Contents I. Introduction
The aim of containment control problems for multiagent systems is to drive all agents into the convex hull of some given leaders. Numerous studies have been carried out for containment control problems [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. Early works [1], [2], [3], [4], [5], [6], [7], [8] mainly focused on the effects of communication graphs on the convergence of the containment control problem. For example, Ji et al. [1] first introduced the concept of containment control and gave a stop-and-go strategy. In [6], necessary and sufficient conditions were given for continuous-time multiagent systems with fixed communication graphs. In [9], [10], [11], and [12], containment control problems were studied for switching communication graphs by using the Lasalle's invariance principle, convex analysis, and nonsmooth analysis. However, the above results for distributed containment control were assumed to be free of constraints. Recently, some researchers have turned their attention to the case with input or state constraints [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. For example, in [13], [14], and [15], containment control problems with input saturation constraints were studied, but the control input of each agent was assumed to stay in a hypercube and the communication graphs were assumed to be fixed. In [16], general nonconvex input constraints and directed switching topologies were taken into account simultaneously for containment control problems. Different from the works of [13], [14], [15], [16], and [17], where the constraints considered were mainly on the velocities and inputs of the agents, the works of [18], [19], [20], [21], and [22] took the position constraints into account for the distributed consensus problems, distributed optimization problems, and containment control problems, respectively. Due to the possibility that the position constraints might not contain the origin, the position constraints have essential difference from the input and state constraints, and therefore the approaches in [13], [14], [15], [16], and [17] cannot be directly applied to deal with the position constraints. In addition, the problems studied in [18], [19], and [20] were the distributed consensus problems and the distributed optimization problems, where each agent is finally driven to a consensus point or an optimal point. In the containment control problem, the final agent positions might not converge to a single point, making the approaches in [18], [19], and [20] fail to tackle the position constraints for the containment control problem. In [21] and [22], containment control problems with position constraints were studied, but it is assumed that the intersection of all position constraints contains the convex hull formed by the leaders. This assumption is a bit strong and might be invalid in some special scenarios including, e.g., a scenario when the position constraint set is far smaller than the convex hull formed by the leaders. Moreover, the analysis approaches in [21] and [22] heavily relied on the convexity of the intersection of all the position constraints, and cannot be used to study the case when the intersection of all constraint sets might be empty.
