## 1 Introduction  
Today, control problems with high dynamics have been of high interest in production and logistic processes or in traffic scenarios. In manufacturing processes, the technical advances have increased the utilisation of robots, especially in assembly lines in the last decades [1]. In logistic processes, autonomous, connected pallet carriers for material have increasingly been used to supply production lines with material. In common traffic scenarios, autonomous connected cars (or mobile robots) have aggregated different research fields, e.g., sensor fusion, modelling, and handling communication. One central issue is to be able to calculate a feasible solution which ensures safety in terms of collision avoidances. Examples for robotic systems can be found in formation control of robots [2], exploration of unknown environment to achieve an optimal covering with limited number of robots [3], or real-time control of distributed helicopters [4]. All these distributed systems are linked by common properties and arising problems: complexity and availability of information. Concerning complexity, these system have to be handled in a distributed manner. One approach is to use multi-agent systems to decrease the complexity and to enable the distributed system to calculate a solution sufficiently fast. The available computational capabilities on distributed systems like robots or cars ease this approach. Additionally, the distributed system is relying on the requirement of an information exchange: In contrast to a central system, where the information about the full system state is globally available, here, each agent has to rely on local information (decentralised system) or some communication methods have to be implemented such that the agents are able to gather information from neighboursor aggregated information about the full system state. If stability is taken into account, such information exchange is essential to achieve and guarantee that stability could be kept; see References [5; 6; 7]. The scenario here describes a group of robots in a shared operation space (spacial set), which shall be steered to individual targets while ensuring collision avoidance. This is implemented in a noncooperative control scheme, where each robot optimises a local optimal control problem (OCP) to minimise a cost function regarding the current distance to an individual target. Therefore, additional communication among the robots is necessary to inform the other robots by the calculated prediction to let them regard the collision avoidance via coupling constraints in the OCP. On the other hand, cooperative control schemes are targetting a common goal, which could be imposed for example via a common objective; see Reference [8] for flight formation of robots. The optimisation is executed in the spacial set, while for the communication among the robots, a quantisation of the continuous space to a grid with equidistant cells is established. The predicted trajectories are quantised obtaining grid indices and used to communicate them between the robots. This approach, introduced in References [9; 10], allows to keep the problem suitable for optimisation algorithms in a continuous setting while taking advantage in the communication reduction via discretisation. The optimisation and therefore the communication is carried out in a sequential manner according to Reference [11]; the future predicted state trajectories of each robot (here, discrete occupancy tuples) are communicated iteratively for each time instant. While in the former work [9; 10] conditions considering initial and recursive feasibility were examined, the focus here is set to derive conditions for convergence and practical stability of the overall system. We focus on theoretical properties of this approach to ensure two aspects: deadlock-free execution and convergence. Considering convergence properties in Distributed Model Predictive Control (DMPC), many advances were made in the last years, c.f. Reference [12].