# ILC with Initial State Learning for Fractional Order Linear Distributed Parameter Systems
## 3 Convergence Analysis for Second Order P-Type ILC  
To obtain the main results, some useful lemmas are introduced. **Lemma 5** (Greens formula [39]). : _Let \(u,\ v\in c^{2}(\Omega)\), then:_  
\[\int_{\Omega}\bigtriangledown u\cdot\bigtriangledown vdx=-\int_{ \Omega}u\bigtriangleup vdx+\int_{\partial\Omega}\frac{\partial v}{\partial v }uds,\]  
_where \(\frac{\partial}{\partial v}\) is the unit outward normal derivative on \(\partial\Omega\)._  
For fractional order linear distribute parameter System (1) with the P-type ILC updating law (7), we first give the following lemmas. **Lemma 6**. : _Let \(\delta u_{k}(x,t)=u_{k}(x,t)-u_{k-1}(x,t)\), \(\delta q_{k}(x,t)=q_{k}(x,t)-q_{k-1}(x,t)\), and:_  
\[Q_{k}(x,t)=\left[ \begin{array}{c}e_{k}(x,t)\\ \delta u_{k}(x,t)\end{array} \right],\] \[\Pi=\left( \begin{array}{cc}G_{11}&G_{12}\\ \Lambda+\Gamma G_{11}&(\gamma-1)I+\Gamma G_{12}\end{array} \right),\] \[F_{k}(x,t)=\left[ \begin{array}{c}F_{1}\delta q_{k+1}(x,t)\\ \Gamma F_{1}\delta q_{k+1}(x,t)\end{array} \right],\]  
_where:_  
\[G_{11}=(I+G(t)\Gamma)^{-1}(I-G(t)\Lambda),\] \[G_{12}=(I+G(t)\Gamma)^{-1}(1-\gamma)G(t),\] \[F_{1}=-(I+G(t)\Gamma)^{-1}C(t),\]  
_then_  
\[Q_{k+1}(x,t)=\Pi Q_{k}(x,t)+F_{k}(x,t),\ \ k\geq 1. \tag{8}\]  
Proof. : It follows from (7) that:  
\[\delta u_{k+1}(x,t) = (\gamma-1)\delta u_{k}(x,t) \tag{9}\] \[+ \Lambda e_{k}(x,t)+\Gamma e_{k+1}(x,t).\]  
Since \(e_{k+1}(x,t)-e_{k}(x,t)=-(y_{k+1}(x,t)-y_{k}(x,t))\), then from (1), it can be obtained that:  
\[e_{k+1}(x,t)-e_{k}(x,t) \tag{10}\] \[= -C(t)\delta q_{k+1}(x,t)-G(t)\delta u_{k+1}(x,t).\]  
From (9) and (10), by some calculation, it can be obtained that:  
\[\left[ \begin{array}{c}e_{k+1}(x,t)\\ \delta u_{k+1}(x,t)\end{array} \right] = \Pi\left[ \begin{array}{c}e_{k}(x,t)\\ \delta u_{k}(x,t)\end{array} \right]+F_{k}(x,t).\]  
The proof is complete. Based on Lemma 6, it is not difficult to prove the following lemma. **Lemma 7**. : _For fractional order linear distribute parameter System (1), if:_  
\[\rho(\Xi)<1, \tag{11}\]  
_where:_  
\[\Xi=\left( \begin{array}{cc}\Xi_{11}&\Xi_{12}\\ \Lambda+\Gamma\Xi_{11}&(\gamma-1)I+\Gamma\Xi_{12}\end{array} \right),\] \[\Xi_{11}=[I+G(0)\Gamma]^{-1}[I-G(0)\Lambda-C(0)L],\] \[\Xi_{12}=[I+G(0)\Gamma]^{-1}(1-\gamma)G(0),\]  
_then for all \(x\in\Omega\), the second order P-type ILC updating law (7) guarantees that:_  
\[\lim_{k\rightarrow\infty}\|Q_{k}(x,0)\|_{L^{2}}^{2}=0. \tag{12}\]  
Proof. : According to Lemma 6, we have:  
\[\left[ \begin{array}{c}e_{k+1}(x,0)\\ \delta u_{k+1}(x,0)\end{array} \right]=\bar{\Pi}\left[ \begin{array}{c}e_{k}( x,0)\\ \delta u_{k}(x,0)\end{array} \right]+\bar{F}_{k}(x,0), \tag{13}\]  
where:  
\[\bar{\Pi}=\left( \begin{array}{cc}\bar{G}_{11}&\bar{G}_{12}\\ \Lambda+\Gamma\bar{G}_{11}&(\gamma-1)I+\Gamma\bar{G}_{12}\end{array} \right),\] \[\bar{F}_{k}(x,0)=\left[ \begin{array}{c}F_{1}\delta q_{k+1}(x,0 )\\ \Gamma F_{1}\delta q_{k+1}(x,0)\end{array} \right],\] \[\bar{G}_{11}=(I+G(0)\Gamma)^{-1}(I-G(0)\Lambda),\] \[\bar{G}_{12}=(I+G(0)\Gamma)^{-1}(1-\gamma)G(0),\] \[\bar{F}_{1}=-(I+G(0)\Gamma)^{-1}C(0),\]  
From (7), we know that:  
\[\delta q_{k+1}(x,0)=Le_{k}(x,0). \tag{14}\]  
Substituting (14) into (13) gives:  
\[\left[ \begin{array}{c}e_{k+1}(x,0)\\ \delta u_{k+1}(x,0)\end{array} \right]=\Xi\left[ \begin{array}{c}e_{k}(x,0)\\ \delta u_{k}(x,0)\end{array} \right]. \tag{15}\]  
Therefore, according to the contracting mapping principle, we have:  
\[\lim_{k\rightarrow\infty}\|Q_{k}(x,0)\|_{L^{2}}^{2}=0. \tag{16}\]  
The proof is complete. From Lemma 6, it can be see that the design problem of a second order P-type ILC updating law (7) for fractional order linear distributed parameter System (1) is equivalent to the stability problem of the discrete System (8).