## 1 Introduction
In these articles, the convergence or stability aspects were shown by utilising either a Lyapunov function or a controller, which guarantees the decrease of costs in any time instant, or of terminal constraints or costs, which then forces the predicted trajectory to achieve an end point or region, or to steer that by a terminal cost penalty or a connective constraint, ensuring that, e.g., a minimum distance or explicit bounds on the optimal value function were used to calculate the suboptimality degree of a finite horizon controller. As the considered scenario here is noncooperative and as a Lyapunov function for one robot may be infeasible if one robot has to increase its costs by taking a detour and if terminal regions or costs will restrict the freedom of feasible trajectories, we use a weaker assumption, which is similar to the movement pattern from former numerical results based on the pattern of a roundabout; see Figures 4 and 7 in Reference [10]. We focus on a noncooperative setting as each robot exhibits an individual target without imposing terminal costs or constraints following Reference [21]. In this article, we derive sufficient conditions to avoid blockings or deadlocks with the assumption that the robots optimise in a fixed order in the spacial set with quantised communication. We avoid imposing Lyapunov conditions for the cost function, which would either impose additional constraints for the OCP or terminal costs. Hence, we implement a switched control, which is inspired from street traffic rules for a roundabout structure. If the robots are below a certain distance to each other, a circle with the mean-based middle point based on the end points of the predicted trajectories of the affected systems is created. Each participating system calculates individual intersection points with this circle and follows the circular curve via a circular control law until the original control law could be reinstated again. Moreover, as a second aspect, we derive a sufficient condition for the prediction horizon length based on a given cost function to examine the sufficient length, which allows the robot to recognise the decrease of costs to take the detour instead of holding their positions probably infinitely long. The paper is structured as follows: In the subsequent section, the problem is stated with the model of the robots including state and control constraints. We recap the construction of the occupancy grid shortly and give an outline on the construction of the constraints including a safety margin regarding the quantised communication. Then, these parts are assembled to define an OCP in Section 3, which is incorporated in the DMPC scheme on each robot. The theoretical contributions concerning a sufficient prediction horizon length and the convergence to achieve practical stability are presented in Section 4. For the convergence part, we consider two possibilities: First a fixed circle size is considered to possibly include all robots on the circle. Second, we extend this approach to a flexible circle, which considers the maximum size without interfering with targets of other robots. To complete the article, a short conclusion and outlook on future extensions is given. **Notation:**\(\mathbb{R}\) and \(\mathbb{N}\) denote the real and natural numbers, respectively. \(\mathbb{N}_{0}:=\mathbb{N}\cup\{0\}\) represents the nonnegative integers, and \(\mathbb{R}_{\geq 0}\) represents the nonnegative real numbers. For integers \(a,b\in\mathbb{N}_{0}\) with \(a\leq b\), the expression \([a:b]\) denotes the sequence \(\{a,a+1,\cdots,b\}\). Moreover, for a vector \(x\in\mathbb{R}^{n}\), \(n\in\mathbb{N}\), we define the infinity norm \(\|x\|_{\infty}:=\max_{i\in[1:n]}|x_{i}|\).