# ILC with Initial State Learning for Fractional Order Linear Distributed Parameter Systems
## 3 Convergence Analysis for Second Order P-Type ILC
\begin{array}{ll}&\|\;F_{k}(x,t)\;\|_{(L^{2},\lambda)}\\ \leq&\frac{\lambda^{\alpha}E_{\alpha}(hT^{\alpha})M_{1}}{\lambda^{\alpha}-E_{ \alpha}(hT^{\alpha})C_{2}}\|Le_{k}(x,0)\|_{L^{2}}^{2}\\ +&\frac{E_{\alpha}(hT^{\alpha})C_{1}M_{1}}{\lambda^{\alpha}-E_{ \alpha}(hT^{\alpha})C_{2}}\cdot\|Q_{k}(x,t)\|_{(L^{2},\lambda)}.\end{array}  \tag{35}\]  
Furthermore, applying the \((L^{2},\lambda,\ \xi)\)-norm to Inequality (35), it yields (17), which completes the proof. In the following, based on Lemmas 6 to 8, a convergent condition for both tracking errors and control input is derived for fractional order linear distributed parameter System (1). **Theorem 1**. : _For fractional order linear distributed parameter System (1) and a given reference \(y_{d}(x,t)\), suppose that the condition in Lemma 7 is satisfied. If:_  
\[\rho(\Pi)\leq\rho_{0}<1, \tag{36}\]  
_where \(\Pi\) is the same as in Lemma 6, then for all \((x,t)\in\Omega\times[0,T]\), the second order \(P\)-type ILC updating law (7) guarantees that:_  
\[\lim_{k\rightarrow\infty}\left\|e_{k}(x,t)\right\|_{L^{2}}=0. \tag{37}\]  
Proof. : It follows from (8) for \(k>1\) that:  
\[Q_{k}(x,t)=\Pi^{k-1}Q_{1}(x,t)+\sum_{i=1}^{k-1}\Pi^{k-i-1}F_{i}(x,t). \tag{38}\]  
Therefore,  
\[\begin{array}{ll}&\|Q_{k}(x,t)\|_{L^{2}}\\ \leq&\left\{\rho(\Pi)\right\}^{k-1}\|Q_{1}(x,t)\|  
Consequently, one obtains from (44) and (45) that:  
\[ \begin{array}{ll}&\parallel Q_{k}(x,t)\parallel_{L^{2}}\\ \leq&\frac{\rho_{0}^{-1}e^{\lambda T}}{(1-\gamma)\xi^{k}}\parallel Q_{1}(x,t) \parallel_{(L^{2},\lambda)}\\ +&\frac{\xi\parallel L\parallel E\rho\rho^{\lambda T}}{\xi^{k}(1-\rho_{0})(1- \gamma)}\\ \times&\frac{\lambda^{k}E_{k}(nT^{k})M_{1}}{\lambda^{k}-E_{k}(nT^{k})C_{2}}. \end{array}  \tag{46}\]  
Note that the condition in Lemma 7 is satisfied. Therefore, \(E_{0}\) is convergent and bounded. Besides, \(r<1,\;\xi>1\), it follows from (46) that:  
\[\lim_{k\to\infty}\parallel Q_{k}(x,t)\parallel_{L^{2}}=0. \tag{47}\]  
Thus, for all \((x,t)\in\Omega\times[0,T]\), \(k=1,2,\ldots\), we have:  
\[\lim_{k\to\infty}\parallel e_{k}(x,t)\parallel_{L^{2}}=0. \tag{48}\]  
The proof is complete. In particular, we have the following corollary. Corollary 1: _For fractional order linear distributed parameter System (1) and a given reference \(y_{d}(x,t)\), if:_  
\[\rho\{(I+G(0)\Gamma)^{-1}(I-G(0)\Lambda-C(0)L)\}<1, \tag{49}\]  
_and:_  
\[\rho\{(I+G(t)\Gamma)^{-1}(I-G(t)\Lambda)\}<1, \tag{50}\]  
_then for all \((x,t)\in\Omega\times[0,T]\), the following P-type ILC updating law:_  
\[\left\{ \begin{array}{ll}u_{k+1}(x,t)=u_{k}(x,t)+\Lambda e_{k}(x,t)+\Gamma e _{k+1}(x,t),\\ q_{k+1}(x,0)=q_{k}(x,0)+Le_{k}(x,0),\end{array} \right. \tag{51}\]  
_guarantees that:_  
\[\lim_{k\to\infty}\|e_{k}(x,t)\|_{L^{2}}=0. \tag{52}\]  
Proof: Setting \(\gamma=1\), from Theorem 1, it can be concluded that if:  
\[\rho(\Xi_{1})<1, \tag{53}\]  
where:  
\[\Xi_{1}=\left( \begin{array}{cc}\Xi_{11}&0\\ \Lambda+\Gamma\Xi_{11}&0\end{array} \right),\] \[\Xi_{11}=[I+G(0)\Gamma]^{-1}[I-G(0)\Lambda-C(0)L],\]  
and:  
\[\rho(\Pi_{1})=\rho\left( \begin{array}{cc}G_{11}&0\\ \Lambda+\Gamma G_{11}&0\end{array} \right)<1, \tag{54}\]where \(G_{11}=(I+G(t)\Gamma)^{-1}(I-G(t)\Lambda)\), then the P-type ILC updating law (51) guarantees that \(\lim_{k\to\infty}\|e_{k}(x,t)\|_{L^{2}}=0\). Noting that:  
\[\det(\Xi_{1})=\left| \begin{array}{cc}\lambda I-\Xi_{11}&0\\ -\Lambda-\Gamma\Xi_{11}&\lambda I\end{array} \right|,\]  
and:  
\[\det(\Pi_{1})=\left| \begin{array}{cc}\lambda I-G_{11}&0\\ -\Lambda-\Gamma G_{11}&\lambda I\end{array} \right|,\]  
it follows from the definition of the spectral radius that:  
\[\rho(\Xi_{1})=\rho(\Pi_{1}),\;\rho(\Pi_{1})=\rho(G_{11}).\]  
The proof is completed. The following corollary is obvious. **Corollary 2**.