## 3 Fixed-Time Stabilizing Controller Design
Now, similarly to the derivation of (16), it is not difficult to derive that  
\[\mathcal{L}_{2}(\mathbf{x}) =\alpha_{3}\Phi^{2+\kappa_{2}}(\mathbf{x})+I_{n}\]0 \[\mathcal{L}_{2}(\mathbf{x}) =\alpha_{3}\Phi^{2+\kappa_{2}}(\mathbf{x})+I_{n}\]1  
which leads to  
\[\mathcal{L}_{2}(\mathbf{x}) =\alpha_{3}\Phi^{2+\kappa_{2}}(\mathbf{x})+I_{n}\]2 \[\mathcal{L}_{2}(\mathbf{x}) =\alpha_{3}\Phi^{2+\kappa_{2}}(\mathbf{x})+I_{n}\]3  
In addition, it can be shown directly by Lemma 2 that  
\[\mathcal{L}_{2}(\mathbf{x}) =\alpha_{3}\Phi^{2+\kappa_{2}}(\mathbf{x})+I_{n}\]4 \[\mathcal{L}_{2}(\mathbf{x}) =\alpha_{3}\Phi^{2+\kappa_{2}}(\mathbf{x})+I_{n}\]5  
With (20)-(22) in mind, we have  
\[\mathcal{L}_{2}(\mathbf{x}) =\alpha_{3}\Phi^{2+\kappa_{2}}(\mathbf{x})+I_{n}\]6 \[\mathcal{L}_{2}(\mathbf{x}) =\alpha_{3}\Phi^{2+\kappa_{2}}(\mathbf{x})+I_{n}\]7  
and  
\[\mathcal{L}_{2}(\mathbf{x}) =\alpha_{3}\Phi^{2+\kappa_{2}}(\mathbf{x})+I_{n}\]8 \[\mathcal{L}_{2}(\mathbf{x}) =\alpha_{3}\Phi^{2+\kappa_{2}}(\mathbf{x})+I_{n}\]9  
which immediately leads to  
\[\Phi(\mathbf{x}) =\mathrm{diag}\left(\phi_{1}(\mathbf{x}),\ldots,\phi_{n}(\mathbf{ x})\right)\]0  
for all \((\mathbf{x},t)\in(\mathbb{R}^{2n}\times\mathbb{R}^{+})\setminus(\mathcal{N}_{1 }\cup\mathcal{N}_{2})\). Notably, it follows from [43] and (23) that with the initial state \(\mathbf{x}_{0}\in\mathbb{R}^{2n}\), the (non-unique) solutions \(\mathbf{x}(t)\) of the closed-loop system (1) under the (piecewise continuous) controller (9) are well-defined on \([0,\infty)\) and locally absolutely continuous; moreover, \(V(\mathbf{x}(t))\) is continuous, decreasing, and satisfies  
\[\Phi(\mathbf{x}) =\mathrm{diag}\left(\phi_{1}(\mathbf{x}),\ldots,\phi_{n}(\mathbf{ x})\right)\]1  
for all \(t\in[  
the origin of the closed-loop system (1) under the (piecewise continuous) controller (9) is fixed-time stable. **Remark 2**. : _Notably, the controller parameters are simply \(\kappa_{1}\) and \(\kappa_{2}\). Once \(\kappa_{1}\) and \(\kappa_{2}\) are determined, the associated settling-time estimate \(T_{max}\) can be computed accordingly. In practice, \(T_{max}\) can be suitably assigned by adjusting the parameters \(\kappa_{1}\) and \(\kappa_{2}\) thereby acquiring a smaller settling-time (convergence time) and its estimate, though this might increase the control effort accordingly._  
**Remark 3**. : _Although the technique of adding a power integrator was also employed in [24] to perform fixed-time stabilization, the system considered in [24] has only a single input. Unlike the results of [24], the approach developed in this paper is applicable, not only to a class of second-order multivariable multi-input systems, but also to systems with a single input. Moreover, the controller designed in [24] is continuous only so that the possible external disturbances were necessarily neglected in [24]. In contrast, by means of the approach presented in this paper, the resultant controller is discontinuous and therefore is capable of handling both the model uncertainties and external disturbances; of course, when there is no uncertainty/disturbance, the resultant controller becomes continuous._  
**Remark 4**. : _The presented controller (9) is constructed with the utilization of fractional powers so that the resultant control efforts provide a finite-time (fixed-time) state convergence; however, the convergence rate will be slower/t worse when the initial state is far away from the origin. As shown in [30], a potential strategy achieving a fast convergence simultaneously for the case of initial states being close to or far way from the origin is to design controllers in a uniform way with considering concurrently the feedbacks of both linear and fractional powers forms. Addressing this issue will be one of our future research directions._  
**Remark 5**.