实际生活中, 许多动态过程都可用偏微分方程模型来刻画, 如特征学习与图像识别^[1]、金融中期权定价^[2]、无人机信道规划^[3]、带柔性连杆的机器人^[4]等. 二阶抛物型方程(热传导方程)和二阶双曲型方程(波动方程), 作为经典的偏微分方程, 常常被应用于固体中的热传导、多孔介质中化学物质的扩散传递、细管串和薄膜中波的传播以及固体的非力学等问题的研究中^[5]. 至今, 已有许多理论分析和数值研究方法被提出来^{[5, 6]}, 用以解决由这两类偏微分方程描述的系统的控制问题.

作为一种新兴的控制技术, 迭代学习控制起源于机器人应用^[7], 考虑的是寻求控制力矩, 通过循环反复利用前一轮得到的信息来改进本轮的信息, 使得机器人于有限时间区间段内按照设定的轨迹运动. 因这类控制方法在具有固定控制目标的循环重复系统上的适用性, 以及对未知模型系统的控制方面有着独到的优势, 吸引了众多学者将其应用到偏微分系统的控制问题中^[8-11]. 文献[8]基于$ P $型学习律讨论了一类抛物型偏微分系统的迭代学习控制问题; 文献[9]借助于泛函分析中的弱收敛性, 通过构建$ D $型学习律, 解决了一类非正则抛物型偏微分系统的迭代学习控制问题; 文献[10]针对一维线性双曲型偏微分系统进行了迭代学习控制算法设计, 构建了$ PD $型迭代学习律, 并给出了系统跟踪误差收敛的充分条件; 文献[11]则将迭代学习控制方法应用到高阶(四阶)偏微分系统上, 拓宽了该控制方法的应用范围.

近年来, 多智能体系统的协调控制是控制领域中的一个热点研究课题, 其主要涉及到无领航者系统的一致性控制、含一个领航者系统的跟踪控制, 以及含多个领航者系统的包容控制等. 一致性控制考虑的是如何使每个智能体的状态(或输出)能够随着时间的增长而趋于同一个值^[12-13]; 跟踪控制考虑的是如何使每个跟随者的状态(或输出)能够随时间的增长而趋于同一个领航者的状态(或输出)^[14-18]; 包容控制考虑的则是如何使每个跟随者的状态(或输出)能够随时间增长而趋于由多个领航者的状态(或输出)构建的凸包内^[19-23]. 上述基于Lyapunov稳定性意义的研究成果涉及的均为常微分多智能体系统. 最近, 有关偏微分多智能体系统的协调控制问题引起了人们的关注^[24-25]. 文献[24]提出并研究了一类二阶抛物型或双曲型偏微分多智能体系统的一致性控制问题, 采用Lyapunov泛函方法, 构建得到反馈控制协议, 证明了系统的一致性误差随时间趋于无穷时于$ {L^2} $空间中收敛到零, 即Lyapunov稳定性意义下的一致性结论成立; 文献[25]进一步研究了该类偏微分多智能体系统的包容控制问题(跟踪控制问题可含于其中), 得到了相应的包容性收敛结论. 另一方面, 从迭代学习控制角度出发, 对偏微分多智能体系统进行控制设计研究, 这方面也取得了一些成果^[26-28]. 文献[26]基于网络拓扑结构, 针对一类二阶抛物型或双曲型偏微分多智能体系统, 构建得到$ P $型迭代学习律, 当该迭代学习律作用于系统时, 随着迭代次数趋于无穷, 系统的一致性误差能在有限区间上于$ {L^2} $空间中收敛到零, 即迭代学习稳定性意义下的一致性结论成立; 文献[27]则将迭代学习算法应用到时滞的偏微分多智能体系统上, 利用构建得到的$ P $型迭代学习控制律, 解决了一类非线性抛物型偏微分多智能体系统的一致性控制问题; 文献[28]进一步将迭代学习算法应用到高阶双曲型偏微分(四阶梁方程)多智能体系统上, 得到了相应的一致性收敛结论. 我们注意到, 上述研究工作涉及的均为一致性控制, 而未涉及到偏微分多智能体系统基于迭代学习的跟踪控制和包容控制问题.

最近, 文献[29]首次将迭代学习算法应用到多智能体系统的包容控制问题中, 并针对一类异构的非线性常微分多智能体系统, 构建得到基于包容性的分布式迭代学习控制律, 证明了当迭代次数趋于无穷时, 系统的包容误差能够在有限时间区间内收敛于零. 这激发了本文的研究工作.

基于文献[26, 29]的研究工作, 本文将迭代学习算法应用到偏微分多智能体系统的包容控制中, 该类偏微分多智能体系统是由二阶抛物型或双曲型偏微分方程(两类最重要的偏微分方程)构建而成. 依据跟随者智能体系统的输出形式, 构建出$ P $型迭代学习律, 并得到相应系统基于迭代学习稳定性意义下的收敛性条件. 利用压缩映射原理, 证明系统的包容误差在有限时间区间内随迭代次数的增加于$ {L^2} $空间中收敛到零.

符号约定: 对于函数$ Q\left( {x, t} \right) $: $ \left[ {0, 1} \right] \times \left[ {0, T} \right] \to {{\rm{R}}^n} $, 取模

$ {\left\| {Q\left( { \cdot , t} \right)} \right\|_{{L^2}}} = \sqrt {\int_0^1 {\left\| {Q\left( {x, t} \right)} \right\|_2^2} {\rm{d}}x} , $

其中$ {\left\| {Q\left( {x, t} \right)} \right\|_2} $为函数$ Q\left( {x, t} \right) $的2范数, 并定义$ {\left\| Q \right\|_{{L^2}, s}} = \sup \limits_{t \in \left[ {0, T} \right]} \left\| {Q\left( { \cdot , t} \right)} \right\|_{{L^2}}^2 $.

用有向图$ G = \left( {V, E, A} \right) $来描述智能体之间的通讯关联, 其结点集合为$ V = \left\{ {1, 2, \cdots , N} \right\} $, 边集合为$ E \subseteq V \times V $. 用第$ i $个结点表示第$ i $个智能体, 那么从$ i $到$ j $的有向边则可表示为有序对$ \left( {i, j} \right) $. 而$ A = \left( {{a_{ij}}} \right) \in {{\rm{R}}^{N \times N}} $为邻接矩阵, 若$ \left( {i, j} \right) \in E $, 则$ {a_{ij}} > 0 $, 意味着第$ j $个智能体能直接从第$ i $个智能体处接受信息; 否则$ {a_{ij}} = 0 $, 另取$ {a_{ii}} = 0 $. 定义图$ G $的Laplacian矩阵为$ {L_G} = D - A $, 这里$ D = {\rm{diag}}\left( {{{\deg }_{in}}\left( 1 \right), {{\deg }_{in}}\left( 2 \right), \cdots , {{\deg }_{in}}\left( N \right)} \right) $为入度矩阵, 而$ {\deg _{in}}\left( i \right) = \sum\limits_{j = 1}^N {{a_{ij}}} $表示第$ i $个结点的入度.

设含有多个领航者的多智能体系统是由$ M $个跟随者和$ N-M $ ($ N>M $)个领航者构建而成, 用有向图$ {G_{fl}} $来描述这$ N $个智能体之间的通讯拓扑, $ {A_{fl}} = \left( {{a_{ij}}} \right) \in {{{{\Bbb R}} }^{N \times N}} $和$ {L_{{G_{fl}}}} \in {{{{\Bbb R}} }^{N \times N}} $分别表示$ {G_{fl}} $的邻接矩阵和Laplacian矩阵, 用$ {L_{{G_f}}} \in {{{{\Bbb R}} }^{M \times M}} $表示由$ M $个跟随者构建而成的子图$ {G_f} $的Laplacian矩阵.

记信息交换矩阵$ H = {L_{{G_f}}} + \tilde D $, 这里$ \tilde D = {\rm{diag}}\left( {{d_1}, {d_2}, \cdots , {d_M}} \right) $, 其中$ {d_i} = \sum\limits_{j = M + 1}^N {{a_{ij}}} $, 而$ {a_{ij}}\left( {i = 1, 2, \cdots , M, j = M + 1, M + 2, \cdots , N} \right) $为矩阵$ {A_{fl}} $的相应元素. 由于领航者与领航者之间无通讯关联, 且领航者与跟随者之间的通讯关联是单向的, 即领航者发送信息, 跟随者接受信息, 故$ {L_{{G_{fl}}}} $可以写成如下分块矩阵的形式

$ {L_{{G_{fl}}}} = \left[ {\begin{array}{cc} {{H_{M \times M}}}{\quad} &{{F_{M \times \left( {N - M} \right)}}}\\ {{0_{\left( {N - M} \right) \times M}}}{\quad} &{{0_{\left( {N - M} \right) \times \left( {N - M} \right)}}} \end{array}} \right]. $

假设2.1^[19]  对于每个跟随者, 至少存在一条由某个领航者出发且连结它的通路.

当假设2.1成立时, 有下列引理.

引理2.1^[19]  矩阵$ - {H^{ - 1}}F $中的每个元素均大于等于零, 且每行元素之和为1.

设由$ M $个跟随者和$ N-M $ ($ N>M $)个领航者构建而成的多智能体系统的描述如下.

含有迭代和控制变量的$ M $个跟随者的动态模型为

(2.1) $ \begin{equation} {\left\{ \begin{array}{l} { } \frac{{{\partial ^\gamma }{Q_{i, k}}\left( {x, t} \right)}}{{\partial {t^\gamma }}} = \frac{{{\partial ^2}{Q_{i, k}}\left( {x, t} \right)}}{{\partial {x^2}}} + {u_{i, k}}\left( {x, t} \right)\\ {z_{i, k}}\left( {x, t} \right) = C{Q_{i, k}}\left( {x, t} \right) + B{u_{i, k}}\left( {x, t} \right) \end{array}, \right.{\rm{ }}i = 1, 2, \cdots , M, \left( {x, t} \right) \in \left[ {0, 1} \right] \times \left[ {0, T} \right], } \end{equation} $

其中, $ k \in {{\Bbb Z}^ + } $表示迭代次数, $ {Q_{i, k}}\left( {x, t} \right) \in {{{\Bbb R}} } $、$ {u_{i, k}}\left( {x, t} \right) \in {{{\Bbb R}} } $、$ {z_{i, k}}\left( {x, t} \right) \in {{{\Bbb R}} } $分别表示第$ k $次迭代时第$ i $个跟随者的状态变量、控制输入和输出.

不含迭代和控制变量的$ N-M $ ($ N>M $)个领航者的动态模型为

(2.2) $ \begin{equation} \left\{ \begin{array}{l} { } \frac{{{\partial ^\gamma }{Q_i}\left( {x, t} \right)}}{{\partial {t^\gamma }}} = \frac{{{\partial ^2}{Q_i}\left( {x, t} \right)}}{{\partial {x^2}}}\\ {z_i}\left( {x, t} \right) = C{Q_i}\left( {x, t} \right) \end{array}, \right.{\rm{ }}i = M + 1, M + 2, \cdots , N, \left( {x, t} \right) \in \left[ {0, 1} \right] \times \left[ {0, T} \right], \end{equation} $

其中, $ {Q_i}\left( {x, t} \right) \in {{{\Bbb R}} } $、$ {z_i}\left( {x, t} \right) \in {{{\Bbb R}} } $分别表示第$ i $个领航者的状态变量和输出. 而$ \gamma = 1 $或$ 2 $, $ B \ne 0 $.

注2.1   当$ \gamma = 1 $时, 系统$ \rm(2.1) $中的每个跟随者是文献[30] 中的一维抛物型系统, 系统$ \rm(2.2) $中的每个领航者则为相应的不含迭代和控制项的一维抛物型系统; 当$ \gamma = 2 $时, 系统$ \rm(2.1) $中的每个跟随者是文献[10] 中的一维双曲型系统, 系统$ \rm(2.2) $中的每个领航者则为相应的不含迭代和控制项的一维双曲型系统.

假设2.2^[26]  对所有的$ k $, 第$ i $个跟随者智能体的初、边值定位条件设定为

$ {Q_{i, k}}\left( {x, 0} \right) = {f_i}\left( x \right)\left( {\gamma = 1, 2} \right), {\left. {\frac{{\partial {Q_{i, k}}\left( {x, t} \right)}}{{\partial t}}} \right|_{t = 0}} = {g_i}\left( x \right)\left( {\gamma = 2} \right), x \in \left[ {0, 1} \right]; {Q_{i, k}}\left( {0, t} \right) = {\sigma _i}\left( t \right) $

(或$ {\left. {\frac{{\partial {Q_{i, k}}\left( {x, t} \right)}}{{\partial x}}} \right|_{x = 0}} = {r_i}\left( x \right) $), $ {Q_{i, k}}\left( {1, t} \right) = {\tau _i}\left( t \right) $ (或$ {\left. {\frac{{\partial {Q_{i, k}}\left( {x, t} \right)}}{{\partial x}}} \right|_{x = 1}} = {v_i}\left( x \right) $), $ t \in (0, T] $, $ i = 1, $$ 2, \cdots , M $, $ k = 0, 1, 2, \cdots $. 第$ i $个领航者智能体的初、边值条件设定为

$ {Q_i}\left( {x, 0} \right) = {f_i}\left( x \right)\left( {\gamma = 1, 2} \right), {\left. {\frac{{\partial {Q_{i, k}}\left( {x, t} \right)}}{{\partial t}}} \right|_{t = 0}} = {g_i}\left( x \right)\left( {\gamma = 2} \right), x \in \left[ {0, 1} \right]; {Q_i}\left( {0, t} \right) = {\sigma _i}\left( t \right) $

(或$ {\left. {\frac{{\partial {Q_i}\left( {x, t} \right)}}{{\partial x}}} \right|_{x = 0}} = {r_i}\left( x \right) $), $ {Q_i}\left( {1, t} \right) = {\tau _i}\left( t \right) $ (或$ {\left. {\frac{{\partial {Q_i}\left( {x, t} \right)}}{{\partial x}}} \right|_{x = 1}} = {v_i}\left( x \right) $), $ t \in (0, T] $, $ i = M + 1, $$ M + 2, $$ \cdots , N $.

基于迭代学习的包容控制的目标是: 构建$ {u_{i, k}}\left( {x, t} \right) $$ (i = 1, 2, \cdots , M $, $ k = 0, 1, 2, \cdots $), 使得

$ \lim \limits_{k \to \infty } {\bigg\| {{z_{i, k}}\left( {x, t} \right) - \sum\limits_{j = M + 1}^N {{c_{i, j}}{z_j}\left( {x, t} \right)} } \bigg\|_{{L^2}, s}} = 0, \; i = 1, 2, \cdots , M. $

其中的常数$ {c_{i, j}} \ge 0 $, 且$ \sum\limits_{j = M + 1}^N {{c_{i, j}} = 1} $ ($ i = 1, 2, \cdots , M $). 即: 随着迭代次数$ k $的增加, 每个跟随者的输出均能于$ {L^2} $空间中在有限时间区间$ [0, T] $上一致收敛于由领航者输出构建的某个凸包内.

记

$ {Q_{f, k}}\left( {x, t} \right) = {\left[ {{Q_{1, k}}\left( {x, t} \right), \cdots , {Q_{M, k}}\left( {x, t} \right)} \right]^{\rm{T}}}, {z_{f, k}}\left( {x, t} \right) = {\left[ {{z_{1, k}}\left( {x, t} \right), \cdots , {z_{M, k}}\left( {x, t} \right)} \right]^{\rm{T}}}, $

$ {u_k}\left( {x, t} \right) = {\left[ {{u_{1, k}}\left( {x, t} \right), \cdots , {u_{M, k}}\left( {x, t} \right)} \right]^{\rm{T}}}, $

则系统(2.1)可以写成如下的紧凑形式

(2.3) $ \begin{equation} \left\{ \begin{array}{l} { } \frac{{{\partial ^\gamma }{Q_{f, k}}\left( {x, t} \right)}}{{\partial {t^\gamma }}} = \frac{{{\partial ^2}{Q_{f, k}}\left( {x, t} \right)}}{{\partial {x^2}}} + {u_k}\left( {x, t} \right)\\ {z_{f, k}}\left( {x, t} \right) = C{Q_{f, k}}\left( {x, t} \right) + B{u_k}\left( {x, t} \right) \end{array}, \right.\left( {x, t} \right) \in \left[ {0, 1} \right] \times \left[ {0, T} \right]. \end{equation} $

记$ {Q_l}\left( {x, t} \right) = {\left[ {{Q_{M + 1}}\left( {x, t} \right), \cdots , {Q_N}\left( {x, t} \right)} \right]^{\rm{T}}} $, $ {z_l}\left( {x, t} \right) = {\left[ {{z_{M + 1}}\left( {x, t} \right), \cdots , {z_N}\left( {x, t} \right)} \right]^{\rm{T}}} $, 则系统(2.2)可以写成如下的紧凑形式

(2.4) $ \begin{equation} \left\{ \begin{array}{l} { } \frac{{{\partial ^\gamma }{Q_l}\left( {x, t} \right)}}{{\partial {t^\gamma }}} = \frac{{{\partial ^2}{Q_l}\left( {x, t} \right)}}{{\partial {x^2}}}\\ {z_l}\left( {x, t} \right) = C{Q_l}\left( {x, t} \right) \end{array}, \right.\left( {x, t} \right) \in \left[ {0, 1} \right] \times \left[ {0, T} \right]. \end{equation} $

定义一个新变量$ {\delta _k}\left( {x, t} \right) = {z_{f, k}}\left( {x, t} \right) + {H^{ - 1}}F{z_l}\left( {x, t} \right) $, 这里$ {\delta _k}\left( {x, t} \right) $称为包容误差(见文献[19]中(15)式). 根据引理2.1, 并结合基于迭代学习的包容控制的目标可知, 若$ \lim \limits_{k \to \infty } {\left\| {{\delta _k}} \right\|_{{L^2}, s}} = 0 $, 则系统的包容控制问题可解.

对于系统(2.3), 构建如下形式的$ P $型迭代学习律

(3.1) $ \begin{equation} {u_{k + 1}}\left( {x, t} \right) = {u_k}\left( {x, t} \right) - \Gamma {\delta _k}\left( {x, t} \right), \end{equation} $

其中$ \Gamma $为学习增益, 并取$ \Gamma $使得

(3.2) $ \begin{equation} \rho = \left| {1 - B\Gamma } \right| < 1. \end{equation} $

根据包容误差$ {\delta _k}\left( {x, t} \right) $的定义, 结合系统(2.3)和(3.1)式, 有

$ \begin{eqnarray*} {\delta _{k + 1}}\left( {x, t} \right) & = & {z_{f, k + 1}}\left( {x, t} \right) + {H^{ - 1}}F{z_l}\left( {x, t} \right)\\ & = & {z_{f, k}}\left( {x, t} \right) + {H^{ - 1}}F{z_l}\left( {x, t} \right) + {z_{f, k + 1}}\left( {x, t} \right) - {z_{f, k}}\left( {x, t} \right)\\ & = &{\delta _k}\left( {x, t} \right) + C\left( {{Q_{f, k + 1}}\left( {x, t} \right) - {Q_{f, k}}\left( {x, t} \right)} \right) + B\left( {{u_{k + 1}}\left( {x, t} \right) - {u_k}\left( {x, t} \right)} \right)\\ & = &\left( {1 - B\Gamma } \right){\delta _k}\left( {x, t} \right) + C\Delta {Q_{f, k}}\left( {x, t} \right), \end{eqnarray*} $

这里$ \Delta {Q_{f, k}}\left( {x, t} \right) = {Q_{f, k + 1}}\left( {x, t} \right) - {Q_{f, k}}\left( {x, t} \right) $. 上式两端取范数、平方并对变量$ x $从$ 0 \to 1 $积分, 再结合(3.2)式, 可得

(3.3) $ \begin{eqnarray} \int_0^1 {\left\| {{\delta _{k + 1}}\left( {x, t} \right)} \right\|_2^2} {\rm{d}}x &\le& {\rho ^2}\int_0^1 {\left\| {{\delta _k}\left( {x, t} \right)} \right\|_2^2} {\rm{d}}x + {C^2}\int_0^1 {\left\| {\Delta {Q_{f, k}}\left( {x, t} \right)} \right\|_2^2} {\rm{d}}x{}\\ &&+ 2\left| C \right|\rho \int_0^1 {{{\left\| {{\delta _k}\left( {x, t} \right)} \right\|}_2}{{\left\| {\Delta {Q_{f, k}}\left( {x, t} \right)} \right\|}_2}} {\rm{d}}x{}\\ & \le& {\rho ^2}\int_0^1 {\left\| {{\delta _k}\left( {x, t} \right)} \right\|_2^2} {\rm{d}}x + {C^2}\int_0^1 {\left\| {\Delta {Q_{f, k}}\left( {x, t} \right)} \right\|_2^2} {\rm{d}}x{}\\ &&+ 2\left| C \right|\int_0^1 {{{\left\| {{\delta _k}\left( {x, t} \right)} \right\|}_2}{{\left\| {\Delta {Q_{f, k}}\left( {x, t} \right)} \right\|}_2}} {\rm{d}}x. \end{eqnarray} $

应用Cauchy-Schwarz积分不等式, 有

$ \int_0^1 {{{\left\| {{\delta _k}\left( {x, t} \right)} \right\|}_2}{{\left\| {\Delta {Q_{f, k}}\left( {x, t} \right)} \right\|}_2}} {\rm{d}}x \le \sqrt {\int_0^1 {\left\| {{\delta _k}\left( {x, t} \right)} \right\|_2^2} {\rm{d}}x} \sqrt {\int_0^1 {\left\| {\Delta {Q_{f, k}}\left( {x, t} \right)} \right\|_2^2} {\rm{d}}x} . $

将上式代入(3.3)式, 可得

$ \left\| {{\delta _{k + 1}}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2 \le {\rho ^2}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2 + {C^2}\left\| {\Delta {Q_{f, k}}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2 + 2\left| C \right|{\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}}{\left\| {\Delta {Q_{f, k}}\left( { \cdot , t} \right)} \right\|_{{L^2}}}. $

取$ \lambda > 0 $, 有

(3.4) $ \begin{eqnarray} &&\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {{\delta _{k + 1}} \left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\}{}\\ & \le& {\rho ^2}\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\} + {C^2}\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {\Delta {Q_{f, k}}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\}{}\\ &&+ 2\left| C \right|\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - \lambda t}}{{\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|}_{{L^2}}}} \right\}\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - \lambda t}}{{\left\| {\Delta {Q_{f, k}}\left( { \cdot , t} \right)} \right\|}_{{L^2}}}} \right\}. \end{eqnarray} $

另一方面, 结合系统(2.3)和(3.1)式, 可知

(3.5) $ \begin{eqnarray} \frac{{{\partial ^\gamma }\Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial {t^\gamma }}} = \frac{{{\partial ^2}\Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial {x^2}}} - \Gamma {\delta _k}\left( {x, t} \right). \end{eqnarray} $

定理3.1   假设2.1、2.2成立, 且(3.2)式成立, 则在迭代学习律(3.1)的作用下, 随着迭代次数$ k $趋于无穷, 系统(2.3)关于系统(2.4)的包容误差$ {\delta _k}\left( {x, t} \right) $在有限时间区间$ [0, T] $上于$ {L^2} $空间中收敛到零, 即$ \lim \limits_{k \to \infty } {\left\| {{\delta _k}} \right\|_{{L^2}, s}} = 0 $.

证   (3.5)式两端左乘$ {\left( {\Delta {Q_{f, k}}\left( {x, t} \right)} \right)^{\rm{T}}} $并对变量$ x $从$ 0 \to 1 $积分, 得

(3.6) $ \begin{eqnarray} && \int_0^1 {\left\{ {{{\left( {\Delta {Q_{f, k}}\left( {x, t} \right)} \right)}^{\rm{T}}}\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial t}}} \right\}} {\rm{d}}x {}\\ & = &\int_0^1 {\left\{ {{{\left( {\Delta {Q_{f, k}}\left( {x, t} \right)} \right)}^{\rm{T}}}\frac{{{\partial ^2}\Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial {x^2}}}} \right\}} {\rm{d}}x - \Gamma \int_0^1 {\left\{ {{{\left( {\Delta {Q_{f, k}}\left( {x, t} \right)} \right)}^{\rm{T}}}{\delta _k}\left( {x, t} \right)} \right\}} {\rm{d}}x. \end{eqnarray} $

而

(3.7) $ \begin{equation} \int_0^1 {\left\{ {{{\left( {\Delta {Q_{f, k}}\left( {x, t} \right)} \right)}^{\rm{T}}}\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial t}}} \right\}} {\rm{d}}x = \frac{1}{2}\frac{{\rm{d}}}{{{\rm{d}}t}}\left\| {\Delta {Q_{f, k}}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2, \end{equation} $

(3.8) $ \begin{equation} - \Gamma \int_0^1 {\left\{ {{{\left( {\Delta {Q_{f, k}}\left( {x, t} \right)} \right)}^{\rm{T}}}{\delta _k}\left( {x, t} \right)} \right\}} {\rm{d}}x \le \frac{{\left| \Gamma \right|}}{2}\left\| {\Delta {Q_{f, k}}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2 + \frac{{\left| \Gamma \right|}}{2}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2. \end{equation} $

分部积分并结合假设2.2中的边值定位条件, 可知

(3.9) $ \begin{eqnarray} &&\int_0^1 {\left\{ {{{\left( {\Delta {Q_{f, k}}\left( {x, t} \right)} \right)}^{\rm{T}}}\frac{{{\partial ^2}\Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial {x^2}}}} \right\}} {\rm{d}}x{}\\ & = & \left. {{{\left( {\Delta {Q_{f, k}}\left( {x, t} \right)} \right)}^{\rm{T}}}\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial x}}} \right|_{x = 0}^{x = 1} - \int_0^1 {\left\{ {{{\left( {\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial x}}} \right)}^{\rm{T}}}\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial x}}} \right\}} {\rm{d}}x{}\\ & = & - \int_0^1 {\left\{ {{{\left( {\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial x}}} \right)}^{\rm{T}}}\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial x}}} \right\}} {\rm{d}}x = - \left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial x}}} \right\|_{{L^2}}^2 \le 0. \end{eqnarray} $

将(3.7)–(3.9)式代入到(3.6)式, 可推得

$ \frac{{\rm{d}}}{{{\rm{d}}t}}\left\| {\Delta {Q_{f, k}}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2 \le \left| \Gamma \right|\left\| {\Delta {Q_{f, k}}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2 + \left| \Gamma \right|\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2. $

应用Gronwall引理, 并结合假设2.2中的初值定位条件, 有

$ \begin{eqnarray*} \left\| {\Delta {Q_{f, k}}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2 &\le& \left| \Gamma \right|\int_0^t {{{\rm{e}}^{\left| \Gamma \right|\left( {t - \eta } \right)}}\left\| {{\delta _k}\left( { \cdot , \eta } \right)} \right\|_{{L^2}}^2} {\rm{d}}\eta \\ &\le& \left| \Gamma \right|{{\rm{e}}^{\left| \Gamma \right|T}}\int_0^t {{{\rm{e}}^{2\lambda \eta }}{{\rm{e}}^{ - 2\lambda \eta }}\left\| {{\delta _k}\left( { \cdot , \eta } \right)} \right\|_{{L^2}}^2} {\rm{d}}\eta \\ & \le& \left| \Gamma \right|{{\rm{e}}^{\left| \Gamma \right|T}}\int_0^t {{{\rm{e}}^{2\lambda \eta }}} {\rm{d}}\eta \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\}\\ & = & \left| \Gamma \right|{{\rm{e}}^{\left| \Gamma \right|T}}\frac{{{{\rm{e}}^{2\lambda t}} - 1}}{{2\lambda }}\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\}. \end{eqnarray*} $

由此

(3.10) $ \begin{eqnarray} \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {\Delta {Q_{f, k}}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\}& \le &\left| \Gamma \right|{{\rm{e}}^{\left| \Gamma \right|T}}\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {\frac{{1 - {{\rm{e}}^{ - 2\lambda t}}}}{{2\lambda }}} \right\}\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\}{}\\ & = & \left| \Gamma \right|{{\rm{e}}^{\left| \Gamma \right|T}}\frac{{1 - {{\rm{e}}^{ - 2\lambda T}}}}{{2\lambda }}\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\}. \end{eqnarray} $

进一步有

(3.11) $ \begin{eqnarray} \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - \lambda t}}{{\left\| {\Delta {Q_{f, k}}\left( { \cdot , t} \right)} \right\|}_{{L^2}}}} \right\} \le \sqrt {\left| \Gamma \right|{{\rm{e}}^{\left| \Gamma \right|T}}\frac{{1 - {{\rm{e}}^{ - 2\lambda T}}}}{{2\lambda }}} \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - \lambda t}}{{\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|}_{{L^2}}}} \right\}. \end{eqnarray} $

将(3.10)–(3.11)式代入到(3.4)式, 可得

$ \begin{eqnarray*} \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {{\delta _{k + 1}}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\} \le \hat \rho \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\}, \end{eqnarray*} $

这里

$ \begin{eqnarray*} \hat \rho = {\rho ^2} + {C^2}\left| \Gamma \right|{{\rm{e}}^{\left| \Gamma \right|T}}\frac{{1 - {{\rm{e}}^{ - 2\lambda T}}}}{{2\lambda }} + 2\left| C \right|\sqrt {\left| \Gamma \right|{{\rm{e}}^{\left| \Gamma \right|T}}\frac{{1 - {{\rm{e}}^{ - 2\lambda T}}}}{{2\lambda }}}. \end{eqnarray*} $

由(3.2)式可知, $ {\rho ^2} < 1 $, 所以当$ \lambda $选取足够大时, 可使$ \hat \rho < 1 $. 利用压缩映射原理可知

$ \mathop {\lim }\limits_{k \to \infty } \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\} = 0, $

从而

$ \mathop {\lim }\limits_{k \to \infty } {\left\| {{\delta _k}} \right\|_{{L^2}, s}} \le {{\rm{e}}^{2\lambda T}}\mathop {\lim }\limits_{k \to \infty } \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\} = 0. $

由此可知, 当迭代次数$ k $趋于无穷时, 抛物型系统的包容误差$ {\delta _k}\left( {x, t} \right) $在有限时间区间$ [0, T] $内于$ {L^2} $空间中收敛到零. 证毕.

定理3.2   假设2.1、2.2成立, 且(3.2)式成立, 则在迭代学习律(3.1)的作用下, 随着迭代次数$ k $趋于无穷, 系统(2.3)关于系统(2.4)的包容误差$ {\delta _k}\left( {x, t} \right) $在有限时间区间$ [0, T] $上于$ {L^2} $空间中收敛到零, 即$ \mathop {\lim }\limits_{k \to \infty } {\left\| {{\delta _k}} \right\|_{{L^2}, s}} = 0 $.

证   (3.5)式两端左乘$ {\left( {\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial t}}} \right)^{\rm{T}}} $并对$ x $从$ 0 \to 1 $积分, 能够得到

(3.12) $ \begin{eqnarray} &&\int_0^1 {\left\{ {{{\left( {\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial t}}} \right)}^{\rm{T}}}\frac{{{\partial ^2}\Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial {t^2}}}} \right\}} {\rm{d}}x {}\\ & = &\int_0^1 {\left\{ {{{\left( {\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial t}}} \right)}^{\rm{T}}}\frac{{{\partial ^2}\Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial {x^2}}}} \right\}} {\rm{d}}x - \Gamma \int_0^1 {\left\{ {{{\left( {\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial t}}} \right)}^{\rm{T}}}{\delta _k}\left( {x, t} \right)} \right\}} {\rm{d}}x. {\qquad} \end{eqnarray} $

而

(3.13) $ \begin{equation} \int_0^1 {\left\{ {{{\left( {\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial t}}} \right)}^{\rm{T}}}\frac{{{\partial ^2}\Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial {t^2}}}} \right\}} {\rm{d}}x = \frac{1}{2}\frac{{\rm{d}}}{{{\rm{d}}t}}\left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial t}}} \right\|_{{L^2}}^2, \end{equation} $

(3.14) $ \begin{equation} - \Gamma \int_0^1 {\left\{ {{{\left( {\frac{{\partial \Delta {Q_{f, k}} \left( {x, t} \right)}}{{\partial t}}} \right)}^{\rm{T}}}{\delta _k}\left( {x, t} \right)} \right\}} {\rm{d}}x \le \frac{{\left| \Gamma \right|}}{2}\left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial t}}} \right\|_{{L^2}}^2 + \frac{{\left| \Gamma \right|}}{2}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2. \end{equation} $

分部积分并结合假设2.2中的边值定位条件, 可知

(3.15) $ \begin{eqnarray} &&\int_0^1 {\left\{ {{{\left( {\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial t}}} \right)}^{\rm{T}}}\frac{{{\partial ^2}\Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial {x^2}}}} \right\}} {\rm{d}}x{}\\ & = & \left. {{{\left( {\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial t}}} \right)}^{\rm{T}}}\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial x}}} \right|_{x = 0}^{x = 1} - \int_0^1 {\left\{ {{{\left( {\frac{{{\partial ^2}\Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial t\partial x}}} \right)}^{\rm{T}}}\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial x}}} \right\}} {\rm{d}}x{}\\ & = &- \int_0^1 {\left\{ {{{\left( {\frac{{{\partial ^2}\Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial x\partial t}}} \right)}^{\rm{T}}}\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial x}}} \right\}} {\rm{d}}x = - \frac{1}{2}\frac{{\rm{d}}}{{{\rm{d}}t}}\left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial x}}} \right\|_{{L^2}}^2. \end{eqnarray} $

将(3.13)–(3.15)式代入(3.12)式, 可推得

$ \begin{eqnarray*} &&\frac{{\rm{d}}}{{{\rm{d}}t}}\left\{ {\left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial t}}} \right\|_{{L^2}}^2 + \left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial x}}} \right\|_{{L^2}}^2} \right\} \le \left| \Gamma \right|\left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial t}}} \right\|_{{L^2}}^2 + \left| \Gamma \right|\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2\\ &\le& \left| \Gamma \right|\left\{ {\left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial t}}} \right\|_{{L^2}}^2 + \left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial x}}} \right\|_{{L^2}}^2} \right\} + \left| \Gamma \right|\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2, \end{eqnarray*} $

应用Gronwall引理, 并结合假设2.2中的初值定位条件, 有

$ \begin{eqnarray*} \left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial t}}} \right\|_{{L^2}}^2 + \left\| {\frac{{\partial \Delta {Q_{f, k}} \left( { \cdot , t} \right)}}{{\partial x}}} \right\|_{{L^2}}^2 &\le &\left| \Gamma \right|\int_0^t {{{\rm{e}}^{\left| \Gamma \right|\left( {t - \eta } \right)}}\left\| {{\delta _k}\left( { \cdot , \eta } \right)} \right\|_{{L^2}}^2} {\rm{d}}\eta \\ & \le &\left| \Gamma \right|{{\rm{e}}^{\left| \Gamma \right|T}}\int_0^t {{{\rm{e}}^{2\lambda \eta }}{{\rm{e}}^{ - 2\lambda \eta }}\left\| {{\delta _k}\left( { \cdot , \eta } \right)} \right\|_{{L^2}}^2} {\rm{d}}\eta \\ & \le &\left| \Gamma \right|{{\rm{e}}^{\left| \Gamma \right|T}}\int_0^t {{{\rm{e}}^{2\lambda \eta }}} {\rm{d}}\eta \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\}\\ & = &\left| \Gamma \right|{{\rm{e}}^{\left| \Gamma \right|T}}\frac{{{{\rm{e}}^{2\lambda t}} - 1}}{{2\lambda }}\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\}. \end{eqnarray*} $

由此

(3.16) $ \begin{eqnarray} &&\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial t}}} \right\|_{{L^2}}^2} \right\}{}\\ &\le& \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial t}}} \right\|_{{L^2}}^2 + {{\rm{e}}^{ - 2\lambda t}}\left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial x}}} \right\|_{{L^2}}^2} \right\}{}\\ & \le& \left| \Gamma \right|{{\rm{e}}^{\left| \Gamma \right|T}}\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {\frac{{1 - {{\rm{e}}^{ - 2\lambda t}}}}{{2\lambda }}} \right\}\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\}{}\\ & = &\left| \Gamma \right|{{\rm{e}}^{\left| \Gamma \right|T}}\frac{{1 - {{\rm{e}}^{ - 2\lambda T}}}}{{2\lambda }}\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\}. \end{eqnarray} $

另一方面, 应用基本不等式, 有

$ \begin{eqnarray*} \frac{{\rm{d}}}{{{\rm{d}}t}}\left\| {\Delta {Q_{f, k}}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2& = & 2\int_0^1 {{{\left( {\Delta {Q_{f, k}}\left( {x, t} \right)} \right)}^{\rm{T}}}\left( {\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial t}}} \right)} {\rm{d}}x\\ & \le& \left\| {\Delta {Q_{f, k}}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2 + \left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial t}}} \right\|_{{L^2}}^2, \end{eqnarray*} $

应用Gronwall引理, 并结合假设2.2中的初值定位条件, 可推得

$ \begin{eqnarray*} \left\| {\Delta {Q_{f, k}}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2 &\le & \int_0^t {{{\rm{e}}^{t - \eta }}\left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , \eta } \right)}}{{\partial \eta }}} \right\|_{{L^2}}^2} {\rm{d}}\eta\\ & = &{{\rm{e}}^t}\int_0^t {{{\rm{e}}^{\left( {2\lambda - 1} \right)\eta }}{{\rm{e}}^{ - 2\lambda \eta }}\left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , \eta } \right)}}{{\partial \eta }}} \right\|_{{L^2}}^2} {\rm{d}}\eta\\ &\le &{{\rm{e}}^t}\int_0^t {{{\rm{e}}^{\left( {2\lambda - 1} \right)\eta }}} {\rm{d}}\eta \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial t}}} \right\|_{{L^2}}^2} \right\}\\ & = & \frac{{{{\rm{e}}^{2\lambda t}} - {{\rm{e}}^t}}}{{2\lambda - 1}}\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial t}}} \right\|_{{L^2}}^2} \right\}, \end{eqnarray*} $

取$ \lambda > \frac{1}{2} $, 则

(3.17) $ \begin{eqnarray} \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {\Delta {Q_{f, k}}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\} & \le& \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {\frac{{1 - {{\rm{e}}^{ - \left( {2\lambda - 1} \right)t}}}}{{2\lambda - 1}}} \right\}\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial t}}} \right\|_{{L^2}}^2} \right\}{}\\ & = & \frac{{1 - {{\rm{e}}^{ - \left( {2\lambda - 1} \right)T}}}}{{2\lambda - 1}}\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial t}}} \right\|_{{L^2}}^2} \right\}. \end{eqnarray} $

将(3.16)式代入(3.17)式得

(3.18) $ \begin{equation} \sup \limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {\Delta {Q_{f, k}}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\} \le \left| \Gamma \right|{{\rm{e}}^{\left| \Gamma \right|T}}\frac{{1 - {{\rm{e}}^{ - \left( {2\lambda - 1} \right)T}}}}{{2\lambda - 1}}\frac{{1 - {{\rm{e}}^{ - 2\lambda T}}}}{{2\lambda }}\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\}, \end{equation} $

进一步有

(3.19) $ \begin{equation} \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - \lambda t}}{{\left\| {\Delta {Q_{f, k}}\left( { \cdot , t} \right)} \right\|}_{{L^2}}}} \right\} \le \sqrt {\left| \Gamma \right|{{\rm{e}}^{\left| \Gamma \right|T}}\frac{{1 - {{\rm{e}}^{ - \left( {2\lambda - 1} \right)T}}}}{{2\lambda - 1}}\frac{{1 - {{\rm{e}}^{ - 2\lambda T}}}}{{2\lambda }}} \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - \lambda t}}{{\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|}_{{L^2}}}} \right\}. \end{equation} $

将(3.18)–(3.19)式代入(3.4)式, 可得

$ \begin{eqnarray*} \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {{\delta _{k + 1}}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\} \le \bar \rho \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {\delta _k}\left( { \cdot , t} \right) \right\|_{{L^2}}^2} \right\}, \end{eqnarray*} $

这里

$ \bar \rho = {\rho ^2} + {C^2}\left| \Gamma \right|{{\rm{e}}^{\left| \Gamma \right|T}}\frac{{1 - {{\rm{e}}^{ - \left( {2\lambda - 1} \right)T}}}}{{2\lambda - 1}}\frac{{1 - {{\rm{e}}^{ - 2\lambda T}}}}{{2\lambda }} + 2\left| C \right|\sqrt {\left| \Gamma \right|{{\rm{e}}^{\left| \Gamma \right|T}}\frac{{1 - {{\rm{e}}^{ - \left( {2\lambda - 1} \right)T}}}}{{2\lambda - 1}}\frac{{1 - {{\rm{e}}^{ - 2\lambda T}}}}{{2\lambda }}} . $

由(3.2)式可知, $ {\rho ^2} < 1 $, 所以当$ \lambda $选取足够大时, 可使$ \bar \rho < 1 $. 利用压缩映射原理可知

$ \mathop {\lim }\limits_{k \to \infty } \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\} = 0, $

从而

$ \mathop {\lim }\limits_{k \to \infty } {\left\| {{\delta _k}} \right\|_{{L^2}, s}} \le {{\rm{e}}^{2\lambda T}}\mathop {\lim }\limits_{k \to \infty } \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\} = 0. $

由此可知, 当迭代次数$ k $趋于无穷时, 双曲型系统的包容误差$ {\delta _k}\left( {x, t} \right) $在有限时间区间$ [0, T] $内于$ {L^2} $空间中收敛到零, 证毕.

注3.1  当$ M = N - 1 $时, 本文中的系统为含有一个领航者的多智能体系统, 即跟踪控制问题.

对系统(2.1)、(2.2), 取$ C = B = 1 $, $ T = 1 $, $ M = 3 $, $ N = 5 $, 则3个跟随者的动态模型为

$ \left\{ \begin{array}{l} { } \frac{{{\partial ^\gamma }{Q_{i, k}}\left( {x, t} \right)}}{{\partial {t^\gamma }}} = \frac{{{\partial ^2}{Q_{i, k}}\left( {x, t} \right)}}{{\partial {x^2}}} + {u_{i, k}}\left( {x, t} \right)\\ {z_{i, k}}\left( {x, t} \right) = {Q_{i, k}}\left( {x, t} \right) + {u_{i, k}}\left( {x, t} \right) \end{array}, \right.i = 1, 2, 3, \; \left( {x, t} \right) \in \left[ {0, 1} \right] \times \left[ {0, 1} \right], $

2个领航者的动态模型为

$ \left\{ \begin{array}{l} { } \frac{{{\partial ^\gamma }{Q_i}\left( {x, t} \right)}}{{\partial {t^\gamma }}} = \frac{{{\partial ^2}{Q_i}\left( {x, t} \right)}}{{\partial {x^2}}}\\ {z_i}\left( {x, t} \right) = {Q_i}\left( {x, t} \right) \end{array}, \right.i = 4, 5, \; \left( {x, t} \right) \in \left[ {0, 1} \right] \times \left[ {0, 1} \right]. $

设多智能体系统的通讯拓扑如图 1所示, 显然, 假设2.1成立.

相应的Laplacian矩阵为

$ {L_{{G_{fl}}}} = \left[ {\begin{array}{ccc} {{H_{3 \times 3}}}{\quad}&{{F_{3 \times 2}}}\\ {{0_{2 \times 3}}}{\quad}&{{0_{2 \times 2}}} \end{array}} \right] = \left[ {\begin{array}{ccccccc} 1&0&{ - 1}&0&0\\ { - 1}&2&0&{\quad}{ - 1}{\quad}&0\\ 0&{\quad}{ - 1}{\quad}&2&0&{ - 1}\\ 0&0&0&0&0\\ 0&0&0&0&0 \end{array}} \right], $

因此

$ {H^{ - 1}}F = \left[ {\begin{array}{ccc} { }{\frac{4}{3}}&{\quad}{ } {\frac{1}{3}}{\quad} &{ }{\frac{2}{3}}\\ { }{\frac{2}{3}}&{ }{\frac{2}{3}}&{ }{\frac{1}{3}}\\ { }{\frac{1}{3}}&{ }{\frac{1}{3}}&{ }{\frac{2}{3}} \end{array}} \right]\left[ {\begin{array}{ccc} 0{\quad}&0\\ { - 1}{\quad}&0\\ 0{\quad}&{ - 1} \end{array}} \right] = \left[ {\begin{array}{ccc} { } { - \frac{1}{3}}{\quad}&{ } { - \frac{2}{3}}\\ { } { - \frac{2}{3}}{\quad}&{ } { - \frac{1}{3}}\\ { } { - \frac{1}{3}}{\quad}&{ } { - \frac{2}{3}} \end{array}} \right], $

从而, 包容误差表示为

$ {\delta _k}\left( {x, t} \right) = \left[ {\begin{array}{ccc} { } {{z_{1, k}}\left( {x, t} \right) - \frac{1}{3}{z_4}\left( {x, t} \right) - \frac{2}{3}{z_5}\left( {x, t} \right)}\\ { } {{z_{2, k}}\left( {x, t} \right) - \frac{2}{3}{z_4}\left( {x, t} \right) - \frac{1}{3}{z_5}\left( {x, t} \right)}\\ { } {{z_{3, k}}\left( {x, t} \right) - \frac{1}{3}{z_4}\left( {x, t} \right) - \frac{2}{3}{z_5}\left( {x, t} \right)} \end{array}} \right]. $

取$ \Gamma = 0.5 $, 则$ \rho = \left| {1 - B\Gamma } \right| = 0.5 < 1 $. 设置$ k $次迭代时跟随者的初、边值如下: $ {Q_{i, k}}\left( {x, 0} \right) = 0.01 \times i \times x \times \left( {1 - x} \right)\left( {\gamma = 1, 2} \right) $, $ {\left. {\frac{{\partial {Q_{i, k}}\left( {x, t} \right)}}{{\partial t}}} \right|_{t = 0}} = 0\left( {\gamma = 2} \right) $, $ x \in \left[ {0, 1} \right] $; $ {Q_{i, k}}\left( {0, t} \right) = 0.01 \times i \times t $, $ {Q_{i, k}}\left( {1, t} \right) = 0.01 \times 2 \times i \times t $, $ t \in \left( {0, 1} \right] $, $ i = 1, 2, 3 $; 设置领航者的初、边值如下: $ {Q_i}\left( {x, 0} \right) = 0.01 \times i \times x \times \left( {1 - x} \right) \left( {\gamma = 1, 2} \right) $, $ {\left. {\frac{{\partial {Q_i}\left( {x, t} \right)}} {{\partial t}}} \right|_{t = 0}} = 0\left( {\gamma = 2} \right) $, $ x \in \left[ {0, 1} \right] $; $ {Q_i}\left( {0, t} \right) = 0.01 \times i \times t $, $ {Q_i}\left( {1, t} \right) = 0.01 \times 2 \times i \times t $, $ t \in \left( {0, 1} \right] $, $ i = 4, 5 $. 并取初始控制为: $ {u_{1, 0}}\left( {x, t} \right) = {u_{2, 0}}\left( {x, t} \right) = {u_{3, 0}}\left( {x, t} \right) \equiv 1 $. 在迭代学习律(3.1)的作用下, 两类系统的包容误差曲面与$ {\left\| {{\delta _k}} \right\|_{{L^2}, s}} $随着$ k $变化时的仿真结果分别见图 2 –图 9.

从图 2 –图 9可看出, 随着迭代次数$ k $变大, 系统的包容误差能够趋于零.

本文研究了一类由二阶抛物型或双曲型偏微分多智能体系统基于迭代学习算法的包容控制问题. 依据跟随者智能体系统的输出形式, 构建了$ P $型迭代学习控制律, 得到了相应系统基于迭代学习稳定性意义下的收敛性条件. 利用压缩映射原理, 证明了系统的包容误差在有限时间区间内能随迭代次数的增加于$ {L^2} $空间中收敛到零, 仿真结果验证了方法的有效性. 如何将迭代学习控制应用到高阶偏微分^{[11, 28]}多智能体系统的包容控制问题上, 有待作进一步的研究.