Comments on “On the Necessity of Diffusive Couplings in Linear Synchronization Problems With Quadratic Cost

In this note, we want to comment on the recent paper “On the necessity of diffusive couplings in linear synchronization problems with quadratic cost” (IEEE Transactions on Automatic Control, vol. 60, pp. 3029-3034, 2015) by Montenbruck et al. on the necessity of diffusiveness of optimal control laws in linear quadratic control problems in the context of synchronization. In the above paper, the authors concentrate on the optimal feedback law associated with the maximal solution of the underlying algebraic Riccati equation (ARE). In this comment, we argue that it is more natural to use the smallest positive semidefinite solution of the ARE. Our approach generalizes the results in the above paper to the case in which the agent dynamics is allowed to have eigenvalues in the open right half plane. Moreover, our proof considerably simplifies the one in the above paper, as it avoids the analysis of the Riccati differential equation. In addition, we propose and solve the zero endpoint version of the linear quadratic problem studied in the above paper.