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Among the most important concerns on the Lyapunov manage is picking an ideal Lyapunov perform to design and style the manage laws. Ordinarily, the control laws as well as the handle results are diverse when the Lyapunov functions are distinct. It is a great concept to decide on the selleck chemical Lyapunov perform based mostly on the geometrical and bodily meanings. Ordinarily, there are actually mainly three Lyapunov functions to become selected: the Lyapunov perform based mostly to the state distance [19�C21, 26�C32, 39�C41], the state error [22, 23, 37, 38], along with the average value of an imaginary mechanical amount [24, 25, 39, 41]. The so-called imaginary mechanical quantity means that it truly is a linear Hermitian operator to become designed and could be not a physically meaningful observable amount this kind of as coordinate and power.

Amid these 3 Lyapunov functions, the Lyapunov-based quantum manage procedures based to the state distance and state error only need to alter the scale factors from the control laws. These two Lyapunov handle techniques are somewhat straightforward and simple to grasp. The Lyapunov-based quantum handle primarily based over the average value of an imaginary mechanical amount is made up of additional adjustable CH5138303parameters. So it's more versatile and in addition a lot more complicated with the same time. Commonly speaking, the Lyapunov-based manage approach can only make sure that the manage program is stable. The probability manage inside the quantum process demands us to style a handle method which may makenearly the method convergent, mainly because a stable quantum management system could result in the manage method can not attain the preferred target state.

As a result an additional significant concern of this handle system for the closed quantum techniques is the convergence on the control methods. To date, there have already been the following study benefits about the convergence of your closed quantum techniques [19, 23�C25, 27�C29, 33, 36].For that Schr?dinger equation, the convergence conditions are as follows. (i) The inner Hamiltonian is strongly regular; (ii) Each of the eigenstates, which are distinctive through the target state, are directly coupled for the target state for the Lyapunov management based to the state distance or even the state error, or any two eigenstates are coupled directly for your Lyapunov control based mostly about the regular worth of an imaginary mechanical amount. For your quantum Liouville equation, the convergence circumstances are as follows. (i) The internal Hamiltonian is strongly regular, and (ii) the handle Hamiltonians are full linked. To start with, the Lyapunov handle strategy which could only ensure the convergence for the target eigenstate was studied [19, 23, 25]. Then, the target mixed state, which commutes with the inner Hamiltonian, was studied [33, 36].