Consider the nonlinear uncertain dynamical system given by (\[unc:sys\]) subject to (\[unc:prm\]), the (modified) reference system given by (\[ref:nonideal\]), and the feedback control law given by (\[total:ctrl\]) along with (\[nom:ctrl\]), (\[adapt:ctrl\]), and (\[update:law\]). Then, the solution $\bigl(e(t),\tilde{W}(t),e_\rom{L}(t),\tilde{x}(t)\bigl)$ given by (\[err:sys:mod\]), (\[filter\]), (\[update:law:error\]), and (\[deviation:error\]) is Lyapunov stable for all $\bigl(e_0,\tilde{W}_0,0,0\bigl) \in \IR^n \times \IR^{(s+n)\times m}\times\IR^n\times\IR^n$ and $t \in \overline{\IR}_+$, and $\lim_{t\rightarrow\infty}\bigl(x(t)-x_\rom{r_i}(t)\bigl)=0$. For $t \in \overline{\IR}_+$, in addition, \_[\_]{}(1+), \[bound:1\] where $\xi\in(0,1)$ and $\epsilon_\mathcal{V}\triangleq\gamma^{-1}\norm{\tilde{W}_0\Lambda^\frac{1}{2}}_\rom{F}^2+\lambda_\rom{max}(P)\norm{e_0}_2^2$.

*Proof*. To show Lyapunov stability of the solution $\bigl(e(t),\tilde{W}(t),e_\rom{L}(t),\tilde{x}(t)\bigl)$ given by (\[err:sys:mod\]), (\[filter\]), (\[update:law:error\]), and (\[deviation:error\]) for all $\bigl(e_0,\tilde{W}_0,0,0\bigl) \in \IR^n \times \IR^{(s+n)\times m}\times\IR^n\times\IR^n$ and $t \in \overline{\IR}_+$, consider the Lyapunov function candidate \^\*(e,,e\_,)&=&(e,)+\^[-1]{}e\_P e\_+2\^[-1]{} \_\^[-1]{}(P)\_(R)P , \[proof:1\]     where $\mathcal{V}\bigl(e,\tilde{W}\bigl)$ is given by (\[lyap:cand\]), and note that $\mathcal{V}^*(0,0,0,0)=0$, $\mathcal{V}^*\bigl(e,\tilde{W},e_\rom{L},\tilde{x}\bigl)>0$ for all $\bigl(e,\tilde{W},e_\rom{L},\tilde{x}\bigl)$ $\neq(0,0,0,0)$, and $\mathcal{V}^*\bigl(e,\tilde{W},e_\rom{L},\tilde{x}\bigl)$ is radially unbounded. Differentiating (\[proof:1\]) along the trajectories of (\[err:sys:mod\]), (\[filter\]), (\[update:law:error\]), and (\[deviation:error\]) yields \^\*()&=&-e(t)Re(t)-\^[-1]{}e\_(t)Re\_(t)-2\^[-1]{}\_\^[-1]{}(P)\_(R)(t) R (t)\
&& -2e(t)Pe\_(t) +2e\_(t)Pe\_(t)+4\_\^[-1]{}(P)\_(R)(t) Pe\_(t)\
&=&-e(t)Re(t)-\^[-1]{}e\_(t)Re\_(t)-2\^[-1]{} \_\^[-1]{}(P)\_(R)(t) R (t)\
&& -2e\_(t)Pe\_(t) +2\_\^[-1]{}(P)\_(R). \[proof:3\] where $e_\rom{H}(t)\triangleq e(t)-e_\rom{L}(t)$[^5]. Now, consider 2P\^P\^e\_&&P+e\_P e\_, \_+,     \[proof:4\] which follows from Young’s inequality \[, Fact 1.4.7\]. Using (\[proof:4\]) in the last term of (\[proof:3\]) and then etting $\mu=\xi\kappa^{-1}\lambda_\rom{max}^{-1}(P) \lambda_\rom{min}(R)$ in (\[proof:4\]) yields \^\*()&& -\_(R)\_2\^2 - \^[-1]{}\_(R)\^2\_2\
&& -2\^[-1]{} \_\^2(R)\_\^[-1]{}(P)\^2\_2, \[proof:10\] and 