## 2 Preliminaries and Problem Description
### A New Integral Barrier Lyapunov Function  
In order to ensure the appointed-time performance constraint condition on tracking error is always satisfied, a new time-varying integral barrier Lyapunov function is constructed as:  
\[V=\int_{0}^{e}\frac{k^{2}\left(t\right)\xi}{k^{2}\left(t\right)-\xi^{2}}d\xi \tag{2}\]  
where \(e=x-x_{d}\) is tracking error with \(x\) and \(x_{d}\) being system state and desired trajectory, respectively. We know that the candidate Lyapunov function \(V\) is positive definite, continuously differentiable, and radially unbounded in the set \(\Omega=\left\{e:\left|e\right|<k\left(t\right)\right\}\); therefore, \(V\) is a valid candidate Lyapunov function. Next, the useful Theorems for stability analysis of the control system are proposed. **Theorem 1**. : _The candidate Lyapunov function \(V\), defined in (2), is positive definite, continuously differentiable, and radially unbounded in the set \(\Omega\). As for \(\left|e\right|<k\left(t\right)\), there is:_  
\[\frac{e^{2}}{2}\leq V\leq\frac{k^{2}\left(t\right)e^{2}}{k^{2}\left(t\right)-e ^{2}} \tag{3}\]  
**Proof of Theorem 1. **  
**Step 1:** We indicate that inequality \(\frac{e^{2}}{2}\leq V\) holds. We define the following function:  
\[g\left(e\right)=\int_{0}^{e}\frac{k^{2}\left(t\right)\xi}{k^{2}\left(t\right)- \xi^{2}}d\xi-\frac{e^{2}}{2} \tag{4}\]  
Calculating the first partial derivative of the \(g\left(e\right)\) with respect to \(e\), the outcome is:  
\[\frac{\partial g\left(e\right)}{\partial e}=\frac{k^{2}\left(t\right)e}{k^{2} \left(t\right)-e^{2}}-e=\frac{e^{3}}{k^{2}\left(t\right)-e^{2}} \tag{5}\]  
_Case 1:_\(e<0\), we have \(\frac{\partial g\left(e\right)}{\partial e}<0\) in the set \(\Omega\). _Case 2:_\(e>0\), we have \(\frac{\partial g\left(e\right)}{\partial e}>0\) in the set \(\Omega\). Furthermore, since \(e=0\), \(g\left(e\right)=0\), we conclude that \(g\left(e\right)\geq 0\), namely, \(\frac{e^{2}}{2}\leq V\). **Step 2:** We prove that inequality \(V\leq\frac{k^{2}\left(t\right)e^{2}}{k^{2}\left(t\right)-e^{2}}\) holds. We define the following function:  
\[p\left(e\right)=\frac{k^{2}\left(t\right)e^{2}}{k^{2}\left(t\right)-e^{2}}- \int_{0}^{e}\frac{k^{2}\left(t\right)\xi}{k^{2}\left(t\right)-\xi^{2}}d\xi \tag{6}\]  
Taking the first partial derivative of \(p\left(e\right)\) with respect to \(e\) yields:  
\[ \begin{split}\frac{\partial p\left(e\right)}{\partial e}& =\frac{2 ek^{2}\left(t\right)\left(k^{2}\left(t\right)-e^{2}\right)- k^{2}\left(t\right)e^{2}\left(-2e\right)}{\left(k^{2}\left(t\right)-e^{2} \right)^{2}}-\frac{k^{2}\left(t\right)e}{k^{2}\left(t\right)-e^{2}}\\ &=\frac{k^{2}\left(t\right)e\left(k^{2}\left(t\right)+e^{2} \right)}{\left(k^{2}\left(t\right)-e^{2}\right)^{2}}\end{split}  \tag{7}\]  
_Case 1:_\(e<0\), we have \(\frac{\partial p\left(e\right)}{\partial e}<0\) in the set \(\Omega\). _Case 2:_\(e>0\), we have \(\frac{\partial p\left(e\right)}{\partial e}>0\) in the set \(\Omega\). Furthermore, since \(e=0\), \(p\left(e\right)=0\), it is straightforward to obtain \(p\left(e\right)\geq 0\), namely \(V\leq\frac{k^{2}\left(t\right)e^{2}}{k^{2}\left(t\right)-e^{2}}\). This completes the proof of Theorem 1. **Remark 2**. : _Unlike the existing works in [23; 24; 25; 26; 27], in this paper, the performance constraint boundary \(k\left(t\right)\) is a time-varying function, and the new integral barrier Lyapunov function can directly handle the tracking errors ofthe nonlinear system. In addition, the proposed Theorem 1, which is different from previous theories, is devoted to stability analysis of the control system._  
Theorem 2: _For any positive bounded function \(k\left(t\right)\), let \(\chi:=\left\{x\in R:\left|x\right|<k\left(t\right)\right\}\subset R\) and \(\mathrm{N}:=R^{l}\times\chi\subset R^{l+1}\) as open sets.