# ILC with Initial State Learning for Fractional Order Linear Distributed Parameter Systems
## 3 Convergence Analysis for Second Order P-Type ILC
Following from Lemma 3 and the definition of the Mittag-Leffler function that for \(\lambda>0\), we have:  
\[\frac{dt^{\alpha}E_{1,1+\alpha}(\lambda t)}{dt}=t^{\alpha-1}E_{1,A}(\lambda t) >0.\]  
As a result,  
\[\frac{g}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}e^{\lambda s}ds=gt^{ \alpha}E_{1,1+\alpha}(\lambda t)\]  
is an increasing function. Using the generalized Gronwall inequality of Inequality (25), it results:  
\[ \begin{array}{l}\|\delta q_{k}(x,t)\|_{L^{2}}^{2}\\ \leq\ \left(\|\delta q_{k}(x,0)\|_{L^{2}}^{2}\right.\\ +\ \frac{g}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}e^{\lambda s}ds\| \delta u_{k}(x,s)\|_{(L^{2},\lambda)}^{2}\right)\\ \times\ E_{\alpha}(ht^{\alpha}).\end{array}  \tag{26}\]Furthermore, it can be easily proven that:  
\[ \begin{array}{ll}&\int_{0}^{t}(t-s)^{a-1}e^{\lambda s}ds\\ &=\int_{0}^{t}w^{a-1}e^{\lambda(t-w)}dw\\ &=e^{\lambda t}\int_{0}^{t}w^{a-1}e^{-\lambda w}dw\\ &=\frac{e^{\lambda t}}{\lambda^{a}}\int_{0}^{\lambda t}s^{a-1}e^{-s}ds<\frac{e ^{\lambda t}}{\lambda^{a}}\Gamma(\alpha).\end{array}  \tag{27}\]  
Substituting it into Inequality (26), one has:  
\[ \begin{array}{ll}&\|\delta q_{k}(x,t)\|_{L^{2}}^{2}\\ &\leq&\left(\|\delta q_{k}(x,0)\|_{L^{2}}^{2}\right.\\ &+&\left.\frac{\mathrm{g}e^{\lambda t}}{\lambda^{a}}\|\delta u_{k}(x,s)\|_{( L^{2},\lambda)}^{2}\right)\times E_{\alpha}(ht^{a}).\end{array}  \tag{28}\]  
Multiplying both sides of the above inequality by \(e^{-\lambda t}\), one has:  
\[ \begin{array}{ll}&\|\delta q_{k}(x,t)\|_{L^{2}}^{2}e^{-\lambda t}\\ &\leq\ e^{-\lambda t}\|\delta q_{k}(x,0)\|_{L^{2}}^{2}\times E_{\alpha}(hT^{ \alpha})\\ &+\ \frac{g}{\lambda^{a}}\|\delta u_{k}(x,s)\|_{(L^{2},\lambda)}^{2}\times E_{ \alpha}(hT^{\alpha})\\ &\leq\ \|\delta q_{k}(x,0)\|_{L^{2}}^{2}\times E_{\alpha}(hT^{\alpha})\\ &+\ \frac{g}{\lambda^{a}}\|\delta u_{k}(x,s)\|_{(L^{2},\lambda)}^{2}\times E_{ \alpha}(hT^{\alpha}).\end{array}  \tag{29}\]  
It follows from (7), and we have \(\|\delta q_{k}(x,0)\|_{L^{2}}^{2}\ \leq\ \|Le_{k}(x,0)\|_{L^{2}}^{2}\). By substituting it into (29), we can obtain:  
\[ \begin{array}{ll}&\|\delta q_{k+1}(x,t)\|_{(L^{2},\lambda)}\\ &\leq\ E_{\alpha}(hT^{\alpha})\times\|Le_{k}(x,0)\|_{L^{2}}^{2}\\ &+\ \frac{g}{\lambda^{a}}\times E_{\alpha}(hT^{\alpha})\|\delta u_{k+1}(x,t) \|_{(L^{2},\lambda)}.\end{array}  \tag{30}\]  
On the other hand, it follows from (9) that:  
\[ \begin{array}{ll}&\|\ \delta u_{k+}(x,t)\|_{(L^{2},\lambda)}\\ &\leq\ \ |(\gamma-1)||\Delta u_{k}(x,t)|_{(L^{2},\lambda)}+\|\Lambda\| \|e_{k}(x,t)\|_{(L^{2},\lambda)}\\ &+\ \ \|\Gamma\|\cdot\|e_{k+1}(x,t)\|_{(L^{2},\lambda)}\\ &\leq\ \ M_{2}\|Q_{k}(x,t)\|_{(L^{2},\lambda)}\\ &+\ \ \|\Gamma\|\cdot\|e_{k+1}(x,t)\|_{(L^{2},\lambda)},\end{array}  \tag{31}\]  
where \(M_{2}=|(\gamma-1)|+\|\Lambda\|\). According to (8), we have:  
\[\begin{array}{ll}&\|e_{k+1}(x,t)\|_{(L^{2},\lambda)}\\ &\leq\ \|G_{11}\|\|\Delta e_{k}(x,t)|_{(L^{2 where \(M_{3}=\|G_{11}\|+\|G_{12}\|\). As a result, we obtain from (30) to (32) that:  
\[ \begin{array}{ll}&\|\delta q_{k+1}(x,t)\|_{(L^{2},\lambda)}\\ \leq&E_{\alpha}(hT^{\alpha})\times\|Le_{k}(x,0)\|_{L^{2}}^{2}\\ +&\frac{g}{\Lambda^{2}}E_{\alpha}(hT^{\alpha})(M_{2}+M_{3}\|\Gamma\|)\cdot\| Q_{k}(x,t)\|_{(L^{2},\lambda)}\\ +&E_{\alpha}(hT^{\alpha})\frac{\delta\|\Gamma\|\cdot\|F_{1}\|}{\Lambda^{2}} \|\delta q_{k+1}(x,t)\|_{(L^{2},\lambda)}.\end{array}  \tag{33}\]  
Thus, by defining \(g(M_{2}+M_{3}\|\Gamma\|)=C_{1}\), \(g\|\Gamma\|\cdot\|F_{1}\|=C_{2}\) and selecting a sufficiently large \(\lambda\) such that \(\lambda^{\alpha}>E_{\alpha}(hT^{\alpha})C_{2}\), we can get:  
\[ \begin{array}{ll}&\|\delta q_{k+1}(x,t)\|_{(L^{2},\lambda)}\\ \leq&\frac{\lambda^{\alpha}E_{\alpha}(hT^{\alpha})}{\lambda^{\alpha}-E_{ \alpha}(hT^{\alpha})C_{2}}\|Le_{k}(x,0)\|_{L^{2}}^{2}\\ +&\frac{E_{\alpha}(hT^{\alpha})C_{1}}{\lambda^{\alpha}-E_{ \alpha}(hT^{\alpha})C_{2}}\cdot\|Q_{k}(x,t)\|_{(L^{2},\lambda)}.\end{array}  \tag{34}\]  
By applying the \((L^{2},\lambda)\)-norm and substituting the above equality into (18), we can obtain:  
\[