# ILC with Initial State Learning for Fractional Order Linear Distributed Parameter Systems
## 3 Convergence Analysis for Second Order P-Type ILC
**Lemma 8**: _Denote that:_  
\[M_{1} = \parallel F_{1}\parallel+\parallel\Gamma F_{1}\parallel,\ M_{2}=| \gamma-1|+\|\Lambda\|,\] \[M_{3} = \|G_{11}\|+\|G_{12}\|,\] \[C_{1} = \|B(t)\|\cdot(M_{2}+M_{3}\|\Gamma\|),\] \[C_{2} = \|B(t)\|\cdot\|\Gamma\|\cdot\|F_{1}\|,\ h=2\|A(t)\|+\|B(t)\|.\]  
_If an appropriate constant \(\lambda\) is chosen such that \(\lambda^{\kappa}>E_{\alpha}(hT^{\alpha})C_{2}\), then:_  
\[ \begin{array}{ll}&\parallel F_{k}(x,t)\parallel_{(L^{2},\lambda,\xi)}\\ \leq&\frac{\lambda^{\kappa}E_{\alpha}(hT^{\alpha})M_{1}}{\lambda^{\kappa}-E_ {\alpha}(hT^{\alpha})C_{2}^{2}}\|L_{\xi}(x,0)\|_{L^{2}}^{2}\\ +&\frac{E_{\alpha}(hT^{\alpha})C_{1}M_{1}}{\lambda^{\kappa}-E_{\alpha}(hT^{ \alpha})C_{2}}\cdot\|Q_{k}(x,t)\|_{(L^{2},\lambda,\xi)}.\end{array}  \tag{17}\]  
It follows the definition of \(F_{k}(x,t)\) and the assumptions that:  
\[ \begin{array}{ll}&\parallel F_{k}(x,t)\parallel_{L^{2}}\\ \leq&(\parallel F_{1}\parallel+\parallel\Gamma F_{1}\|)\cdot\|\delta q_{k+1}( x,t)\|_{L^{2}}\\ =&M_{1}\|\delta q_{k+1}(x,t)\|_{L^{2}}.\end{array}  \tag{18}\]  
From (1), we have:  
\[ \begin{array}{ll}&\mathcal{D}_{t}^{\alpha}\delta q_{k}(x,t)=D\triangle\delta q _{k}(x,t)+A(t)\delta q_{k}(x,t)\\ &+B(t)\delta u_{k}(x,t).\end{array}  \tag{19}\]  
Note that:  
\[ \begin{array}{ll}&\mathcal{D}_{t}^{\alpha}(\|\delta q_{k}(x,t)\|_{L^{2}}^{2} )\\ =&\int_{\Omega}\mathcal{D}_{t}^{\alpha}\Big{(}\delta q_{k}^{T}(x,t)\delta q_{ k}(x,t)\Big{)}dx\\ \leq&2\int_{\Omega}\delta q_{k}^{T}(x,t)\mathcal{D}_{t}^{\alpha}\delta q_{k}(x, t)dx\\ =&2\int_{\Omega}\delta q_{k}^{T}(x,t)D\triangle\delta q_{k}(x,t)dx\\ +&\int_{\Omega}\delta q_{k}^{T}(x,t)(A^{T}(t)+A(t))\delta q_{k}(x,t)dx\\ +&2\int_{\Omega}\delta q_{k}^{T}(x,t)B(t)\delta u_{k}(x,t)dx\\ :=&I_{1}+I_{2}+I_{3}.\end{array}  \tag{20}\]  
Then, using the Green formula and the boundary condition (2) to \(I_{1}\), we have:  
\[ \begin{array}{llll}I_{1}&=&2\sum_{i=1}^{n}d_{i}\int_{\Omega} \delta q_{ki}^{T}(x,t)\triangle\delta q_{ki}(x,t)dx\\ &=&2\sum_{i=1}^{n}d_{i}\int_{\Omega}\delta q_{ki}^{T}(x,t)\frac{\partial \delta q_{ki}(x,t)}{\partial v}dx\\ &-&2\sum_{i=1}^{n}d_{i}\int_{\Omega}\
abla q_{ki}^{T}(x,t)\
abla q_{ki}(x,t)dx \\ &=&2\sum_{i=1}^{n}d_{i}\int_{\Omega}\delta q_{ki}^{T}(x,t)(-\frac{\alpha_{i} }{\beta_{i}}\delta q_{ki}(x,t))dx\\ &-&2\sum_{i=1}^{n}d_{i}\int_{\Omega}\
abla q_{ki}^{T}(x,t)\
abla q_{ki}(x,t)dx \\ &\leq&0.\end{array}  \tag{21}\]Clearly,  
\[ \begin{array}{l}I_{2}\leq 2\parallel A(t)\parallel\int_{\Omega}\delta q_{k}^{T}(x,t)\delta q_{k}(x,t)dx\\ =2\parallel A(t)\parallel\cdot\|\delta q_{k}(x,t)\|_{L^{2}}^{2}.\end{array}  \tag{22}\]  
Using the H\(\ddot{o}\)lder inequality of \(I_{3}\), we get:  
\[ \begin{array}{l}I_{3}\ \ \leq\parallel B(t)\|\delta q_{k}(x,t)\|_{L^{2}}^{2} \\ +\ \parallel B(t)\|\delta u_{k}(x,t)\|_{L^{2}}^{2}.\end{array}  \tag{23}\]  
Therefore, from (20) to (23), we obtain:  
\[ \begin{array}{l}\mathcal{D}_{t}^{\alpha}(\|\delta q_{k}(x,t)\|_{L^{2}}^{2} )\\ \leq(2\parallel A(t)\parallel+\|B(t)\|)\delta q_{k}(x,t)\|_{L^{2}}^{2}\\ +\ \parallel B(t)\|\delta u_{k}(x,t)\|_{L^{2}}^{2}.\end{array}  \tag{24}\]  
Integrating both sides of (24) above \(t\), we can get:  
\[ \begin{array}{l}\|\delta q_{k}(x,t)\|_{L^{2}}^{2}\\ \leq\ \|\delta q_{k}(x,0)\|_{L^{2}}^{2}\\ +\ \frac{g}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}\|\delta u_{k}(x,s)\|_{ L^{2}}^{2}ds\\ +\ \frac{h}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}\parallel\delta q_{k}(x,s )\|_{L^{2}}^{2}ds,\\ \leq\ \|\delta q_{k}(x,0)\|_{L^{2}}^{2}\\ +\ \frac{g}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}e^{\lambda s}ds\| \delta u_{k}(x,s)\|_{(L^{2},\lambda)}^{2}\\ +\ \frac{h}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}\parallel\delta q_{k}(x,s )\|_{L^{2}}^{2}ds\end{array}  \tag{25}\]  
where \(g=\|B(t)\|\), \(h=2\parallel A(t)\parallel+\|B(t)\|\).