 $, $ \mu\in (0, 1) $时,有

$ D^{0, \mu}_{0^{+}}f(t) = \frac{\rm d}{{\rm d}t}I^{1-\mu}_{0^{+}}f(t) = ^{L}D^{\mu}_{0^{+}}f(t), $

即Hilfer分数阶导数$ D^{0, \mu}_{0^{+}}f(t) $是Riemann-Liouville分数阶导数$ ^{L}D^{\mu}_{0^{+}}f(t) $.

(ⅱ)当$ \nu = 1 $, $ \mu\in (0, 1) $时,有

$ D^{1, \mu}_{0^{+}}f(t) = I^{1-\mu}_{0^{+}}\frac{\rm d}{{\rm d}t}f(t) = ^{C}D^{\mu}_{0^{+}}f(t), $

即Hilfer分数阶导数$ D^{1, \mu}_{0^{+}}f(t) $是Caputo分数阶导数$ ^{C}D^{\mu}_{0^{+}}f(t) $.

受文献[7]的启发,定义系统(1.1)的温和解.

定义2.5^[7]  函数$ x\in C_{1-\alpha}(J, X) $是系统(1.1)的一个温和解,如果$ I^{(1-\nu)(1-\mu)}_{0^{+}}x(t)|_{t = 0} = x_{0} $,并且满足

(2.1) $ \begin{eqnarray} x(t) = S_{\nu, \mu}(t)x_{0}+\int_{0}^{t}T_{\mu}(t-s)[f\left(s, x(s), (\phi x)(s), (\varphi x)(s)\right)+Bu(s)]{\rm d}s, t\in J', \end{eqnarray} $

其中

$ S_{\nu, \mu}(t) = I^{\nu(1-\mu)}_{0^{+}}T_{\mu}(t), \; T_{\mu}(t) = t^{\mu-1}P_{\mu}(t), \;P_{\mu}(t) = \int_{0}^{\infty}\mu\theta M_{\mu}(\theta)S(t^{\mu}\theta){\rm d}\theta, $

$ M_{\mu}(\theta) = \sum\limits_{n = 1}^{\infty}\frac{(-\theta)^{n-1}}{(n-1)\Gamma(1-n\mu)}, \;\theta\in {\Bbb C} $

是Wright函数,并且

$ \int_{0}^{\infty}\theta^{q}M_{\mu}(\theta){\rm d}\theta = \frac{\Gamma(1+q)}{\Gamma(1+\mu q)}, \; \; \theta\geq 0. $

为了描述方便,将(2.1)式记为

$ x(t) = S_{\nu, \mu}(t)x_{0}+\int_{0}^{t}(t-s)^{\mu-1}P_{\mu}(t-s)[f\left(s, x(s), (\phi x)(s), (\varphi x)(s)\right)+Bu(s)]{\rm d}s, t\in J'. $

引理2.1给出了推广的Gronwall不等式.

引理2.1^[24-25]  假设$ a:J\rightarrow {{\Bbb R}} ^{+} $是一个局部可积函数, $ b:J\rightarrow {{\Bbb R}} ^{+} $是一个非减连续函数, $ b(t)\leq C $ ($ C $是一个常数).如果$ y:J\rightarrow {{\Bbb R}} ^{+} $局部可积,并且

$ y(t)\leq a(t)+b(t)\int_{0}^{t}(t-s)^{\beta-1}y(s){\rm d}s, \;t\in J, \;\beta>0. $

那么

$ y(t)\leq a(t)+\int_{0}^{t}\left[\sum\limits_{n = 1}^{\infty}\frac{[b(t)\Gamma(\beta)]^{n}}{\Gamma(n\beta)}(t-s)^{n\beta-1}a(s)\right]{\rm d}s, \; t\in J. $

此外,如果$ a $是一个非减函数,那么

$ y(t)\leq a(t)E_\beta\left(b(t)\Gamma(\beta)t^{\beta}\right), $

其中$ E_\beta $是Mittag-Leffler函数,定义为

$ E_\beta(z) = \sum\limits_{k = 0}^{\infty}\frac{z^{k}}{\Gamma(k\beta+1)}. $

我们需要如下假设.

$ (H_{0}) $当$ t>0 $时, $ S(t) $在一致算子拓扑下是连续的,且存在$ M>1 $,使得$ \sup\limits_{t\in[0, \infty)}|S(t)|<M $.

引理2.2^[7]  如果$ (H_{0}) $成立,则如下性质成立.

(ⅰ) $ P_{\mu}(t) $, $ T_{\mu}(t) $和$ S_{\nu, \mu}(t) $是有界线性算子.也就是说,对于$ \forall \;t>0, \;x\in X $,

$ \|P_{\mu}(t)x\|\leq\frac{M\|x\|}{\Gamma(\mu)}, \;\|T_{\mu}(t)x\|\leq\frac{Mt^{\mu-1}\|x\|}{\Gamma(\mu)}, \;\|S_{\nu, \mu}(t)x\|\leq \frac{Mt^{\alpha-1}\|x\|}{\Gamma(\alpha)}, \alpha = \nu+\mu-\nu\mu. $

(ⅱ)算子$ P_{\mu}(t) $, $ T_{\mu}(t) $和$ S_{\nu, \mu}(t) $是强连续的.

定义2.6  如果$ \overline{K_{b}(f)} = X $,那么系统(1.1)在$ J $上是逼近能控的,其中$ \overline{K_{b}(f)} $是$ K_{b}(f) $的闭包,并且

$ K_{b}(f) = \left\{x(b;u):x(\cdot;u) {\rm{是系统(1.1)关于控制函数}}u {\rm{的温和解}}\right\}. $

本节研究系统(1.1)的温和解的存在唯一性.首先给出如下假设.

$ (H_1) $存在函数$ \psi\in L^{p}(J, {{\Bbb R}} ^{+}), \;p>\frac{1}{\mu} $和常数$ c_{1}>0 $,使得

$ \|f(t, x, y, z)\|\leq \psi(t)+c_{1}(\|x\|+\|y\|+\|z\|), \;\forall t\in J, \; \forall x, y, z\in X. $

$ (H_2) $存在常数$ l_{1}>0 $,使得对于$ \forall t\in J, \;\forall x_{1}, y_{1}, z_{1}, x_{2}, y_{2}, z_{2}\in X $,都有

$ \|f(t, x_{1}, y_{1}, z_{1})-f(t, x_{2}, y_{2}, z_{2})\|\leq l_{1}(\|x_{1}-x_{2}\|+\|y_{1}-y_{2}\|+\|z_{1}-z_{2}\|). $

定理3.1  假设$ (H_{0}) $–$ (H_{2}) $成立,对于$ \forall\;u(\cdot)\in L^{p}(J, U) $,系统(1.1)在$ C_{1-\alpha}(J, X) $上具有唯一温和解,如果

(3.1) $ \begin{eqnarray} \frac{b^{\mu}Ml_{1}B(\mu, \alpha)}{\Gamma(\mu)} +\frac{b^{1+\mu}Ml_{1}B(\mu, \alpha+1)(h_{0}+k_{0})}{\alpha\Gamma(\mu)}<1, \end{eqnarray} $

其中$ B(\cdot, \cdot) $是Beta函数.

证  定义$ C_{1-\alpha}(J, X) $上的算子$ T $为

$ (Tx)(t) = S_{\nu, \mu}(t)x_{0}+\int_{0}^{t}(t-s)^{\mu-1}P_{\mu}(t-s)[f\left(s, x(s), (\phi x)(s), (\varphi x)(s)\right)+Bu(s)]{\rm d}s, \;t\in J'. $

以下将通过两个步骤来证明$ T $在$ C_{1-\alpha}(J, X) $有一个不动点.

步骤1  $ T $将$ C_{1-\alpha}(J, X) $映射到$ C_{1-\alpha}(J, X) $.

需要证明对于$ \forall x\in C_{1-\alpha}(J, X) $, $ t^{1-\alpha}(Tx)(t)\in C(J, X) $.令$ y(t) = t^{1-\alpha}x(t)\in C(J, X) $,定义算子$ F $为

$ \begin{eqnarray*} (Fy)(t) & = &t^{1-\alpha}(Tx)(t)\\ & = &t^{1-\alpha}S_{\nu, \mu}(t)x_{0} +t^{1-\alpha}\int_{0}^{t}(t-s)^{\mu-1}P_{\mu}(t-s)Bu(s){\rm d}s\\ &&+t^{1-\alpha}\int_{0}^{t}(t-s)^{\mu-1}P_{\mu}(t-s)f\left(s, x(s), (\phi x)(s), (\varphi x)(s)\right){\rm d}s, \;t\in J. \end{eqnarray*} $

通过引理2.2和$ (H_{1}) $可得

$ \left\|t^{1-\alpha}\int_{0}^{t}(t-s)^{\mu-1}P_{\mu}(t-s)f\left(s, x(s), (\phi x)(s), (\varphi x)(s)\right){\rm d}s\right\|\rightarrow 0, \;\mbox{当}\; t\rightarrow 0^{+}, $

$ \left\|t^{1-\alpha}\int_{0}^{t}(t-s)^{\mu-1}P_{\mu}(t-s)Bu(s){\rm d}s\right\|\rightarrow 0, \;\mbox{当}\; t\rightarrow 0^{+}, $

并且

$ \lim\limits_{t\rightarrow 0^{+}}(Fy)(t) = \lim\limits_{t\rightarrow 0^{+}}t^{1-\alpha}S_{\nu, \mu}(t)x_{0} = \frac{x_{0}}{\Gamma(\alpha)}. $

因此,可以定义$ (Fy)(0) = \frac{x_{0}}{\Gamma(\alpha)} $.下面通过证明$ F:C(J, X)\rightarrow C(J, X) $来证明$ T:C_{1-\alpha}(J, X)\rightarrow C_{1-\alpha}(J, X) $,共包括两个断言.

断言1  对于任意$ y(t) = t^{1-\alpha}x(t) $, $ \sup\limits_{t\in J}\|(Fy)(t)\|<\infty $.

对于$ \forall\; x\in C_{1-\alpha}(J, X) $,有

(3.2) $ \begin{equation} \|(\phi x)(s)\| = \left\|\int_{0}^{s}K(s, \tau)x(\tau){\rm d}\tau\right\| \leq k_{0}\int_{0}^{s}\tau^{\alpha-1}\tau^{1-\alpha}\|x(\tau)\|{\rm d}\tau\leq \frac{k_{0}s^{\alpha}\|x\|_{C_{1-\alpha}}}{\alpha}, \end{equation} $

(3.3) $ \begin{equation} \|(\varphi x)(s)\| = \left\|\int_{0}^{s}H(s, \tau)x(\tau){\rm d}\tau\right\| \leq h_{0}\int_{0}^{s}\tau^{\alpha-1}\tau^{1-\alpha}\|x(\tau)\|{\rm d}\tau \leq \frac{h_{0}s^{\alpha}\|x\|_{C_{1-\alpha}}}{\alpha}. \end{equation} $

对于任意$ y(t) = t^{1-\alpha}x(t)\in C(J, X) $,利用引理2.2, $ (H_{1}) $, (3.2)–(3.3)式和Hölder不等式,可得

$ \begin{eqnarray*} &&\|(Fy)(t)\|\\ &\leq&t^{1-\alpha}\|S_{\nu, \mu}(t)x_{0}\| +t^{1-\alpha}\left\|\int_{0}^{t}(t-s)^{\mu-1}P_{\mu}(t-s)f\left(s, x(s), (\phi x)(s), (\varphi x)(s)\right){\rm d}s\right\| \\&&+t^{1-\alpha}\left\|\int_{0}^{t}(t-s)^{\mu-1}P_{\mu}(t-s)Bu(s){\rm d}s\right\| \\ &\leq&\frac{t^{1-\alpha}Mt^{\alpha-1}\|x_{0}\|}{\Gamma(\alpha)} +\frac{t^{1-\alpha} M}{\Gamma(\mu)}\int_{0}^{t}(t-s)^{\mu-1}\|f\left(s, x(s), (\phi x)(s), (\varphi x)(s)\right)\|{\rm d}s\\ &&+\frac{t^{1-\alpha}M}{\Gamma(\mu)}\int_{0}^{t}(t-s)^{\mu-1}\|Bu(s)\|{\rm d}s\\ &\leq&\frac{M\|x_{0}\|}{\Gamma(\alpha)}+\frac{t^{1-\alpha}Mc_{1}}{\Gamma(\mu)}\int_{0}^{t}(t-s)^{\mu-1}\psi(s){\rm d}s +\frac{t^{1-\alpha}Mc_{1}}{\Gamma(\mu)}\int_{0}^{t}(t-s)^{\mu-1}s^{\alpha-1}s^{1-\alpha}\|x(s)\|{\rm d}s\\ &&+\frac{t^{1-\alpha}Mc_{1}}{\Gamma(\mu)}\int_{0}^{t}(t-s)^{\mu-1}\left(\|(\phi x)(s)\|+\|(\varphi x)(s)\|\right){\rm d}s\\ &\leq&\frac{M\|x_{0}\|}{\Gamma(\alpha)}+\frac{b^{1-\alpha+\frac{p\mu-1}{p}}Mc_{1}}{\Gamma(\mu)}\cdot \left(\frac{p-1}{p\mu-1}\right)^{\frac{p-1}{p}}\cdot\|\psi\|_{L^{p}(J, {{\Bbb R}} ^{+})} +\frac{b^{\mu}Mc_{1}B(\mu, \alpha)\|x\|_{C_{1-\alpha}}}{\Gamma(\mu)}\\ &&+\frac{b^{\mu+1}Mc_{1}(k_{0}+h_{0})B(\mu, \alpha+1)\|x\|_{C_{1-\alpha}}}{\alpha\Gamma(\mu)} +\frac{Mb^{1-\alpha+\frac{p\mu-1}{p}}}{\Gamma(\mu)}\left(\frac{p-1}{p\mu-1}\right)^{\frac{p-1}{p}}\|Bu\|_{L^{p}(J, U)}. \end{eqnarray*} $

因此, $ \sup\limits_{t\in J}\|(Fy)(t)\|<\infty $.

断言2  对于任意$ y(t) = t^{1-\alpha}x(t) $, $ t\rightarrow (Fy)(t) $在$ J $上是连续的.

对于$ t_{1} = 0 $, $ 0<t_{2}\leq b $,易知当$ t_{2}\rightarrow t_{1} $时, $ \|(Fy)(t_{2})-(Fy)(0)\|\rightarrow 0. $

对于$ 0<t_{1}<t_{2}\leq b $,有

$ \begin{eqnarray*} \|(Fy)(t_{2})-(Fy)(t_{1})\| &\leq&\|t^{1-\alpha}_{2}S_{\nu, \mu}(t_{2})x_{0}-t^{1-\alpha}_{1}S_{\nu, \mu}(t_{1})x_{0}\|\\ &&+\bigg\|t^{1-\alpha}_{2}\int_{0}^{t_{2}}(t_{2}-s)^{\mu-1}P_{\mu}(t_{2}-s)Bu(s){\rm d}s\\ && -t^{1-\alpha}_{1}\int_{0}^{t_{1}}(t_{1}-s)^{\mu-1}P_{\mu}(t_{1}-s)Bu(s){\rm d}s\bigg\|\\ &&+\|t^{1-\alpha}_{2}\int_{0}^{t_{2}}(t_{2}-s)^{\mu-1}P_{\mu}(t_{2}-s)f\left(s, x(s), (\phi x)(s), (\varphi x)(s)\right){\rm d}s\\ &&-t^{1-\alpha}_{1}\int_{0}^{t_{1}}(t_{1}-s)^{\mu-1}P_{\mu}(t_{1}-s)f\left(s, x(s), (\phi x)(s), (\varphi x)(s)\right){\rm d}s\| \\&: = &I_{1}+I_{2}+I_{3}. \end{eqnarray*} $

根据$ t^{1-\alpha}S_{\nu, \mu}(t) $的强连续性可得, $ \lim\limits_{t_{2}\rightarrow t_{1}}I_1 = 0 $.

对于$ I_{2} $,我们有

$ \begin{eqnarray*} I_{2} &\leq& \left\|\int_{t_{1}}^{t_{2}}t^{1-\alpha}_{2}(t_{2}-s)^{\mu-1}P_{\mu}(t_{2}-s)Bu(s){\rm d}s\right\| \\ &&+\bigg\|\int_{0}^{t_{1}}t^{1-\alpha}_{2}(t_{2}-s)^{\mu-1}P_{\mu}(t_{2}-s)Bu(s){\rm d}s\\&& -\int_{0}^{t_{1}}t^{1-\alpha}_{1}(t_{1}-s)^{\mu-1}P_{\mu}(t_{2}-s)Bu(s){\rm d}s\bigg\| \\ &&+\bigg\|\int_{0}^{t_{1}}t^{1-\alpha}_{1}(t_{1}-s)^{\mu-1}P_{\mu}(t_{2}-s)Bu(s){\rm d}s\\&& -\int_{0}^{t_{1}}t^{1-\alpha}_{1}(t_{1}-s)^{\mu-1}P_{\mu}(t_{1}-s)Bu(s){\rm d}s\bigg\|\\ &: = & I_{21}+I_{22}+I_{23}. \end{eqnarray*} $

利用引理2.2和Hölder不等式,我们有

$ I_{21}\leq\frac{Mt^{1-\alpha}_{2}\|Bu\|_{L^{p}(J, U)}}{\Gamma(\mu)}\left[\frac{p-1}{p\mu-1} (t_{2}-t_{1})^{\frac{p\mu-1}{p-1}}\right]^{\frac{p-1}{p}}\rightarrow0, \ {\rm{当}}\;t_{2}\rightarrow t_{1}\; {\rm{时}}. $

当$ t_{2}\rightarrow t_{1} $时,

$ I_{22}\leq\frac{M\|Bu\|_{L^{p}(J, U)}}{\Gamma(\mu)}\left(\int_{0}^{t_{1}} \left|t^{1-\alpha}_{2}(t_{2}-s)^{\mu-1}-t^{1-\alpha}_{1}(t_{1}-s)^{\mu-1}\right|^{\frac{p}{p-1}}{\rm d}s \right)^{\frac{p-1}{p}}\rightarrow 0. $

对于充分小的$ \varepsilon>0 $,有

$ \begin{eqnarray*} I_{23}&\leq&\int_{0}^{t_{1}-\varepsilon}t^{1-\alpha}_{1}(t_{1}-s)^{\mu-1}\|P_{\mu}(t_{2}-s)-P_{\mu}(t_{1}-s)\|\|Bu(s)\|{\rm d}s\\ &&+\int_{t_{1}-\varepsilon}^{t_{1}}t^{1-\alpha}_{1}(t_{1}-s)^{\mu-1}\|P_{\mu}(t_{2}-s)-P_{\mu}(t_{1}-s)\|\|Bu(s)\|{\rm d}s\\ &\leq&t^{1-\alpha}_{1}\left(\frac{p-1}{p\mu-1}\left(t_{1}^{\frac{p\mu-1}{p-1}}-\varepsilon^{\frac{p\mu-1}{p-1}}\right)\right)^{\frac{p-1}{p}} \|Bu\|_{L^{p}(J, U)}\\&& \times\sup\limits_{s\in[0, t_{1}-\varepsilon]}\|P_{\mu}(t_{2}-s)-P_{\mu}(t_{1}-s)\|\\ &&+\frac{2Mt^{1-\alpha}_{1}\|Bu\|_{L^{p}(J, U)}}{\Gamma(\mu)}\left(\frac{p-1}{p\mu-1}\varepsilon^{\frac{p\mu-1}{p-1}}\right)^{\frac{p-1}{p}} \rightarrow0, \; {\rm{当}}\;t_{2}\rightarrow t_{1}, \;\varepsilon\rightarrow 0 {\rm{时}}. \end{eqnarray*} $

因此, $ \lim\limits_{t_{2}\rightarrow t_{1}}I_2 = 0 $.

对于$ I_{3} $,我们有

$ \begin{eqnarray*} I_{3}&\leq& \left\|\int_{t_{1}}^{t_{2}}t^{1-\alpha}_{2}(t_{2}-s)^{\mu-1}P_{\mu}(t_{2}-s)f\left(s, x(s), (\phi x)(s), (\varphi x)(s)\right){\rm d}s\right\| \\ &&+\bigg\|\int_{0}^{t_{1}}t^{1-\alpha}_{2}(t_{2}-s)^{\mu-1}P_{\mu}(t_{2}-s)f\left(s, x(s), (\phi x)(s), (\varphi x)(s)\right){\rm d}s\\ && -\int_{0}^{t_{1}}t^{1-\alpha}_{1}(t_{1}-s)^{\mu-1}P_{\mu}(t_{2}-s)f\left(s, x(s), (\phi x)(s), (\varphi x)(s)\right){\rm d}s\bigg\| \\ &&+\bigg\|\int_{0}^{t_{1}}t^{1-\alpha}_{1}(t_{1}-s)^{\mu-1}P_{\mu}(t_{2}-s)f\left(s, x(s), (\phi x)(s), (\varphi x)(s)\right){\rm d}s\\ && -\int_{0}^{t_{1}}t^{1-\alpha}_{1}(t_{1}-s)^{\mu-1}P_{\mu}(t_{1}-s)f\left(s, x(s), (\phi x)(s), (\varphi x)(s)\right){\rm d}s\bigg\|. \end{eqnarray*} $

利用与$ I_{2} $类似的推导,可得$ \lim\limits_{t_{2}\rightarrow t_{1}}I_3 = 0 $.因此, $ t\rightarrow (Fy)(t) $在$ J $上是连续的.通过断言1–2可知, $ F $将$ C(J, X) $映射到$ C(J, X) $,即$ T $将$ C_{1-\alpha}(J, X) $映射到$ C_{1-\alpha}(J, X) $.

步骤2  $ T:C_{1-\alpha}(J, X)\rightarrow C_{1-\alpha}(J, X) $是一个压缩映射.

对于$ \forall x, y\in C_{1-\alpha}(J, X) $,有

(3.4) $ \begin{equation} \|(\phi x)(s)-(\phi y)(s)\| = \left\|\int_{0}^{s}K(s, \tau)\tau^{\alpha-1}\tau^{1-\alpha}(x(\tau)-y(\tau)){\rm d}\tau\right\| \leq \frac{k_{0}s^{\alpha}\|x-y\|_{C_{1-\alpha}}}{\alpha}, \end{equation} $

(3.5) $ \begin{equation} \|(\varphi x)(s)-(\varphi y)(s)\| = \left\|\int_{0}^{s}H(s, \tau)\tau^{\alpha-1}\tau^{1-\alpha}(x(\tau)-y(\tau)){\rm d}\tau\right\| \leq \frac{h_{0}s^{\alpha}\|x-y\|_{C_{1-\alpha}}}{\alpha}. \end{equation} $

利用引理2.2,条件$ (H_{2}) $, (3.4)和(3.5)式,可得

$ \begin{eqnarray*} &&t^{1-\alpha}\|(Tx)(t)-(Ty)(t)\|\\ &\leq&\frac{t^{1-\alpha}M}{\Gamma(\mu)} \int_{0}^{t}(t-s)^{\mu-1}\left\|f\left(s, x(s), (\phi x)(s), (\varphi x)(s)\right)- f\left(s, y(s), (\phi y)(s), (\varphi y)(s)\right)\right\|{\rm d}s\\ &\leq&\frac{t^{1-\alpha}Ml_{1}}{\Gamma(\mu)}\int_{0}^{t}(t-s)^{\mu-1}\|x(s)-y(s)\|{\rm d}s\\&& +\frac{t^{1-\alpha}Ml_{1}}{\Gamma(\mu)}\int_{0}^{t}(t-s)^{\mu-1}\|(\phi x)(s)-(\phi y)(s)\|{\rm d}s\\ &&+\frac{t^{1-\alpha}Ml_{1}}{\Gamma(\mu)}\int_{0}^{t}(t-s)^{\mu-1}\|(\varphi x)(s)-(\psi y)(s)\|{\rm d}s\\ &\leq&\frac{t^{1-\alpha}Ml_{1}}{\Gamma(\mu)}\int_{0}^{t}(t-s)^{\mu-1}s^{\alpha-1}s^{1-\alpha}\|x(s)-y(s)\|{\rm d}s\\ &&+\frac{t^{1-\alpha}Ml_{1}k_{0}\|x-y\|_{C_{1-\alpha}}}{\alpha\Gamma(\mu)}\int_{0}^{t}(t-s)^{\mu-1}s^{\alpha}{\rm d}s\\&& +\frac{t^{1-\alpha}Ml_{1}h_{0}\|x-y\|_{C_{1-\alpha}}}{\alpha\Gamma(\mu)}\int_{0}^{t}(t-s)^{\mu-1}s^{\alpha}{\rm d}s\\ &\leq&\frac{b^{\mu}Ml_{1}B(\mu, \alpha)\|x-y\|_{C_{1-\alpha}}}{\Gamma(\mu)} +\frac{b^{1+\mu}Ml_{1}B(\mu, \alpha+1)(h_{0}+k_{0})\|x-y\|_{C_{1-\alpha}}}{\alpha\Gamma(\mu)}. \end{eqnarray*} $

因此

$ \|Tx-Ty\|_{C_{1-\alpha}}\leq \left[ \frac{b^{\mu}Ml_{1}B(\mu, \alpha)}{\Gamma(\mu)} +\frac{b^{1+\mu}Ml_{1}B(\mu, \alpha+1)(h_{0}+k_{0})}{\alpha\Gamma(\mu)}\right]\|x-y\|_{C_{1-\alpha}}. $

根据步骤1–2以及Banach压缩映射原理可知, $ T $在$ C_{1-\alpha}(J, X) $上具有唯一的不动点.因此,系统(1.1)有唯一的温和解.证毕.

本节给出系统(1.1)的逼近能控性.

定义算子$ \widehat{F}:C_{1-\alpha}(J, X)\rightarrow L^{p}(J, X) $为

$ \begin{eqnarray*} (\widehat{F}x)(t) = f\left(t, x(t), (\phi x)(t), (\varphi x)(t)\right), \;t\in J. \end{eqnarray*} $

有界线性算子$ {\cal L}:L^{p}(J, X)\rightarrow X(p>\frac{1}{\mu}) $定义为

$ \begin{eqnarray*} {\cal L}\rho = \int_{0}^{b}(b-s)^{\mu-1}P_{\mu}(b-s)\rho(s){\rm d}s, \;t\in J. \end{eqnarray*} $

需要如下假设.

$ (H_{3}) $存在一个常数$ l'_{1}>0 $,使得对于$ \forall t\in J, \forall x_{1}, y_{1}, z_{1}, x_{2}, y_{2}, z_{2}\in X $,

$ \|f(t, x_{1}, y_{1}, z_{1})-f(t, x_{2}, y_{2}, z_{2})\|\leq l'_{1}t^{1-\alpha}\|x_{1}-x_{2}\|, $

$ (H_{4}) $对于$ \forall \varepsilon>0 $, $ \rho\in L^{p}(J, X) $,存在$ u(\cdot)\in L^{p}(J, U) $和一个常数$ c_{2}>0 $,使得

(4.1) $ \begin{equation} \left\|{\cal L}\rho-{\cal L}Bu\right\|<\varepsilon, \end{equation} $

(4.2) $ \begin{equation} \|Bu\|_{L^{p}(J, U)}<c_{2}\|\rho\|_{L^{p}(J, X)}, \end{equation} $

并且

(4.3) $ \begin{equation} \frac{b^{1-\alpha+\mu}l'_{1}c_{2}M}{\Gamma(\mu)}\left(\frac{p-1}{p\mu-1}\right)^{1-\frac{1}{p}} E_{\mu}(b^{1-\alpha+\mu}Ml'_{1})<1, \end{equation} $

其中$ c_{2} $是一个独立于$ \rho $的常数.

引理4.1  如果条件$ (H_{0}), (H_{1}), (H_{3}) $和

(4.4) $ \begin{eqnarray} \frac{b^{1-\alpha+\mu}Ml'_{1}}{\Gamma(1+\mu)}<1 \end{eqnarray} $

成立,那么系统(1.1)在$ C_{1-\alpha}(J, X) $上具有唯一的温和解.令$ u_{1}, u_{2}\in L^{p}(J, U) $,则

(4.5) $ \begin{eqnarray} \|x_{1}-x_{2}\|_{C_{1-\alpha}} \leq \frac{b^{1-\alpha+\mu-\frac{1}{p}}M}{\Gamma(\mu)}\left(\frac{p-1}{p\mu-1}\right)^{\frac{p-1}{p}} E_{\mu}\left(b^{1-\alpha+\mu}Ml'_{1}\right)\|Bu_{1}-Bu_{2}\|_{L^{p}(J, U)}, \end{eqnarray} $

其中$ x_{i}, i = 1, 2 $是系统(1.1)关于$ u_{i}, i = 1, 2 $的温和解.

证  根据$ (H_{0}) $, $ (H_{1}) $和定理3.1可得,算子$ T $将$ C_{1-\alpha}(J, X) $映射到$ C_{1-\alpha}(J, X) $.对于$ \forall x, y\in C_{1-\alpha}(J, X) $,利用$ (H_{3}) $,我们有

$ \begin{eqnarray*} &&t^{1-\alpha}\|(Tx)(t)-(Ty)(t)\|\\ &\leq&\frac{t^{1-\alpha} M}{\Gamma(\mu)}\int_{0}^{t}(t-s)^{\mu-1} \left\|f\left(s, x(s), (\phi x)(s), (\varphi x)(s)\right)-f\left(s, y(s), (\phi y)(s), (\varphi y)(s)\right)\right\|{\rm d}s\\ &\leq&\frac{t^{1-\alpha} Ml'_{1}}{\Gamma(\mu)}\int_{0}^{t}(t-s)^{\mu-1}s^{1-\alpha}\|x(s)-y(s)\|{\rm d}s\\ &\leq&\frac{t^{1-\alpha+\mu}Ml'_{1}\|x-y\|_{C_{1-\alpha}}}{\Gamma(1+\mu)}. \end{eqnarray*} $

因此

$ \|Tx-Ty\|_{C_{1-\alpha}}\leq \frac{b^{1-\alpha+\mu}Ml'_{1}}{\Gamma(1+\mu)}\|x-y\|_{C_{1-\alpha}}. $

根据(4.4)式和Banach压缩映射原理可知, $ T $在$ C_{1-\alpha}(J, X) $上具有唯一不动点.因此,系统(1.1)具有唯一的温和解.

令$ x_{i}, \;i = 1, 2 $是系统(1.1)关于$ u_{i}, \;i = 1, 2 $的温和解,根据引理2.2和$ (H_{3}) $可得

$ \begin{eqnarray*} t^{1-\alpha}\|x_{1}(t)-x_{2}(t)\| &\leq&\frac{t^{1-\alpha} M}{\Gamma(\mu)}\int_{0}^{t}(t-s)^{\mu-1} \bigg\|f\left(s, x_{1}(s), (\phi x_{1})(s), (\varphi x_{1})(s)\right)\\ &&-f\left(s, x_{2}(s), (\phi x_{2})(s), (\varphi x_{2})(s)\right)\bigg\|{\rm d}s\\ &&+\frac{t^{1-\alpha} M}{\Gamma(\mu)}\int_{0}^{t}(t-s)^{\mu-1} \|Bu_{1}(s)-Bu_{2}(s)\|{\rm d}s\\ &\leq&\frac{b^{1-\alpha} Ml'_{1}}{\Gamma(\mu)}\int_{0}^{t}(t-s)^{\mu-1}s^{1-\alpha}\|x_{1}(s)-x_{2}(s)\|{\rm d}s\\ &&+\frac{b^{1-\alpha}M}{\Gamma(\mu)}\left(\frac{p-1}{p\mu-1}b^{\frac{p\mu-1}{p-1}}\right)^{\frac{p-1}{p}} \|Bu_{1}-Bu_{2}\|_{L^{p}(J, U)}. \end{eqnarray*} $

令$ Z(t) = t^{1-\alpha}\|x_{1}(t)-x_{2}(t)\| $,根据引理2.1可得

$ Z(t)\leq\frac{b^{1-\alpha}M}{\Gamma(\mu)}\left(\frac{p-1}{p\mu-1}b^{\frac{p\mu-1}{p-1}}\right)^{\frac{p-1}{p}} E_{\mu}\left(b^{1-\alpha}Ml'_{1}t^{\mu}\right)\|Bu_{1}-Bu_{2}\|_{L^{p}(J, U)}. $

因此

$ \|x_{1}-x_{2}\|_{C_{1-\alpha}}\leq \frac{b^{1-\alpha+\mu-\frac{1}{p}}M}{\Gamma(\mu)}\left(\frac{p-1}{p\mu-1}\right)^{\frac{p-1}{p}} E_{\mu}\left(b^{1-\alpha+\mu}Ml'_{1}\right)\|Bu_{1}-Bu_{2}\|_{L^{p}(J, U)}. $

证毕.

接下来,我们利用序方法来证明系统(1.1)的逼近能控性.

定理4.1  如果条件$ (H_{0}), (H_{1}), (H_{3}), (H_{4}) $和(4.4)成立,那么系统(1.1)是逼近能控的.

证  由于$ \overline{D(A)} = X $,只要证明$ D(A)\subset \overline{K_{b}(f)} $就够了,即证明对于$ \forall \varepsilon>0 $和$ \xi\in D(A) $,存在控制函数$ u\in L^{p}(J, U) $,使得

$ \|\xi-x(b)\| = \|\xi-S_{\nu, \mu}(b)x_{0}-{\cal L}(\widehat{F}x)-{\cal L}Bu\|<\varepsilon, $

其中$ x(\cdot) $是系统(1.1)关于$ u $的唯一温和解.

对于$ \forall x_{0}\in X, \;\forall \xi\in D(A) $, $ \xi-S_{\nu, \mu}(b)x_{0}\in D(A) $.因此,存在函数$ \rho\in L^{p}(J, X) $,使得$ {\cal L}\rho = \xi-S_{\nu, \mu}(b)x_{0} $.对于$ \forall \varepsilon>0 $和$ u_{1}\in L^{p}(J, U) $,由$ (H_{4}) $知,存在控制函数$ u_{2}\in L^{p}(J, U) $使得

$ \|\xi-S_{\nu, \mu}(b)x_{0}-{\cal L}(\widehat{F}x_{1})-{\cal L}Bu_{2}\| = \|{\cal L}[\rho-(\widehat{F}x_{1})]-{\cal L}Bu_{2}\| <\frac{\varepsilon}{2^{2}}, $

其中$ x_{1} $是系统(1.1)关于$ u_{1} $的唯一温和解.令$ x_{2} $为系统(1.1)关于$ u_{2} $的唯一温和解,根据$ (H_{4}) $和引理4.1可得,存在$ \omega_{2}\in L^{p}(J, U) $,使得

$ \|{\cal L}[\widehat{F}x_{2}-\widehat{F}x_{1}]-{\cal L}B\omega_{2}\|<\frac{\varepsilon}{2^{3}}, $

并且

$ \begin{eqnarray*} &&\|B\omega_{2}\|_{L^{p}(J, U)}\\&\leq&c_{2}\|\widehat{F}x_{2}-\widehat{F}x_{1}\|_{L^{p}(J, X)}\\ & = &c_{2}\left(\int_{0}^{b}\left\|f\left(s, x_{2}(s), (\phi x_{2})(s), (\varphi x_{2})(s)\right)-f\left(s, x_{1}(s), (\phi x_{1})(s), (\varphi x_{1})(s)\right)\right\|^{p}{\rm d}s\right)^{\frac{1}{p}}\\ &\leq&c_{2}\left(\int_{0}^{b}\left(l'_{1}s^{1-\alpha}\|x_{2}(s)-x_{1}(s)\|\right)^{p}{\rm d}s\right)^{\frac{1}{p}}\\ &\leq&l'_{1}c_{2}b^{\frac{1}{p}}\|x_{2}-x_{1}\|_{C_{1-\alpha}}\\ &\leq&\frac{b^{1-\alpha+\mu}l'_{1}c_{2}M}{\Gamma(\mu)}\left(\frac{p-1}{p\mu-1}\right)^{1-\frac{1}{p}} E_{\mu}(b^{1-\alpha+\mu}Ml'_{1})\|Bu_{2}-Bu_{1}\|_{L^{p}(J, U)}. \end{eqnarray*} $

令$ u_{3} = u_{2}-\omega_{2}\in L^{p}(J, U) $,则

$ \begin{eqnarray*} &&\|\xi-S_{\nu, \mu}(b)x_{0}-{\cal L}(\widehat{F}x_{2})-{\cal L}Bu_{3}\|\\ &\leq&\|\xi-S_{\nu, \mu}(b)x_{0}-{\cal L}(\widehat{F}x_{1})-{\cal L}Bu_{2}\|+\|{\cal L}B\omega_{2}-{\cal L}[\widehat{F}x_{2}-\widehat{F}x_{1}]\|\\ &\leq&\left(\frac{1}{2^{2}}+\frac{1}{2^{3}}\right)\varepsilon. \end{eqnarray*} $

通过归纳法可得,存在$ u_{n}\in L^{p}(J, U) $,使得

$ \|\xi-S_{\nu, \mu}(b)x_{0}-{\cal L}(\widehat{F}x_{n})-{\cal L}Bu_{n+1}\|\leq\left(\frac{1}{2^{2}}+\frac{1}{2^{3}} +\cdots+\frac{1}{2^{n+1}}\right)\varepsilon, $

并且

$ \begin{eqnarray*} &&\|Bu_{n+1}-Bu_{n}\|_{L^{p}(J, U)}\\& = &\|B\omega_{n}\|_{L^{p}(J, U)}\\ &<&\frac{b^{1-\alpha+\mu}l'_{1}c_{2}M}{\Gamma(\mu)} \left(\frac{p-1}{p\mu-1}\right)^{1-\frac{1}{p}} E_{\mu}(b^{1-\alpha+\mu}Ml'_{1})\|Bu_{n}-Bu_{n-1}\|_{L^{p}(J, U)}, \end{eqnarray*} $

其中$ x_{n} $是系统(1.1)关于$ u_{n} $的唯一温和解.根据$ (H_{4}) $可知,序列$ \{Bu_{n}\}_{n\geq1} $是$ L^{p}(J, X) $中的一个Cauchy序列.因此,存在$ \varsigma\in L^{p}(J, X) $,使得

$ \lim\limits_{n\rightarrow\infty}Bu_{n} = \varsigma\;\mbox{在}\;L^{p}(J, X)\;\mbox{中}. $

因此,对于$ \forall \varepsilon>0 $,存在正整数$ N>0 $,使得

$ \left\|{\cal L}Bu_{N+1}-{\cal L}Bu_{N}\right\|<\frac{\varepsilon}{2}. $

因此,我们有

$ \begin{eqnarray*} &&\|\xi-S_{\nu, \mu}(b)x_{0}-{\cal L}(\widehat{F}x_{N})-{\cal L}Bu_{N}\|\\ &\leq&\|\xi-S_{\nu, \mu}(b)x_{0}-{\cal L}(\widehat{F}x_{N})-{\cal L}Bu_{N+1}\|+\|{\cal L}Bu_{N+1}-{\cal L}Bu_{N}\| \\ &\leq&\left(\frac{1}{2^{2}}+\frac{1}{2^{3}} +\cdots+\frac{1}{2^{N+1}}\right)\varepsilon+\frac{\varepsilon}{2}\\ &<&\varepsilon, \end{eqnarray*} $

其中$ x_{N} $是系统(1.1)关于$ u_{N} $的唯一温和解.这也意味着$ D(A)\subset \overline{K_{b}(f)} $.因此,系统(1.1)是逼近能控的.证毕.

注4.1  定理4.1是文献[17-21]结果的一个推广.我们的结果是在不需要非线性项是一致有界的,并且相应的分数阶线性系统是逼近能控的假设条件下取得的.

考虑如下分数阶控制系统

(5.1) $ \begin{eqnarray} \left\{\begin{array}{ll} { } D^{\nu, \frac{3}{4}}_{0^{+}}z(t, \xi) = \frac{\partial^{2}}{\partial \xi^{2}}z(t, \xi) +f(t, z(t, \xi), (\phi z)(t, \xi), (\varphi z)(t, \xi))+Bu(t, \xi), \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad t\in (0, 1], \;\xi\in[0, \pi], \\ z(t, 0) = z(t, \pi) = 0, {\qquad}{\quad}{\qquad}{\qquad}{\qquad}{\qquad} t\in (0, 1], \\ I^{\frac{1}{4}(1-\nu)}_{0^{+}}z(t, \xi)|_{t = 0} = z_{0}(\xi), \; {\qquad}{\qquad}{\qquad}{\qquad}\, \xi\in[0, \pi], \end{array}\right. \end{eqnarray} $

其中$ D^{\nu, \frac{3}{4}}_{0^{+}} $表示Hilfer分数阶导数, $ \nu\in[0, 1] $.令$ X = L^{2}(0, \pi), \;J' = (0, 1] $, $ J = [0, 1] $, $ \alpha = \frac{1}{4}\nu+\frac{3}{4} $, $ e_{n}(\xi) = \sqrt{\frac{2}{\pi}}\sin n\xi, n = 1, 2, \cdots $.显然, $ \{e_{n}\}_{n\geq 1} $是$ X $的一个正交基.定义算子$ A:D(A)\subset X\rightarrow X $为$ Az = \frac{\partial^{2}z}{\partial\xi^{2}} $,其中

$ D(A) = \left\{z\in X:z, \frac{\partial z}{\partial\xi}\; {\rm{是绝对连续的, }}\;\frac{\partial^{2}z}{\partial\xi^{2}}\in X, \;z(0) = z(\pi) = 0\right\}. $

易知, $ A $生成一个强连续半群$ \{S(t)\}_{t\geq0} $,并且是紧的、解析的和自共轭的^[5].因此, $ (H_{0}) $成立.

定义无穷维空间$ U $为

$ U = \bigg\{u:u = \sum\limits_{n = 2}^{\infty}u_{n}e_{n}, \;\sum\limits_{n = 2}^{\infty}u^{2}_{n}<\infty\bigg\}. $

空间$ U $内的范数定义为$ \|u\|_{U} = \left(\sum\limits_{n = 2}^{\infty}u^{2}_{n}\right)^{\frac{1}{2}} $.定义有界线性算子$ B:U\rightarrow X $为

$ Bu = 2u_{2}e_{1}+\sum\limits_{n = 2}^{\infty}u_{n}e_{n}, \;u = \sum\limits_{n = 2}^{\infty}u_{n}e_{n}\in U. $

定义

$ f(t, z(t, \xi), (\phi z)(t, \xi), (\varphi z)(t, \xi)) = \frac{t^{1-\alpha}e^{-t}|z(t, \xi)|}{(1+e^{t})(1+|z(t, \xi)|)}. $

注意到$ \|f(t, z(t, \xi), (\phi z)(t, \xi), (\varphi z)(t, \xi))\|\leq e^{-t} $,并且

$ \begin{eqnarray*} &&\|f(t, z_{1}(t, \xi), (\phi z_{1})(t, \xi), (\varphi z_{1})(t, \xi))-f(t, z_{2}(t, \xi), (\phi z_{2})(t, \xi), (\varphi z_{2})(t, \xi))\|\\ & = &\frac{t^{1-\alpha}e^{-t}\big||z_{1}(t, \xi)|-|z_{2}(t, \xi)|\big|}{(1+e^{t})(1+|z_{1}(t, \xi)|)(1+|z_{2}(t, \xi)|)}\\ &\leq&\frac{t^{1-\alpha}e^{-t}}{1+e^{t}}|z_{1}(t, \xi)-z_{2}(t, \xi)|\\ &\leq&\frac{1}{2}t^{1-\alpha}|z_{1}(t, \xi)-z_{2}(t, \xi)|. \end{eqnarray*} $

因此, $ (H_{1}) $和$ (H_{3}) $成立.

令

$ x(t)(\xi) = z(t, \xi), \; f(t, x(t), (\phi x)(t), (\varphi x)(t))(\xi) = f(t, z(t, \xi), (\phi z)(t, \xi), (\varphi z)(t, \xi)), $

$ u(t)(\xi) = u(t, \xi). $

显然,可以将系统(5.1)写成系统(1.1)的抽象形式.如果条件$ (H_{4}) $和(4.4)成立,根据定理4.1可知,系统(5.1)是逼近能控的.