In this paper, by using the optimization idea, a new ultimate bound for the complex Lorenz system is derived. It is shown that a hyperelliptic estimate of the ultimate bound set can be analytically calculated based on the optimization method and the Lagrange multiplier method. And based on the ellipsoidal bound set and set operations, one further obtains a more conservative boundary for each variable in the complex system, which only relies on the system parameters. Afterwards, the estimated results are applied to the exponential finite-time synchronization of the complex Lorenz system. Especially, the designed control depends on the parameters of the exponential convergence rate, the finite-time convergence rate, the bound of the initial states of the master system, and the system parameter. Finally, numerical simulations are given to verify the effectiveness and correctness of the obtained results

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