## Appendix B Proof of Theorem 3  
Let \(\delta_{k}=U^{k-1}+\bigl{(}\mathbf{W}_{c}^{k-1}\bigr{)}^{T}\bigl{(}\gamma \mathbf{\[PERSON]}_{c}^{k}-\mathbf{\[PERSON]}_{c}^{k-1}\bigr{)}\) be expressed further as \(\delta_{k}=\underbrace{U^{k-1}+(\mathbf{W}_{c}^{*})^{T}\bigl{(}\gamma\mathbf{ \[PERSON]}_{c}^{k}-\mathbf{\[PERSON]}_{c}^{k-1}\bigr{)}}_{E1}+\biggl{(}\widetilde{ \mathbf{W}}_{c}^{k-1}\biggr{)}^{T}\bigl{(}\gamma\mathbf{\[PERSON]}_{c}^{k}-\mathbf{ \[PERSON]}_{c}^{k-1}\bigr{)}\). Let \(\mathbf{\[PERSON]}_{c}^{k}=\mathbf{\[PERSON]}_{c}^{k-1}+\Delta\mathbf{\[PERSON]}_{c}^{k}\) make \(\delta_{k}=E1+(\gamma-1)\zeta_{c}^{k-1}+\gamma\biggl{(}\widetilde{\mathbf{W}}_{c}^ {k-1}\biggr{)}^{T}\Delta\mathbf{\[PERSON]}_{c}^{k-1}\).  
Define the Lyapunov function  
\[L^{k-1}=\frac{1}{\alpha_{c}}tr\left(\left(\overset{-k-1}{\mathbf{W}}_{c}^{-1} \right)^{T}\overset{-k-1}{\mathbf{W}}_{c}^{-1}\right)+\frac{1}{\alpha_{a}}tr \left(\left(\overset{-k-1}{\mathbf{W}}_{a}^{-1}\right)^{T}\overset{-k-1}{ \mathbf{W}}_{a}^{-1}\right)=L_{1}+L_{2}. \tag{111}\]  
(\(tr|\)) meaning the matrix trace operator) leading to the first order differences in \(L_{1}\) and \(L_{2}\)  
\[ \begin{array}{l}\Delta L_{1}=\frac{1}{\alpha_{c}}[tr\left(\left(\overset{-k }{\mathbf{W}}_{c}^{k}\right)^{T}\overset{-k}{\mathbf{W}}_{c}^{k}\right)-tr \left(\left(\overset{-k-1}{\mathbf{W}}_{c}^{-1}\right)^{T}\overset{-k-1}{ \mathbf{W}}_{c}^{-1}\right)],\\ \Delta L_{2}=\frac{1}{\alpha_{a}}[tr\left(\left(\overset{-k}{\mathbf{W}}_{a}^{ k}\right)^{T}\overset{-k}{\mathbf{W}}_{a}^{-k}\right)-tr\left(\left(\overset{-k-1}{ \mathbf{W}}_{a}^{-1}\right)^{T}\overset{-k-1}{\mathbf{W}}_{a}^{-k-1}\right)]. \end{array}  \tag{112}\]  
Using estimation error dynamics (32), \(\Delta L_{1},\Delta L_{2}\) are refined further. Let  
\[ \begin{array}{l}\Delta L_{1}=\frac{1}{\alpha_{c}}[tr|2\alpha_{c}\delta_{k} \underbrace{\left(\Phi_{c}^{k-1}\right)^{T}}_{\overset{-k}{\mathbf{W}}_{c}^{ k}}+\alpha_{c}\delta_{k}^{2}\left(\[PERSON]_{c}^{k-1}\right)^{T}\Phi_{c}^{k-1}]=2\alpha_{c} \left(E1+(\gamma-1)\zeta_{c}^{k-1}+\gamma\left(\overset{-k-1}{\mathbf{W}}_{c }^{k-1}\right)^{T}\Delta\[PERSON]_{c}^{k-1}\right)\zeta_{c}^{k-1}+\\ +\alpha_{c}\left(E1+(\gamma-1)\zeta_{c}^{k-1}+\gamma\left(\overset{-k-1}{ \mathbf{W}}_{c}^{k-1}\right)^{T}\Delta\Phi_{c}^{k-1}\right)^{2}\|\Phi_{c}^{k-1 }\|^{2}\leq 2\alpha_{c}(\gamma-1)\left(\zeta_{c}^{k-1}\right)^{2}+2\alpha_{c} \left(E1+\gamma\left(\overset{-k-1}{\mathbf{W}}_{c}^{k-1}\right)^{T}\Delta \Phi_{c}^{k-1}\right)\zeta_{c}^{k-1}\right.\\ +\alpha_{c}\overline{\varphi}_{c}\left(2(\gamma-1)^{2}\left(\zeta_{c}^{k-1} \right)^{2}+2\left(E1+\gamma\left(\overset{-k-1}{\mathbf{W}}_{c}^{k-1}\right) ^{T}\Delta\Phi_{c}^{k-1}\right)\right)^{2}=2\alpha_{c}(\gamma-1)\left(\zeta_{c }^{k-1}\right)^{2}-\left(\zeta_{c}^{k-1}-\alpha_{c}\left(E1+\gamma\left( \overset{-k-1}{\mathbf{W}}_{c}^{k-1}\right)^{T}\Delta\Phi_{c}^{k-1}\right) \right)^{2}\right.\\ +2\alpha_{c}\overline{\varphi}_{c}(\gamma-1)^{2}\left(\zeta_{c}^{k-1}\right)^ {2}+2\alpha_{c}\overline{\varphi}_{c}\left(E1+\gamma\left(\overset{-k-1}{ \mathbf{W}}_{c}^{k-1}\right)^{T}\Delta\Phi_{c}^{k-1}\right)^{2}+\left(\zeta_{c }^{k-1}\right)^{2}+\alpha_{c}^{2}\left(E1+\gamma\left(\overset{-k-1}{ \mathbf{W}}_{c}^{k-1}\right)^{T}\Delta\Phi_{c}^{k-1}\right)^{2}=\\ ==[1+2\alpha_{c}(\gamma-1)+2\alpha_{c}\overline{\varphi}_{c}(\gamma-1)^{2}] \left(\zeta_{c}^{k-1}\right)^{2}-E2+\underbrace{\left(2\alpha_{c}\overline{ \varphi}_{c}+\alpha_{c}^{2}\right)\left(E1+\gamma\left(\overset{-k-1}{ \mathbf{W}}_{c}^{k-1}\right)^{T}\Delta\Phi_{c}^{k-1}\right)}_{E3}.\end{array}  \tag{113}\]  
One can show that the coefficient of \((\zeta_{c}^{k-1})\) in the first term, is a function of \(\gamma\), i.e., \(\widetilde{f}(\gamma)=1+2\alpha_{c}(\gamma-1)+2\alpha_{c}\overline{\varphi}_{c}( \gamma-1)^{2}\), which will have two real roots for \(\alpha_{c}>2\overline{\varphi}_{c}\). Let these roots be \(\gamma_{1}<\gamma_{2}\).