  Maintaining the stability of synchronization state is crucial for the
functioning of many natural and artificial systems. In this study, we develop
methods to optimize the synchronization stability of the Kuramoto model by
minimizing the dominant Lyapunov exponent. With the help of the recently
proposed cut-set space approximation of the steady states, we greatly simplify
the objective function, and further derive its gradient and Hessian with
respect to natural frequencies, which leads to an efficient algorithm with the
quasi-Newton's method. The optimized systems are demonstrated to achieve better
synchronization stability for the Kuramoto model with or without inertia in
certain regimes. Hence our method is applicable in improving the stability of
power grids. It is also viable to adjust the coupling strength of each link to
improve the stability of the system. Various operational constraints can also
be easily integrated into our scope by employing the interior point method in
convex optimization. The properties of the optimized networks are also
discussed.
