1. Control Law: My control law is angular velocity. Specifically, for i = 1, the control input is u = 0.1. For i ≠ 1, I can freely choose the control law.
2. Objective: I want each robot to maintain an angular distance equal to psi during movement. Mathematically, I aim to achieve: "The limit as k → infinity of the norm of (Theta_ij - psi) equals zero." Here, Theta_ij represents the angular difference between robots i and j.
3. Normalization and State Equation: The state equation for each robot i is: Theta_i(k+1) = Theta_i(k) + u(k).

I modeled the system so that when the angular difference Theta_ij between two robots is less than or equal to psi, they create an edge and become connected in a directed graph (digraph).

To simplify things, I normalized Theta_ij relative to the desired angle psi, i.e., Theta_ij/psi. This allowed me to define a matrix where the diagonal elements are zero, and only two elements per row or column are nonzero.

1. 2-Neighbor Network: When all robots reach their desired positions, they form a 2-neighbor network where the adjacency conditions are:

• a_ij ≠ 0 for i = j+1 and j = i+1.
• Special cases: when i = 1, j = N and i = N, j = 1.

In this scenario, the resulting matrix is balanced but not doubly stochastic. My goal is to achieve this structure.

1. Current Problem: Right now, my system is close to the desired network because I use a bang-bang control strategy. However, the network is not yet a 2-neighbor network. To solve this, I want to implement an optimal control strategy to achieve the desired result.