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Certainly one of the key worries from the Lyapunov control is selecting an acceptable Lyapunov perform to design the control laws. Ordinarily, the management laws and also the manage effects are various once the Lyapunov functions are distinct. It's a good idea to choose the free copy Lyapunov perform based mostly to the geometrical and physical meanings. Normally, you can find largely three Lyapunov functions to be picked: the Lyapunov function based mostly about the state distance [19�C21, 26�C32, 39�C41], the state error [22, 23, 37, 38], and also the normal worth of an imaginary mechanical quantity [24, 25, 39, 41]. The so-called imaginary mechanical amount signifies that it's a linear Hermitian operator to be intended and may be not a physically meaningful observable amount this kind of as coordinate and energy.

Amongst these 3 Lyapunov functions, the Lyapunov-based quantum control strategies based within the state distance and state error only have to have to modify the scale things of the management laws. These two Lyapunov manage solutions are comparatively simple and easy to grasp. The Lyapunov-based quantum manage primarily based about the normal value of an imaginary mechanical quantity includes far more adjustable So it is actually extra flexible as well as more complex on the very same time. Normally speaking, the Lyapunov-based control method can only ensure that the control system is secure. The probability management in the quantum program necessitates us to style and design a manage strategy which can makeCH5138303 the procedure convergent, since a secure quantum handle strategy might lead to that the control system can't reach the desired target state.

Therefore a further important concern of this manage approach for that closed quantum methods could be the convergence in the management systems. Up to now, there have been the following investigate final results around the convergence with the closed quantum methods [19, 23�C25, 27�C29, 33, 36].To the Schr?dinger equation, the convergence ailments are as follows. (i) The inner Hamiltonian is strongly frequent; (ii) All of the eigenstates, which are diverse from your target state, are right coupled towards the target state for that Lyapunov handle based mostly within the state distance or even the state error, or any two eigenstates are coupled right for the Lyapunov management primarily based within the common value of an imaginary mechanical quantity. For the quantum Liouville equation, the convergence conditions are as follows. (i) The internal Hamiltonian is strongly common, and (ii) the manage Hamiltonians are full connected. Initially, the Lyapunov manage technique which could only assure the convergence towards the target eigenstate was studied [19, 23, 25]. Then, the target mixed state, which commutes together with the inner Hamiltonian, was studied [33, 36].