## 2 Preliminaries  
### Problem Formulation  
Consider a class of nonlinear systems described by  
\[\dot{\mathbf{x}}_{1} =\mathbf{x}_{2}\] \[\dot{\mathbf{x}}_{2} =\mathbf{f}(\mathbf{x},t)+G(\mathbf{x},t)\mathbf{u}+\mathbf{d}( \mathbf{x},t) \tag{1}\]where \(\mathbf{x}_{1}=(x_{11},\ldots,x_{1n})^{T}\in\mathbb{R}^{n}\), \(\mathbf{x}_{2}=(x_{21},\ldots,x_{2n})^{T}\in\mathbb{R}^{n}\) and \(\mathbf{x}=(\mathbf{x}_{1}^{T},\mathbf{x}_{2}^{T})^{T}\in\mathbb{R}^{2n}\) denote the system states, \(\mathbf{u}\in\mathbb{R}^{m}\) is the control input, \(\mathbf{d}(\mathbf{x},t)=(d_{1}(\mathbf{x},t),\ldots,d_{n}(\mathbf{x},t))^{T} \in\mathbb{R}^{n}\) describes the model uncertainties and/or external disturbances, and \(\mathbf{f}(\mathbf{x},t)\) and \(G(\mathbf{x},t)\) are smooth functions with \(rank(G(\mathbf{x},t))=n\) for all \((\mathbf{x},t)\in\mathbb{R}^{2n}\times\mathbb{R}^{+}\), which in turn ensures the controllability of system (1) (see, e.g., [35]). The initial time described by \(t_{0}\) is set to be zero, i.e., \(t_{0}=0\), and the initial state of system (1) is denoted by \(\mathbf{x}(0)=\mathbf{x}_{0}\in\mathbb{R}^{2n}\). It is worth mentioning that a very large class of physical systems can be represented by system (1), including spacecrafts [36], robotic manipulators [37], etc. Additionally, the solutions of system (1) are understood in the sense of Filippov [38] since the control input \(\mathbf{u}=\mathbf{u}(\mathbf{x},t)\) is admitted to be discontinuous (piecewise continuous) and \(\mathbf{d}(\mathbf{x},t)\) is also assumed to be piecewise continuous and bounded as follows.  
**Assumption 1**.: _There exists a known constant \(\rho>0\) such that_  
\[|d_{i}(\mathbf{x},t)|\leq\rho\]  
_for all \((\mathbf{x},t)\in\mathbb{R}^{2n}\times\mathbb{R}^{+}\) and \(i=1,\ldots,n\)._  
Under Assumption 1, the main objective of this paper is to design a controller \(\mathbf{u}=\mathbf{u}(\mathbf{x},t)\) that renders the origin of system (1) fixed-time stable in the sense of the following definition.  
**Definition 1** ([23]).: _Consider the nonlinear system_  
\[\dot{\mathbf{x}}=\mathbf{h}(\mathbf{x},t) \tag{2}\]  
_where \(\mathbf{x}\in\mathbb{R}^{n}\), \(t\in\mathbb{R}^{+}\), and \(\mathbf{h}:\mathbb{R}^{n}\times\mathbb{R}^{+}\rightarrow\mathbb{R}^{n}\) is discontinuous (piecewise continuous). The initial time is \(t_{0}=0\) and the initial state is \(\mathbf{x}(0)=\mathbf{x}_{0}\). The solutions of system (2) are understood in the sense of [PERSON] [38]. Then, the origin of system (2) is said to be fixed-time stable if it is globally uniformly finite-time stable (see, e.g., [39]) and the settling-time function \(T(\mathbf{x}_{0})\) is globally uniformly bounded by a positive constant; i.e., there exists a positive constant \(T_{max}>0\) such that \(T(\mathbf{x}_{0})\leq T_{max}\) for all \(\mathbf{x}_{0}\in\mathbb{R}^{n}\)._  
**Remark 1**.: _Compared to global uniform finite-time stability, the key feature of fixed-time stability is the uniformity of its settling time. To see this point more clearly, the following two examples are considered. First, it is easy to see that the origin of the system_  
\[\dot{x}=-\frac{  
_which means that the origin of system (4) is fixed-time stable with the settling-time function \(T(x(0))\) satisfying \(T(x(0))\leq\pi/2\) uniformly in \(x(0)\)._