# Fixed-Time Optimization of Perturbed Multi-Agent Systems under the Resource Constraints
## 3 Main Results
### Three-Stage Fixed-Time Robust MSCO Algorithm  
By choosing a suitable \(\gamma\), solving the problem (10) yields an \(e\)-approximate solution to the original problem (5). Using the fixed-time integral sliding mode control [28], the fixed-time robust MSCO algorithm for solving the problem (10) under the undirected connected graph is proposed as follows:  
\[\left\{ \begin{aligned} u_{i}=& u_{i1}+u_{i2},\\ u_{i1}=&-\xi_{1}\text{sign}^{n}(e_{i})-\xi_{2}\text{ sign}^{n}(e_{i})+\eta_{i}\\ e_{i}=& x_{i}-d_{i}-\int_{0}^{t}\eta_{i}d\tau,\\ \eta_{i}=&\zeta_{1}\sum_{j\in\mathcal{N}_{i}}\text{ sign}^{n}(\phi_{j}-\phi_{i})+\zeta_{2}\sum_{j\in\mathcal{N}_{i}}\text{sign}^{n}( \phi_{j}-\phi_{i}),\\ \phi_{i}=&\frac{\partial f_{i}^{p}}{\partial x_{i}}+ \sum_{j\in\mathcal{N}_{i}}\frac{\partial f_{j}^{p}}{\partial x_{i}},\\ u_{i2}=&-\kappa_{0}s_{i}-\kappa_{1}\text{sgn}(s_{i} )-\kappa_{2}\text{sign}^{n}(s_{i}),\\ s_{i}=& x_{i}-\int_{0}^{t}u_{i1}(\tau)d\tau,\end{aligned} \right. \tag{16}\]  
where, \(\kappa_{0},\kappa_{2},\xi_{1},\xi_{2},\zeta_{1},\zeta_{2}\in\mathbb{R}_{>0}\), \(\kappa_{1}>\overline{\delta}\), \(\alpha>1\), \(0<\mu<1\), \(\
u>1\) are adjustable parameters, \(s_{i}\) denotes the integral sliding manifold, \(\phi_{i}\) denotes the summation of the local and neighboring gradient information of agent \(i\), and \(u_{i1}\), \(u_{i2}\), \(e_{i}\), \(\eta_{i}\) are auxiliary variables. **Remark 5**: _The Algorithm (16) is based on a three-stage strategy design. Inspired by the integral sliding mode control, the main task of the first stage is to complete the rejection of the external disturbance \(w_{i}(t)\). Using the symmetry of the undirected graph, the second stage is designed so that the global resource demand constraint is satisfied. The third stage drives the agents to the global optimum \(\hat{x}^{\star}\). In summary, since the above three stages are completed in fixed-time, the Algorithm (16) will drive the agents to solve the problem (10) in fixed-time._  
**Theorem 2**: _Assuming that Assumptions 1-4 hold, the problem (5) can be solved by distributed Algorithm (16) in fixed-time._  
The proof can be divided into three steps. 1. [label=Step 1]
2. Reaching law \(u_{i2}(t)\) is used to guarantee the fixed-time convergence of sliding manifold \(s_{i}\), where the effect of the external perturbation \(\omega_{i}(t)\) will be totally eliminated thereafter. The dynamics of \(s_{i}\) are obtained from the algorithm (16)  
\[s_{i}(t) =\dot{x}_{i}(t)-u_{i1}(t)\] \[=u_{i2}(t)+\omega_{i}(t)\] \[=-\kappa_{0}s_{i}-\kappa_{1}\text{sgn}(s_{i})-\kappa_{2}\text{ sign}^{a}(s_{i})+\omega_{i}(t). \tag{17}\]  
Consider the Lyapunov function \(V_{1,i}(t)=s_{i}^{2}\). The derivative of \(V_{1,i}(t)\) along the dynamics (17) is  
\[\dot{V}_{1,i}(t) =2s_{i}\dot{s}_{i}\] \[=-2\kappa_{0}s_{i}^{2}-2\kappa_{1}|s_{i}|-2\kappa_{2}|s_{i}|^{a+1 }+2s_{i}\omega_{i}\] \[\leq-2\kappa_{0}s_{i}^{2}-2(\kappa_{1}-\overline{\delta})(s_{i}^{ 2})^{1/2}-2\kappa_{2}(s_{i}^{2})^{\frac{a+1}{2}}\] \[=-2\kappa_{0}V_{1,i}-2(\kappa_{1}-\overline{\delta})V_{1,i}^{1/2}- 2\kappa_{2}V_{1,i}^{\frac{a+1}{2}}. \tag{18}\]By applying the Cauchy-Schwarz inequality \(s_{i}\omega_{i}\leq|s_{i}||\omega_{i}|\) and leveraging Lemma 4, it can be shown that inequality (18) holds. Furthermore, Lemma 1 states that if \(V_{1,i}(t)=0\) implies \(s_{i}=0\), then \(s_{i}\) can converge to \(0\) in fixed-time \(T_{1}\) even in the presence of external perturbations \(\omega_{i}(t)\). As a result, we can derive the following:  
\[T_{1}\leq\frac{\ln\Bigl{(}1+\frac{\kappa_{0}}{\kappa_{1}-\delta}\Bigr{)}}{ \kappa_{0}}+\frac{\ln\Bigl{(}1+\frac{\kappa_{0}}{\kappa_{2}}\Bigr{)}}{\kappa_{ 0}(\alpha-1)}. \tag{19}\]  
Thus, we have \(\dot{x}_{i}(t)=u_{i1}(t),\forall t\geq T_{1}\). 1. By using the symmetry property of control input \(u_{i1}\), it is proven that the global resource constraint \(\sum_{x_{i}}=D\) will be satisfied in fixed time.