## 2 Main Result
### Robustness Lyapunov Analysis for Error System
#### 2.5.5. Combining Bounds to Determine Radius of Convergence \((R_{*})\)  
Using the result from Equations (38) and (45) in Equation (32) results in the following:  
\[\begin{split}\dot{V}_{el}+2 aV_{el}&\leq|(P_{x}e_{x },v)|+a|\operatorname{tr}(L_{*}\gamma_{y}^{-1}\Delta L^{\dagger})|\\ &\leq\lambda_{\max}(P_{x})^{\frac{1}{2}}M_{v}(2V_{el})^{\frac{1}{ 2}}+a\bigg{(}\frac{M_{v}}{a}\bigg{)}(2V_{el})^{\frac{1}{2}}\\ &\leq\Big{(}\lambda_{\max}(P_{x})^{\frac{1}{2}}+1\Big{)}M_{v}(2V_ {el})^{\frac{1}{2}}.\end{split} \tag{46}\]Following, Equation (46) can also be expressed as follows:  
\[\frac{\dot{V}_{eL}+2 aV_{eL}}{V_{eL}^{\frac{1}{2}}}\leq\sqrt{2}(\lambda_{\max}(P_{ x})^{\frac{1}{2}}+1)M_{v}. \tag{47}\]  
Consider an alternative expression for the left-hand side of Equation (47):  
\[\begin{split}\frac{d}{dt}(2e^{\mu t}V_{eL}^{\frac{1}{2}})& =e^{\mu t}V_{eL}^{-\frac{1}{2}}\dot{V}_{eL}+2a^{\mu t}V_{e}^{\frac{1}{2}}\\ &=\frac{\dot{V}_{eL}+2 aV_{eL}}{V_{eL}^{\frac{1}{2}}}e^{\mu t}. \end{split} \tag{48}\]  
Therefore, Equation (47) can be expressed as follows:  
\[\frac{d}{dt}(2e^{\mu t}V_{eL}^{\frac{1}{2}})\leq\sqrt{2}(\lambda_{\max}(P_{x}) ^{\frac{1}{2}}+1)M_{v}e^{\mu t}. \tag{49}\]  
Integrating and evaluating both sides of Equation (49) from 0 to \(\tau\) results in the following:  
\[(2e^{\mu\tau}V_{eL}(\tau)^{\frac{1}{2}}-2V_{eL}(0)^{\frac{1}{2}})\leq\frac{ \sqrt{2}(\lambda_{\max}(P_{x})^{\frac{1}{2}}+1)M_{v}}{a}(e^{\mu\tau}-1). \tag{50}\]  
Solving for \(V_{eL}(\tau)^{\frac{1}{2}}\):  
\[V_{eL}(\tau)^{\frac{1}{2}}\leq e^{-\pi\tau}V_{eL}(0)^{\frac{1}{2}}+\frac{ \sqrt{2}(\lambda_{\max}(P_{x})^{\frac{1}{2}}+1)M_{v}}{2a}(1-e^{-\sigma\tau}). \tag{51}\]  
From Equation (51), \(V_{eL}^{\frac{1}{2}}\) can be expressed as being bounded for all \(\tau\geq 0\), which indicates that \(V_{eL}\) is also bounded. Given \(V_{eL}\) is a function of \(\{e_{x},\Delta L\}\), it follows that both \(\{e_{x},\Delta L\}\) are likewise bounded.  
Continuing, \(V_{eL}(\tau)^{\frac{1}{2}}\) can be lower bounded by considering the following:  
\[\frac{1}{\sqrt{2}}\lambda_{\min}(P_{x})^{\frac{1}{2}}\|e_{x}\|\leq V_{e}(\tau )^{\frac{1}{2}}\leq V_{e}(\tau)^{\frac{1}{2}}+V_{L}(\tau)^{\frac{1}{2}}\leq V _{eL}(\tau)^{\frac{1}{2}}. \tag{52}\]  
Substituting the lower bound for Equation (52) lower bound into Equation (51),  
\[\frac{1}{\sqrt{2}}\lambda_{\min}(P_{x})^{\frac{1}{2}}\|e_{x}\|\leq e^{-\sigma \tau}V_{eL}(0)^{\frac{1}{2}}+\frac{\sqrt{2}(\lambda_{\max}(P_{x})^{\frac{1}{2 }}+1)M_{v}}{2a}(1-e^{-\sigma\tau}). \tag{53}\]  
Solving for \(\|e_{x}\|\) results in the following:  
\[\|e_{x}\|\leq\frac{\sqrt{2}V_{eL}(0)^{\frac{1}{2}}}{\lambda_{\min}(P_{x})^{ \frac{1}{2}}}e^{-\sigma\tau}+\frac{(\lambda_{\max}(P_{x})^{\frac{1}{2}}+1)M_{ v}}{\lambda_{\min}(P_{x})^{\frac{1}{2}}a}(1-e^{-\sigma\tau}). \tag{54}\]  
Taking the _supremum_ of \(\|e_{x}\|\) as time approaches infinity results in the following:  
\[\limsup_{t\to\infty}\|e_{x}\|\leq\frac{1+\sqrt{\lambda_{\max}(P_{x})}}{a\sqrt {\lambda_{\min}(P_{x})}}M_{v}\equiv R_{*}. \tag{55}\]  
Therefore, upper bounding the error state norm to a _radius of convergence_\((R_{*})\) around a neighborhood of zero.  
Alternatively, if the following adaptive law is considered,  
\[\Delta L=-\delta_{y}y^{\dagger}\gamma_{y}, \tag{56}\]where \(aL=0\) from Equation (18), robustness proof can be modified accordingly, resulting in the following radius of convergence:  
\[\limsup_{t\to\infty}\|e_{x}\|\leq\frac{\sqrt{\lambda_{\max}(P_{x})}}{a\sqrt{ \lambda_{\min}(P_{x})}}M_{v}\equiv R_{*}. \tag{57}\]  
This version of the robustness proof has looser constraints, as Equation (43) need not be satisfied. Regardless of the adaptive law selected, robustness proof can be extended without loss of generality to consider the error system with fixed gains, as in Equation (11).