## Appendix A Proof of Theorem 3
For the term \(2\mathbf{e}^{\top}\mathbf{P_{1}}\mathbf{B}\mathbf{K_{id}}^{\top}\mathbf{\ddot{\zeta}}\) in (103), applying the inequality of Lemma 1 for this term, we obtain  
\[2\mathbf{e}^{\top}\mathbf{P_{1}}\mathbf{B}\mathbf{K_{id}}^{\top}\mathbf{\ddot{\zeta}}\leq\mathbf{e}^{ \top}\mathbf{P_{1}}\mathbf{\Delta_{1}}\mathbf{P_{1}}\mathbf{e}+\mathbf{\dot{\zeta}}^{\top}\mathbf{K_{ id}}\mathbf{B}^{\top}\mathbf{\Delta_{1}}^{-1}\mathbf{B}\mathbf{K_{id}}^{\top}\mathbf{\ddot{\zeta}} \tag{104}\]  
and, substituting (104) in (102), we obtain  
\[\dot{V}\leq \mathbf{e}^{\top}\Big{(}\mathbf{A_{1}}^{\top}\mathbf{P_{1}}+\mathbf{P_{1}}\mathbf{A_ {1}}+\mathbf{P_{1}}\mathbf{\Delta_{1}}\mathbf{P_{1}}\Big{)}\mathbf{e}+\mathbf{\ddot{\zeta}}^{\top} \Big{(}\mathbf{A_{2}}^{\top}\mathbf{P_{2}}+\mathbf{P_{2}}\mathbf{A_{2}}+\mathbf{K_{id}}\mathbf{B}^{ \top}\mathbf{\Delta_{1}}^{-1}\mathbf{B}\mathbf{K_{id}}^{\top}\Big{)}\mathbf{\ddot{\zeta}} \tag{105}\] \[+2\mathbf{\dot{\zeta}}^{\top}\mathbf{P_{2}}\mathbf{B}\mathbf{\ddot{W}}(t)^{\top} \mathbf{\Psi}\Big{(}\mathbf{\dot{\zeta}}\Big{)}+2\mathbf{\dot{\zeta}}^{\top}\mathbf{P_{2}}\mathbf{B} \mathbf{W_{0}}^{\top}\mathbf{\ddot{\Psi}}+2\mathbf{\dot{\zeta}}^{\top}\mathbf{P_{2}}\mathbf{B}\mathbf{ \ddot{f}}_{\rm e}(\mathbf{\zeta},t)+\frac{1}{\gamma_{1}}\mathbf{\ddot{W}}(t)^{\top} \mathbf{\dot{W}}(t)\] \[+\mathbf{\ddot{b}}(t)\bigg{(}2\mathbf{\ddot{\zeta}}^{\top}\mathbf{P_{2}}\mathbf{B}u (\mathbf{\zeta})+\frac{1}{\gamma_{2}}\mathbf{\dot{b}}(t)\bigg{)}.\]  
Following the same procedure, for the term \(2\mathbf{\dot{\zeta}}^{\top}\mathbf{P_{2}}\mathbf{B}\mathbf{W_{0}}^{\top}\mathbf{\ddot{\Psi}}\), we obtain  
\[\dot{V}= 2\mathbf{e}^{\top}\mathbf{P_{1}}\Big{(}\mathbf{A}\mathbf{e}+\mathbf{B}\Big{[}a_{1} \zeta_{1},+a_{2}\zeta_{2}+\mathbf{W}(t)^{\top}\mathbf{\Psi}\big{(}\mathbf{\dot{\zeta}} \big{)}+\mathbf{\dot{b}}(t)u(\mathbf{\zeta})+\mathbf{K_{id}}^{\top}\mathbf{\ddot{\zeta}}-f_{2} \left(\zeta_{1},\zeta_{2},t\right)\Big{]}\Big{)} \tag{102}\]0  
and, under Assumptions 4 and 6, the inequality (106) is rewritten as  
\[\dot{V}= 2\mathbf{e}^{\top}\mathbf{P_{1}}\Big{(}\mathbf{A}\mathbf{e}+\mathbf{B}\Big{[}a_{1} \zeta_{1},+a_{2}\zeta_{2}+\mathbf{W}(t)^{\top}\mathbf{\Psi}\big{(}\mathbf{\dot{\zeta}} \big{)}+\mathbf{\dot{b}}(t)u(\mathbf{\zeta})+\mathbf{K_{id}}^{\top}\mathbf{\ddot{\zeta}}-f_{2} \left(\zeta_{1},\zeta_{2},t\right)\Big{]}\Big{)} \tag{102}\]1  
Additionally, notice that the term \(2\mathbf{\dot{\zeta}}^{\top}\mathbf{P_{2}}\mathbf{B}\mathbf{\ddot{W}}(t)^{\top}\mathbf{\Psi}\Big{(}\mathbf{ \dot{\zeta}}\Big{)}\) can be written as  
\[\dot{V}= 2\mathbf{e}^{\top}\mathbf{P_{1}}\Big{(}\mathbf{A}\mathbf{e}+\mathbf{B}\Big{[}a_{1} \zeta_{1},+a_{2}\zeta_{2}+\mathbf{W}(t)^{\top}\mathbf{\Psi}\big{(}\mathbf{\dot{\zeta}} \big{)}+\mathbf{\dot{b}}(t)u(\mathbf{\zeta})+\mathbf{K_{id}}^{\top}\mathbf{\ddot{\zeta}}-f_{2} \left(\zeta_{1},\zeta_{2},t\right)\Big{]}\Big{)} \tag{102}\]2  
Then, substituting Equations (107) and (108) in (105), we obtain  
\[\dot{V}= 2\mathbf{e}^{\top}\mathbf{P_{1}}\Big{(}\mathbf{A}\mathbf{e}+\mathbf{B}\Big{[}a_{1} \zeta_{1},+a_{2}\zeta_{2}+\mathbf{W}(t)^{\top}\mathbf{\Psi}\big{(}\mathbf{\dot{\zeta}} \big{)}+\mathbf{\dot{b}}(t)u(\mathbf{\zeta})+\mathbf{K_{id}}^{\top}\mathbf{\ddot{\zeta}}-f_{2} \left(\zeta_{1},\zeta_{2},t\right)\Big{]}\Big{)} \tag{102}\]3 (109) \[\dot{V}= 2\mathbf{e}^{\top}\mathbf{P_{1}}\Big{(}\mathbf{A}\mathbf{e}+\mathbf{B}\Big{[}a_{1} \zeta_{1},+a_{2}\zeta_{2}+\mathbf{W}(t)^{\top}\mathbf{\Psi}\big{(}\mathbf{\dot{\zeta}} \big{)}+\mathbf{\dot{b}}(t)u(\mathbf{\zeta})+\mathbf{K_{id}}^{\top}\mathbf{\ddot{\zeta}}-f_{2} \left(\zeta_{1},\zeta_{2},t\right)\Big{]}\Big{)} \tag{102}\]4 (A10)  
Notice that, from Assumption 5 and the Rayleigh-Ritz inequality, we obtain  
\[\dot{V}= 2\mathbf{e}^{\top}\mathbf{P_{1}}\Big{(}\mathbf{A}\mathbf{e}+\mathbf{B}\Big{[}a_{1} \zeta_{1},+a_{2}\zeta_{2}+\mathbf{W}(t)^{\top}\mathbf{\Psi}\big{(}\mathbf{\dot{\zeta}} \big{)}+\mathbf{\dot{b}}(t)u(\mathbf{\zeta})+\mathbf{K_{id}}^{\top}\mathbf{\ddot{\zeta}}-f_{2} \left(\zeta_{1},\zeta_{2},t\right)\Big{]}\Big{)} \tag{102}\]5 (A11)  
Then, substituting the adaptive law (46) for the weights of the dynamic neural network as well as (A10) and (A11), we obtain for (A9) the following:  
\[\dot{V}= 2\mathbf{e}^{\top}\mathbf{P_{1}}\Big{(}\mathbf{A}\mathbf{e}+\mathbf{B}\Big{[}a_{1} \zeta_{1},+a_{2}\zeta_{2}+\mathbf{W}(t)^{\top}\mathbf{\Psi}\big{(}\mathbf{\dot{\zeta}} \big{)}+\mathbf{\dot{b}}(t)u(\mathbf{\zeta})+\mathbf{K_{id}}^{\top}\mathbf{\ddot{\zeta}}-f_{2} \left(\zeta_{1},\zeta_{2},t\right)\Big{]}\Big{)} \tag{102}\]6 (A19)  
Finally, we obtain for (A19) the following:  
\[\dot{V}= 2\mathbf{e}^{\top}\mathbf{P_{1}}\Big{(}\mathbf{A}\mathbf{e}+\mathbf{B}\Big{[}a_{1} \zeta_{1},+a_{2}\zeta_{2}+\mathbf{W}(t)^{\top}\mathbf{\Psi}\big{(}\mathbf{\dot{\zeta}} \big{)}+\mathbf{\dot{b}}(t)u(\mathbf{\zeta})+\mathbf{K_{id}}^{\top}\mathbf{\ddot{\zeta}}-f_{2} \left(\zeta_{1},\zeta_{2},t\right)\Big{]}\Big{)} \tag{102}\]7 (A20)  
where \(\alpha\) is defined as  
\[\dot{V}= 2\mathbf{e}^{\top}\mathbf{P_{1}}\Big{(}\mathbf{A}\mathbf{e}+\mathbf{B}\Big{[}a_{1} \zeta_{1},+a_{2}\zeta_{2}+\mathbf{W}(t)^{\top}\mathbf{\Psi}\big{(}\mathbf{\dot{\zeta}} \big{)}+\mathbf{\dot{b}}(t)u(\mathbf{\zeta})+\mathbf{K_{id}}^{\top}\mathbf{\ddot{\zeta}}-f_{2} \left(\zeta_{1},\zeta_{2},t\right)\Big{]}\Big{)} \tag{102}\]8 (A21)  
By using the comparison principle, we can solve the differential inequality (A20), leading to  
\[\dot{V}= 2\mathbf{e}^{\top}\mathbf{P_{1}}\Big{(}\mathbf{A}\mathbf{e}+\mathbf{B}\Big{[}a_{1} \zeta_{1},+a_{2}\zeta_{2}+\mathbf{W}(t)^{\top}\mathbf{\Psi}\big{(}\mathbf{\dot{\zeta}} \big{)}+\mathbf{\dot{b}}(t)u(\mathbf{\zeta})+\mathbf{K_{id}}^{\top}\mathbf{\ddot{\zeta}}-f_{2} \left(\zeta_{1},\zeta_{2},t\right)\Big{]}\Big{)} \tag{102}\]9 (A22)  
In consequence,  
\[+2\mathbf{\dot{\zeta}}^{\top}\mathbf{P_{2}}\Big{(}\mathbf{A}\mathbf{\dot{\zeta}} +\mathbf{B}\mathbf{\ddot{W}}(t)u(\mathbf{\zeta})+\mathbf{B}\mathbf{\ddot{W}}(t)^{\top}\mathbf{\Psi} \Big{(}\mathbf{\dot{\zeta}}\Big{)}+\mathbf{B}\mathbf{W_{0}}\mathbf{\ddot{\Psi}}+\mathbf{B}\mathbf{\dot{ f}}_{\rm e}(\mathbf{\zeta},t)-\mathbf{B}\mathbf{K_{id}}^{\top}\mathbf{\ddot{\zeta}}\Big{)}\]0 (A23)  
In order to obtain the size of the convergence region around the origin for the identification and tracking errors as well as the weight \(\tilde{W}(t)\) and estimation \(\tilde{b}\) error trajectories, we use the Raleigh-Ritz inequality, and the Lyapunov functions (51) and (A23), obtaining that these deviations are _ultimately bounded_[53] as in (48) and (49).