the third result of this paper.

*Theorem 7.1*. Consider the nonlinear uncertain dynamical system given by (\[unc:sys\]) subject to (\[unc:prm:rlx\]), the (modified) reference system given by (\[ref:nonideal\]), and the feedback control law given by (\[total:ctrl\]) along with (\[nom:ctrl\]), (\[adapt:ctrl\]), and (\[update:law:2\]). Then, the solution $\bigl(e(t),\tilde{W}(t),e_\rom{L}(t),\tilde{x}(t)\bigl)$ given by (\[err:sys:mod\]), (\[filter\]), (\[update:law:err2\]), and (\[deviation:error\]) is uniformly bounded 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 $t \in \overline{\IR}_+$, with ultimate bound \_[\_]{}(1+), \[ultim:bound\] where $\xi\in(0,1)$ and \_&&\^[-1]{}\_\^2\_( 1+4\^[-1]{}\_\^2\_\_\^[-2]{}(R) ).

*Proof.* Consider the Lyapunov-like function candidate given by (\[proof:1\]). Then, using the property of the projection operator given by (\[defnt:b\]) and following similar steps in the proof of Theorem 5.1, we have \^\*()&&-c\_1\_2\^2-c\_2\^2\_2-c\_3\_2\^2+c\_4,      \[pr:1\] where $c_1\triangleq\lambda_\rom{min}(R)$, $c_2 \triangleq \eta^{-1}\kappa \lambda_\rom{min}(R)$, $c_3 \triangleq 2 \xi \kappa^{-1}$ $\cdot \lambda_\rom{min}^2(R)\lambda_\rom{max}^{-1}(P)(1-\xi)$, and $c_4 \triangleq 2 \gamma^{-1} \norm{\Lambda}_\rom{F} \tilde{w}_\rom{max} \dot{w}_\rom{max}$, and hence, either $\norm{e(t)}_2 \ge \psi_1 \triangleq \bigl(c_4/c_1\bigl)^2$ or $\norm{e_\rom{L}(t)}_2 \ge \psi_2 \triangleq \bigl(c_4/c_2\bigl)^2$ or $\norm{\tilde{x}(t)}_2 \ge \psi_3 \triangleq \bigl(c_4/c_3\bigl)^2$ yields $\dot{\mathcal{V}}^*(\cdot)\le0$. Since the update law for $\hat{W}(t)$ is predicated on the projection operator, $\norm{\tilde{W}(t)}\le\tilde{w}_\rom{max}$. That is, $\dot{\mathcal{V}}^*(\cdot)\le0$ outside the compact set defined by \_ && {(e(t),(t),e\_(t),(t))\^n \^[(s+n) m]{}\^n\^n: \_2\_1\
&&  \_2\_2  \_2\_3}   {(e(t),(t),e\_(t),(t))\^n\
&& \^[(s+n) m]{}\^n\^n: \_}. \[set:set\] Therefore, $\mathcal{V}^*\bigl(e(t),\tilde{W}(t),e_\rom{L}(t),\tilde{x}(t)\bigl)$ cannot grow outside $\mathcal{D}_\mathcal{V}$, and hence, 