# Robust Adaptive Synchronization of a Class of Uncertain Chaotic Systems with Unknown Time-Delay
## 2 Preliminaries and Problem Formulation
** Based on the direct method of the Lyapunov theory, the following function is proposed:  
\[V(e_{n},\widetilde{\alpha}_{1},\widetilde{\alpha}_{2},\widetilde{\gamma}_{1 },\widetilde{\gamma}_{2},\overline{\tau}_{1},\overline{\tau}_{2})=\frac{1}{2 }(e_{n}{}^{2}+\widetilde{\alpha}_{1}{}^{2}+\widetilde{\alpha}_{2}{}^{2}+ \widetilde{\gamma}_{1}{}^{2}+\widetilde{\gamma}_{2}{}^{2}+m_{1}\overline{ \tau}_{1}{}^{2}+m_{2}\overline{\tau}_{2}{}^{2}). \tag{13}\]  
The derivative of the proposed Lyapunov function (13) is yielded in the following form:  
\[\dot{V}=e_{n}\dot{e}_{n}+\sum_{i=1}^{2}(\widetilde{\alpha}_{i}\dot{\widetilde{ \alpha}}_{i}+\widetilde{\gamma}_{i}\dot{\widetilde{\gamma}}_{i}+m_{i}\overline{ i}\dot{\widetilde{\tau}}_{i}). \tag{14}\]Based on the error dynamics \(\dot{e}_{n}\) in the differential equations of synchronization error (7) and its substitution in (14), the proposed derivative of the proposed Lyapunov function can be illustrated in the following new form:  
\[ \begin{array}{c}\dot{V}=e_{n}(g(y(t-\tau_{2}))+\Delta g(x(t))+d_{2}(t)-f(x(t- \tau_{1}))-\Delta f(x(t))-d_{1}(t)+u(t))\\ \hskip 113.811024 pt+\sum\limits_{i=1}^{2}(\widetilde{\alpha}_{i}\dot{ \widetilde{\alpha}}_{i}+\widetilde{\gamma}_{i}\dot{\widetilde{\gamma}}_{i}+m_{ i}\widetilde{\tau}_{i}\dot{\widetilde{\tau}}_{i}).\end{array}  \tag{15}\]  
If in Equation (15), \(u(t)\) is expressed as follows:  
\[u(t)=-g(y(t-\hat{\tau}_{2}))+f(x(t-\hat{\tau}_{1}))-k_{0}e_{n}+\overline{u}(t), \tag{16}\]  
where \(\hat{\tau}_{1}\) and \(\hat{\tau}_{2}\) denote estimation of the unknown delays \(\tau_{1}\) and \(\tau_{2}\) in the master and slave systems (1) and (2), and \(k_{0}\) is an arbitrary positive constant, then (15) is rewritten as follows:  
\[ \begin{array}{c}\dot{V}=e_{n}(g(y(t-\tau_{2}))-g(y(t-\hat{\tau}_{2}))+\Delta g (x(t))+d_{2}(t)+f(x(t-\hat{\tau}_{1}))\\ \hskip 113.811024 pt-f(x(t-\tau_{1}))-\Delta f(x(t))-d_{1}(t)-k_{0}e_{n}+ \overline{u}(t))\\ \hskip 113.811024 pt+\sum\limits_{i=1}^{2}(\widetilde{\alpha}_{i}\dot{\widetilde {\alpha}}_{i}+\widetilde{\gamma}_{i}\dot{\widetilde{\gamma}}_{i}+m_{i} \widetilde{\tau}_{i}\dot{\widetilde{\tau}}_{i}).\end{array}  \tag{17}\]  
Applying the first norm \(l_{1}\), it is explicit that (17) can be rewritten as follows:  
\[ \begin{array}{c}\dot{V}\leq|e_{n}|\Big{(}\big{|}g(y(t-\tau_{2}))-g(y(t-\hat {\tau}_{2}))\big{|}+\big{|}\Delta g(x(t))\big{|}+\big{|}d_{2}(t)\big{|}\\ \hskip 113.811024 pt+\big{|}f(x(t-\hat{\tau}_{1}))-f(x(t-\tau_{1}))\big{|}+ \big{|}\Delta f(x(t))\big{|}+\big{|}d_{1}(t)\big{|}\big{)}-k_{0}e_{n}^{2}\\ \hskip 113.811024 pt+e_{n}\overline{u}(t)+\sum\limits_{i=1}^{2}(\widetilde{ \alpha}_{i}\dot{\widetilde{\alpha}}_{i}+\widetilde{\gamma}_{i}\dot{\widetilde {\gamma}}_{i}+m_{i}\widetilde{\tau}_{i}\dot{\widetilde{\tau}}_{i}).\end{array}  \tag{18}\]  
Based on the assumed conditions for disturbances and uncertain uncertainties in the master and slave systems provided in Assumption 1, (18) is revised as follows:  
\[ \begin{array}{c}\dot{V}\leq|e_{n}|\Big{(}\big{|}g(y(t-\tau_{2}))-g(y(t-\hat {\tau}_{2}))\big{|}+\alpha_{2}h_{2}(y)+\gamma_{2}\\ \hskip 113.811024 pt+\big{|}f(x(t-\hat{\tau}_{1}))-f(x(t-\tau_{1}))\big{|}+ \alpha_{1}h_{1}(x)+\gamma_{1}\Big{)}-k_{0}e_{n}^{2}+e_{n}\overline{u}(t)\\ \hskip 113.811024 pt+\sum\limits_{i=1}^{2}(\widetilde{\alpha}_{i}\dot{\widetilde {\alpha}}_{i}+\widetilde{\gamma}_{i}\dot{\widetilde{\gamma}}_{i}+m_{i} \widetilde{\tau}_{i}\dot{\widetilde{\tau}}_{i}).\end{array}  \tag{19}\]  
Due to the fulfillment of Lipschitz conditions (6) by non-linear functions withunknown time delays in the master and slave systems in accordance with Assumption 2, (19) is presented as follows:  
\[