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Laplacian matrices play an important role in linear-consensus algorithms. This paper studies optimal linear-consensus algorithms for multivehicle systems with single-integrator dynamics in both continuous-time and discrete-time settings. We propose two global cost functions, namely, interaction-free and interaction-related cost functions. With the interaction-free cost function, we derive the optimal (nonsymmetric) Laplacian matrix by using a linear-quadratic-regulator-based method in both continuous-time and discrete-time settings. It is shown that the optimal (nonsymmetric) Laplacian matrix corresponds to a complete directed graph. In addition, we show that any symmetric Laplacian matrix is inverse optimal with respect to a properly chosen cost function. With the interaction-related cost function, we derive the optimal scaling factor for a prespecified symmetric Laplacian matrix associated with the interaction graph in both continuous-time and discrete-time settings. Illustrative examples are given as a proof of concept.
Cao et al. (2009) studied this question.