## 3 Fixed-Time Stabilizing Controller Design  
We first summarize our approach to the construction of a fixed-time stabilizing controller for system (1) as follows. Theorem 1: _Under Assumption 1, the origin of system (1) is fixed-time stable with the settling-time estimate_  
\[T(\mathbf{x}_{0})\leq T_{max}:=\frac{(2n)^{\frac{\kappa_{2}}{2}}}{2^{\frac{(2 +\kappa_{1})\kappa_{1}}{2}-1}\kappa_{2}}-\frac{1}{2^{\frac{(2+\kappa_{1}) \kappa_{1}}{2}-1}\kappa_{1}} \tag{8}\]_if the controller \(\mathbf{u}(\mathbf{x},t)=\mathbf{u}\) is designed as (Because \(\kappa_{1}=\kappa_{11}/\kappa_{12}\), \(\kappa_{2}=\kappa_{21}/\kappa_{22}\) and \(\kappa_{i1}\) and \(\kappa_{i2}\), for \(i=1,2\), are positive even integers and positive odd integers, respectively, the controller (9) together with its parameters is well-defined. )_  
\[\mathbf{u}=-G^{+}(\mathbf{x},t)\left(\mathcal{L}_{1}(\mathbf{x})\xi_{2}^{2 \kappa_{1}+1}+\mathcal{L}_{2}(\mathbf{x})\xi_{2}^{\kappa_{1}+\kappa_{2}+1}+ \mathbf{f}(\mathbf{x},t)+\rho\,\mathbf{sign}\left(\xi_{2}\right)\right) \tag{9}\]  
_with_  
\[\xi_{2}=\mathbf{x}_{2}^{\frac{1}{1+\kappa_{1}}}+\left(2\mathbf{x}_{1}^{1+ \kappa_{1}}+2\mathbf{x}_{1}^{1+\kappa_{2}}\right)^{\frac{1}{1+\kappa_{1}}}\]  
_where \(\kappa_{1}=\kappa_{11}/\kappa_{12}\in(-1/2,0)\) and \(\kappa_{2}=\kappa_{21}/\kappa_{22}\in(0,1)\) are parameters with \(\kappa_{i1}\) and \(\kappa_{i2}\), for \(i=1,2\), being positive even integers and positive odd integers, respectively, and \(\mathcal{L}_{1}(\mathbf{x})\in\mathbb{R}^{n\times n}\) and \(\mathcal{L}_{2}(\mathbf{x})\in\mathbb{R}^{n\times n}\) are square matrices defined as_  
\[\mathcal{L}_{1}(\mathbf{x}) =\alpha_{1}\Phi^{2+\kappa_{1}}(\mathbf{x})+\alpha_{2}I_{n}+I_{n} +2^{-\kappa_{1}}(1-\kappa_{1})\Phi(\mathbf{x})\] \[\mathcal{L}_{2}(\mathbf{x}) =\alpha_{3}\Phi^{2+\kappa_{2}}(\mathbf{x})+I_{n}\] \[\Phi(\mathbf{x}) =\mathrm{diag}\left(\phi_{1}(\mathbf{x}),\ldots,\phi_{n}(\mathbf{ x})\right)\]  
_with \(\phi_{i}(\mathbf{x})\), \(\alpha_{1}\), \(\alpha_{2}\), and \(\alpha_{3}\) of the following form_  
\[\phi_{i}(\mathbf{x}) =\frac{2^{\frac{1}{(1+\kappa_{1})}}(\kappa_{2}-\kappa_{1})}{(1+ \kappa_{1})}x_{1i}^{\kappa_{2}-\kappa_{1}}\left(1+x_{1i}^{\kappa_{2}-\kappa_{ 1}}\right)^{\frac{-\kappa_{1}}{1+\kappa_{1}}}+2^{\frac{1}{1+\kappa_{1}}} \left(1+x_{1i}^{\kappa_{2}-\kappa_{1}}\right)^{\frac{1}{1+\kappa_{1}}}\] \[\alpha_{1} =\frac{2^{3-\kappa_{1}^{2}}(1+\kappa_{1})^{1+\kappa_{1}}(1-\kappa _{1})^{2+\kappa_{1}}}{(2+\kappa_{1})^{2+\kappa_{1}}}\] \[\alpha_{2} =\frac{2^{\frac{2}{(1+\kappa_{1})}-(1+\kappa_{1})}(1+\kappa_{1}) }{(2+\kappa_{1})^{1+\frac{1}{(1+\kappa_{1})}}}\] \[\alpha_{3} =\frac{2^{2+2\kappa_{2}-2\kappa_{1}-\kappa_{1}\kappa_{2}}(1+ \kappa_{2})^{1+\kappa_{2}}(1-\kappa_{1})^{2+\kappa_{2}}}{(2+\kappa_{2})^{2+ \kappa_{2}}}.\]  
Proof. : A new design philosophy for constructing a fixed-time stabilizer is presented in the proof. To be more specific, the technique of adding a power integrator [34] is skillfully modified and revamped by introducing extra manipulations in the construction of virtual controls so that a two-step design approach is developed whereby a fixed-time stabilizing controller is explicitly designed. Details are as follows. _Step 1:_ Choose the scalar function \(V_{1}:\mathbb{R}^{n}\rightarrow\mathbb{R}\) as below  
\[\mathbf{u}=-G^{+}(\mathbf{x},t)\left(\mathcal{L}_{1}(\mathbf{x})\xi_{2}^{2 \kappa_{1}+1}+\mathcal{L}_{2}(\mathbf{x})\xi_{2}^{\kappa_{1}+\kappa_{2}+1}+ \mathbf{f}(\mathbf{x},t)+\rho\,\mathbf{sign}\left(\xi_{2}\right)\right) \tag{9}\]0  
which is obviously positive definite, proper (A scalar function \(\gamma:\mathbb{R}^{n}\rightarrow\mathbb{R}^{+}\) is said to be proper if for any \(c>0\), the set \(\gamma^{-1}\left([0,c]\right)\) is compact in \(\mathbb{R}^{n}\).) and continuously differentiable.