# Fixed-Time Optimization of Perturbed Multi-Agent Systems under the Resource Constraints
## 2 Preliminary and Problem Description  
### Some Related Lemmas  
Consider the following system:  
\[\dot{x}=g(t,x),x(0)=x_{0}, \tag{2}\]  
where \(x\in\mathbb{R}^{n}\), \(g:\mathbb{R}^{n}\to\mathbb{R}^{n}\) is a nonlinear function. Assume that the solution of the system (2) exists and is unique, and is continuous for all initial values \(x(0)\in\mathbb{R}^{n}\) of the system. The origin is the equilibrium point of the system (2), i.e., \(g(0_{n})=0_{n}\).  
Definition 1: If the equilibrium point \(x=0_{n}\) of the system (2) is globally asymptotically stable and any trajectory of the system (2) arrives at the equilibrium point in fixed-time, i.e., \(x(t,x_{0})=0_{n},\forall t\geq T(x_{0})\), then the equilibrium point \(x=0_{n}\) is said to be the global fixed-time stable equilibrium point, and \(T:\mathbb{R}_{n}\to\mathbb{R}_{>0}\) is the settling time.  
Definition 2: If any trajectory of the system (2) arrives at the set \(M\subset\mathbb{R}^{n}\) and stays in \(M\) in fixed-time, i.e., \(x(t,x_{0})\in M,\forall t\geq T(x_{0})\), then \(M\) is said to be the global fixed-time attraction domain of the system (2), and \(T:\mathbb{R}_{n}\to\mathbb{R}_{>0}\) is the settling time.  
Lemma 1 ([17,19]): If there exists a radial unbounded function \(V\in C^{1}(\mathbb{R}^{n},\mathbb{R}_{<0})\), such that \(V(x)=0\Rightarrow x\in M\). The time derivative of \(V\) along the system (2) satisfies for any \(c_{0}>0\), \(c_{1}>0\), \(c_{2}>0\), \(0<\mu<1\), \(\
u>1\) with  
(1) \(\dot{V}(x(t))\leq-c_{1}V^{\mu}(x(t))-c_{2}V^{\
u}(x(t))\), then the set \(M\) is a global fixed-time attraction domain of the system (2), and settling time \(T\leq\frac{1}{c_{1}(1-\mu)}+\frac{1}{c_{2}(\
u-1)}\);  
(2) \(\dot{V}(x(t))\leq-c_{0}V(x(t))-c_{1}V^{\mu}(x(t))-c_{2}V^{\
u}(x(t))\) holds, then the set \(M\) is a global fixed-time attraction domain of the system (2), and \(T\leqslant\frac{\ln(1+c_{0}/c_{1})}{c_{0}(1-\mu)}+\frac{\ln(1+c_{0}/c_{2})}{c_ {0}(\
u-1)}\).  
Definition 3: If function \(f\in C^{1}(\mathbb{R}^{n},\mathbb{R})\) is \(m\)-strongly convex, then there exists \(m>0\) such that \(f(y)\geq f(x)+\
abla f(x)^{\top}(y-x)+\frac{m}{2}\|x-y\|^{2},\forall x,y\in \mathbb{R}^{n}\).  
Lemma 2 ([23]): For any \(z_{i}\in\mathbb{R}_{\geq 0}\), \(i\in\{1,2,\cdots,N\}\), the following inequalities holds:  
\[\sum_{i=1}^{N}z_{i}^{p} \geq\left(\sum_{i=1}^{N}z_{i}\right)^{p},0<p\leq 1, \tag{3a}\] \[\sum_{i=1}^{N}z_{i}^{p} \geq N^{1-p}\left(\sum_{i=1}^{N}z_{i}\right)^{p},p>1. \tag{3b}\]  
Lemma 3 ([24]): The graph \(\mathcal{G}=(A,\mathcal{V})\) is an undirected connected graph. The Laplacian matrix of \(\mathcal{G}\)\(L_{A}\triangleq[l_{ij}]\in\mathbb{R}^{N\times N}\) is given by the following equation:\[l_{ij}=\begin{cases}\sum_{k=1,k\
eq i}a_{ik},&i=j,\\ -a_{ij}\quad,&i\
eq j.\end{cases} \tag{4}\]  
_The Laplacian matrix_ \(L_{A}\) _has the following properties:_  
1. \(L_{A}\) _is a symmetric semi-positive definite matrix,_ \(L_{A}\mathbf{1}_{N}=0_{N}\)_,_ \(\lambda_{2}(L_{A})>0\)_._
2. \(x^{\top}L_{A}x=\frac{1}{2}\sum_{i,j=1}^{n}a_{ij}(x_{i}-x_{j})^{2}\)_, if_ \(\mathbf{1}_{N}^{\top}x=0\)_, we have_ \(x^{\top}L_{A}x\geq\lambda_{2}(L_{A})x^{\top}x\)_._