## 2 Preliminaries
### Model Description  
In this study, we consider a multi-agent system consisting of \(B+P\) agents in which the dynamics of the agents are designed as  
\[ \begin{cases}\dot{x}_{i}(t)=f(x_{i}(t))+u_{i}(t)+\sigma\omega_{i}(t),\\ y_{i}(t)=Cx_{i}(t),\end{cases}  \tag{1}\]  
for \(i=1,2,\ldots,B,B+1,\ldots,B+P\), where \(x_{i}(t)\in R^{n}\), \(u_{i}(t)\in R^{n}\), \(\omega_{i}(t)\in R^{n}\), and \(y_{i}(t)\in R^{m}\) are the system state, control protocol, bounded exogenous disturbance, and output of agent \(i\), respectively. Matrix \(C\in R^{m\times n}\) is a constant matrix. \(\sigma\) is a detective mechanism that can be used to detect whether the system is affected by exogenous disturbances and is defined as follows:  
\[\sigma= \begin{cases}1,&\omega_{i}(t)\ \textit{exists}\\ 0,&\ \ \textit{otherwise}.\end{cases}  \tag{2}\]  
Let us suppose that a disturbance (\(\omega_{i}(t),\ i=1,2,\ldots,B,B+1,\ldots,B+P\)) is generated by the following linear exogenous system [33]:  
\[ \begin{cases}d_{i}(t)=Qd_{i}(t),\\ \omega_{i}(t)=Kd_{i}(t),\end{cases}  \tag{3}\]where \(d_{i}(t)\in R^{l}\) denotes the state of the exogenous disturbance and \(Q\in R^{l\times l}\) and \(K\in R^{n\times l}\) are known constant matrices of the exogenous system. The continuously vector-valued function (\(f:R^{n}\to R^{n}\)) represents a nonlinear term satisfying the following Lipschitz conditions. **Assumption 1**. : _For a known nonlinear function (\(f\)), there exists a non-negative constant (\(\rho\)) such that_  
\[\parallel f(a)-f(b)\parallel\leqslant\rho\parallel a-b\parallel,\ \forall a,b\in R^{n}. \tag{4}\]  
_It should be noted that \(\rho\) does not need to be given or known exactly, and an upper bound of \(\rho\) should be estimated in this study._  
We define \(\mathcal{V}_{1}=\{1,2,\ldots,B\}\) as group 1 and \(\mathcal{V}_{2}=\{B+1,B+2,\ldots,B+P\}\) as group 2. This clearly satisfies \(\mathcal{V}=\mathcal{V}_{1}\cup\mathcal{V}_{2}\) and \(\mathcal{V}_{1}\cap\mathcal{V}_{2}=\emptyset\). In this study, the state information of each agent is supposed to be unavailable, which is common in many scenarios and adds to the complexity of control. For the purpose of designing a prescribed-time bipartite output control protocol, we introduce a time-varying scaling function [25] as  
\[\mu(t)= \begin{cases}\dfrac{T^{h}}{(T+t_{0}-t)^{h}},&t\in[t_{0},t_{0}+T)\\ 1,&t\in[t_{0}+T,\infty)\end{cases}  \tag{5}\]  
where \(T>0\) is the prescribed time and \(h>2\). We define the right-hand derivative of \(\mu(t)\) at \(t=t_{0}+T\) as \(\dot{\mu}(t_{0}+T)\). Then,  
\[\dot{\mu}(t)= \begin{cases}\dfrac{h}{T}\mu^{1+\frac{1}{h}},&t\in[t_{0},t_{0}+T) \\ 0,&t\in[t_{0}+T,\infty).\end{cases}  \tag{6}\]  
**Remark 1**. : _The core role of \(\mu(t)\) is to ensure that the multi-agent system achieves bipartite output consensus within a user-defined time (\(T\)) by dynamically adjusting the gain of the control protocol. Its design guarantees accelerated convergence during \(t\in[t_{0},t_{0}+T)\) and maintains a stability of \(\mu(t)=1\) for \(t\in[t_{0},t_{0}+T)\). Compared with traditional finite-time or fixed-time control frameworks, this mechanism eliminates the dependence on initial conditions or system parameters, allowing the convergence time to be fully specified by the user. As \(t\) approaches \(t_{0}+T\), the denominator (\(T+t_{0}-t\)) approaches zero, causing \(\mu(t)\) to grow rapidly. This amplifies the gain term in the control protocol, accelerating system convergence. For \(t>t_{0}+T\), \(\mu(t)\equiv 1\), ensuring the system remains stable after consensus is achieved. The condition of \(h>2\) ensures that \(\mu(t)\) becomes sufficiently steep near \(t=t_{0}+T\), enhancing control over convergence speed and avoiding overshoot or delays caused by insufficient gain._  
**Lemma 1** ([34]). : _Let us consider a non-negative continuously differentiable function (\(V(t)\)).