A Survey of Quantum Lyapunov Control Methods

The condition of a quantum Lyapunov-based control which can be well used in a closed quantum system is that the method can make the system convergent but not just stable. In the convergence study of the quantum Lyapunov control, two situations are classified: nondegenerate cases and degenerate cases. For these two situations, respectively, in this paper the target state is divided into four categories: the eigenstate, the mixed state which commutes with the internal Hamiltonian, the superposition state, and the mixed state which does not commute with the internal Hamiltonian. For these four categories, the quantum Lyapunov control methods for the closed quantum systems are summarized and analyzed. Particularly, the convergence of the control system to the different target states is reviewed, and how to make the convergence conditions be satisfied is summarized and analyzed.


Introduction
The theory of quantum mechanics is one of the major discoveries in the history of science in the 20th century. It is a very important issue to study the properties of the quantum mechanical systems and their control. According to whether the system is isolated or not, a quantum mechanical system can be a closed system or an open system. In a closed quantum system, the evolution of the state is unitary. There are mainly two methods to describe the evolution of a closed quantum system's states. They are the Schrödinger equation In an open quantum system, the system interacts with the surroundings, thus the loss of the system's information leads to the non-unitary evolution of the state. The most common method to describe the open system is the Lindblad master equation:̇= [ 0 +∑ =1 ( ), ]+ ( ), which is in fact the sum of a closed system and a dissipative term caused by the loss of the information or energy. Obviously, the research of the properties of the closed quantum system and its control is relatively simple. Moreover, there is a more important fact: the research of the closed quantum system is the basis of that of the open quantum system. Quantum control has attracted much attention in recent years and it has been found the potential applications in many fields such as atomic physics [1][2][3][4], molecular chemistry [5][6][7][8][9] and quantum information [10,12]. Up to now, there have been many quantum methods, such as quantum optimal control [13][14][15], adiabatic control [16][17][18], the Lyapunovbased control , and optimal Lyapunov-based quantum control [42]. For the Lyapunov-based quantum control, it is relatively easy to design an analytical but not numerical control law, and the control system based on this control method is at least stable, so it has been a common control method.
One of the major concerns of the Lyapunov control is choosing an appropriate Lyapunov function to design the control laws. Ordinarily, the control laws and the control effects are different when the Lyapunov functions are distinct. It's a good idea to choose the Lyapunov function based on the geometrical and physical meanings. Usually, there are mainly three Lyapunov functions to be selected: the Lyapunov function based on the state distance [19-21, 26-32, 39-41], the state error [22,23,37,38], and the average value of an imaginary mechanical quantity [24,25,39,41]. The socalled imaginary mechanical quantity means that it is a linear Hermitian operator to be designed and may be not a physically meaningful observable quantity such as coordinate and energy. Among these three Lyapunov functions, the Lyapunov-based quantum control methods based on the state 2 The Scientific World Journal distance and state error only need to adjust the scale factors of the control laws. These two Lyapunov control methods are relatively simple and easy to grasp. The Lyapunov-based quantum control based on the average value of an imaginary mechanical quantity contains more adjustable parameters. So it is more flexible and also more complex at the same time.
Generally speaking, the Lyapunov-based control method can only ensure that the control system is stable. The probability control in the quantum system requires us to design a control strategy which can make the system convergent, because a stable quantum control method may result in that the control system cannot reach the desired target state. Therefore another major concern of this control strategy for the closed quantum systems is the convergence of the control systems. So far, there have been the following research results on the convergence of the closed quantum systems [19, 23-25, 27-29, 33, 36]. At first, the Lyapunov control method which could only ensure the convergence to the target eigenstate was studied [19,23,25]. Then, the target mixed state, which commutes with the internal Hamiltonian, was studied [33,36]. Later, the convergence to the target superposition state was solved by means of designing the control laws in the interaction picture of the control system, and the convergence to the target mixed state which does not commute with the internal Hamiltonian was solved by using a unitary transformation.
In fact, many actual systems do not satisfy the convergence conditions mentioned above, such as the time domain model of the selective excitation of the stimulated Raman scattering [43], the coupled two spin systems, and onedimensional oscillators [44]. These systems are called nonideal systems and in the degenerate cases. For the degenerate cases, the convergence of the control systems was solved by introducing a series of implicit function perturbations and choosing an implicit Lyapunov function [37][38][39][40]. At first, the convergence to the target eigenstate was only guaranteed [37][38][39][40]. Then the convergence to the target mixed state which commutes with the internal Hamiltonian was solved by introducing a series of implicit function perturbations, and the convergence to the target superposition state and the target mixed state which does not commute with the internal Hamiltonian was solved by introducing a series of constant disturbances.
The aim of this paper is to summarize and analyze the existing Lyapunov control methods for the nondegenerate and degenerate cases, respectively. Dividing the target state into four categories: the eigenstate, the mixed state which commutes with the internal Hamiltonian, the superposition state, and the mixed state which does not commute with the internal Hamiltonian state, we summarize the design methods of the control laws, analyze the convergence to the target state, and investigate how to make these conditions of the convergence be satisfied.
The remainder of this paper is arranged as follows. In Section 2, the research results for the nond-egenerate cases are summarized and analyzed. In Section 3, the research results for the degenerate cases are summarized and analyzed. Some concluding remarks are drawn in Section 4.

Non-Degenerate Cases
The design of the control laws and the analysis of the convergence are very important in the Lyapunov control method. The design of the control laws is based on the Lyapunov stability theorem, which is to design the control laws to make the selected positive semi-definite Lyapunov function ( ) satisfẏ( ) ≤ 0. The convergence analysis of this control method is mainly based on the LaSalle invariance principle [44] for the autonomous systems, or the improved Barbalat lemma [45] for the non-autonomous systems.
In this Section, we will summarize and analyze the research results on the convergence of the control system in the non-degenerate cases for the target state being an eigenstate, a mixed state which commutes with the internal Hamiltonian, the superposition state, or/and a mixed state which does not commute with the internal Hamiltonian state, respectively.

Target Eigenstate.
It is convenient to use the bilinear Schrödinger equation to describe the control systems if the target state is a pure state. Consider the N-level closed quantum system governed by the following bilinear Schrödinger equation:̇( where | ( )⟩ is the quantum state vector, 0 is the internal Hamiltonian, , ( = 1, . . . , ) are control Hamiltonians, and ( ), ( = 1, . . . , ) are scalar and real control laws.
In the following sections, in the case that the target state | ⟩ is an eigenstate, that is, 0 | ⟩ = | ⟩, where is the eigenvalue of the internal Hamiltonian 0 , the research results on the convergence are summarized and analyzed for the Lyapunov control based on the state distance, the state error, and the average value of an imaginary mechanical quantity, respectively.

Lyapunov Control Based on State
Distance. Consider the following Lyapunov function based on the state distance: The Scientific World Journal 3 The time derivative of the Lyapunov function (2) iṡ The control laws ( ) which can makė1 ≤ 0 hold can be designed as where > 0, and = ( ), ( = 1, . . . , ) are monotonic increasing functions through the coordinate origin of the plane − .
The control laws designed in (4) can only ensure the control system (1) to be stable. One needs to do further study on the convergence of the control system. The control system governed by (1) is an autonomous system, whose convergence can be analyzed based on the LaSalle invariance principle [44]. According to the LaSalle invariance principle, as → ∞, any state trajectory will converge to the largest invariant set contained in the set in which the states satisfy that the first order derivative of the Lyapunov function equals zero. In fact, the set contains not only the target state but also other states, thus the system may converge to other states rather than the target state. The main idea to solve this problem is to add restrictions to make the set as small as possible. Based on the LaSalle invariance principle, the convergence of the control system governed by (1) can be depicted by Theorem 1.
Theorem 1 (see [19]). Consider the control system governed by (1) with control fields ( ) designed in (4). If (i) The system is strongly regular, that is, From Theorem 1, one can see that in the case that the target state is an eigenstate, if the control system governed by (1) satisfies the conditions (i)-(ii), the control system can converge to the equivalent state of the target eigenstate | ⟩ from any initial pure state. These two conditions are relevant to the internal Hamiltonian and the control Hamiltonians, which are system parameters. Once the control system is determined, the Hamiltonians are fixed and cannot be changed by designing the control laws.

Lyapunov Control Based on State
Error. Consider the following Lyapunov function based on the state error: In the case of selecting the Lyapunov function based on the state error defined by (5), in order to facilitate to design the control laws based on the Lyapunov stability theorem, the drift item appeared in the first order time derivative of Lyapunov function, which is caused by the internal Hamiltonian, is needed to be eliminated. The existing solution is to add a global phase control item into the control system governed by (1). This method will not change the population distribution of the control system. Thus the dynamical equation (1) becomeṡ After some deduction, one can obtain the time derivative of the Lyapunov function (5) aṡ The control laws which can makė2 ≤ 0 hold can be designed as where > 0, and = ( ), ( = 0, . . . , ) are the monotonic increasing functions through the coordinate origin of the plane − .
Based on the LaSalle invariance principle, the convergence of the control system governed by (6) can be depicted by Theorem 2.
Theorem 2 (see [19,23]). Consider the control system governed by (6) with control fields ( ) designed in (9) and designed in (8). From Theorem 2, one can see that for the case that the target state is an eigenstate, if the control system governed by (6) satisfies the conditions (i)-(ii), the control system can also converge to the equivalent state of the target eigenstate | ⟩ from any initial pure state.

Lyapunov Control Based on Average
Value of an Imaginary Mechanical Quantity. Consider the following Lyapunov function based on the average value of an imaginary mechanical quantity: where the imaginary mechanical quantity is a positive definite Hermitian operator.
The first order time derivative of the Lyapunov function (10) can be obtained aṡ The Scientific World Journal Set [ 0 , ] = 0 such that the drift term in the right side of (11) can be eliminated. In order to ensurė3 ≤ 0, one can design ( ) as  (1), if the control system is strongly regular and any eigenstate is directly coupled to all other eigenstates, that is, for any ̸ = , , ∈ {1, 2, . . . , }, there exists a ∈ {1, . . . , } such that ⟨ | | ⟩ ̸ = 0, then one can deduce thaṫ3( ) = 0 holds for all ≥ 0 , 0 ∈ is equivalent to where and are the lth and jth eigenvalues of , respectively.
Equation (14) implies that there is at most one ̸ = 0, ( = 1, . . . , ); that is, the system will converge to an eigenstate with → ∞. Thus the convergence of the control system (1) can be depicted by Theorem 3.
From Theorem 3, one can see that the control system will converge from any initial pure state to an eigenstate which may not be the target eigenstate. In order to make the system converge to the target eigenstate | ⟩ from any initial pure state | 0 ⟩, aṡ3( ) ≤ 0, one can add a restriction as where | 0 ⟩ is the initial state, | other ⟩ represents any other state in the set 3 except the target state.
In such a way, any state trajectory of the system will converge to | ⟩ from any initial pure state | 0 ⟩.
Next, let us analyze how to make these convergence conditions be satisfied. Conditions (i) and (ii) are only relevant to the internal Hamiltonian 0 and the control Hamiltonians , ( = 1, . . . , ) which cannot be changed by designing appropriate control laws. Condition (iii) means that and 0 have the same eigenstates. In order to make condition (iii) be satisfied, the eigenvalues of can be designed as Because should be positive definite, one needs to design > 0, ( = 1, . . . , ). The restriction (15) can be satisfied by means of designing an appropriate . For the restriction (15), Grivopoulos and Bamieh proposed a design principle of to make 3 (| ⟩) < 3 (| other ⟩) hold. This design principle of can be depicted by Proposition 4.
Proposition 4 (see [25]). With the constraint condition ⟨ | ⟩ = 1, the set of critical points of the Lyapunov function According to Proposition 4, in order to make 3 (| ⟩) < 3 (| other ⟩) hold, > , ( = 1, . . . , , ̸ = ) needs to be designed, where is the eigenvalue of corresponding to | ⟩. Then let us consider the whole restriction (15); it is an attraction problem. If the eigenvalues of except are close together, the attraction region will be very large. For the limiting case = > > 0, ( ̸ = , , ̸ = ), the attraction region will be the whole state space. Thus the design principle of is > , ( = 1, . . . , , ̸ = ) and to make , ̸ = close together. In conclusion, for the target state being an eigenstate, the design principle of is as follows: From the above analyses, we can conclude that the Lyapunov control method based on the imaginary mechanical quantity proposed in [19,24,25] can only ensure the convergence to an eigenstate, but cannot guarantee the convergence to the target eigenstate from any initial pure state. However, if there exists a to make the restriction (15) hold, then any state trajectory of the system will converge to the equivalent state of the target eigenstate | ⟩ from any initial pure state.
The Scientific World Journal 5

Relations between Three Lyapunov Functions.
In the Liouville space, the Hilbert-Schmidt distance between two density operators 1 and 2 is The inner product of two operators and is defined as ⟨⟨ | ⟩⟩ = tr( † ), where the operation † refers to the conjugate transpose of . Because = | ⟩⟨ |, the square of the Hilbert-Schmidt distance between the density operator and the target density operator can be deduced as 2 One can conclude from (18) that the Lyapunov function based on the state distance 1 (| ⟩) = (1/2)(1 − |⟨ | ⟩| 2 ) and the state error For the Lyapunov function based on the average value of an imaginary mechanical quantity 3 [19]. Therefore 1 (| ⟩) is formally a special case of 3 (| ⟩).
From the above analyses, we can conclude that all these three Lyapunov control methods can converge to the equivalent state of the target eigen state from any initial pure state. The Lyapunov control methods based on the state distance and the state error have only one adjustable parameter. So these two methods are very easy to grasp and very simple. The Lyapunov control based on the average value of an imaginary mechanical quantity has more adjustable parameters. So it is more flexible and also more complex at the same time. For the target eigenstate, 1 (| ⟩) and 2 (| ⟩) are equivalent, so the Lyapunov control based on the state distance and the state error have similar control effects. Because 1 (| ⟩) is formally a special case of 3 (| ⟩), generally, the control effect of the Lyapunov control based on the average value of an imaginary mechanical quantity is better than that of the state distance. At least, it can get the same control effect as the Lyapunov control of the state distance.

Target Mixed
where ( ) is the density operator.
Consider the Lyapunov function based on the average value of an imaginary mechanical quantity: By means of setting [ 0 , ] = 0, the first order time derivative of the Lyapunov function (21) can be deduced aṡ In order to ensurė5 ≤ 0, one can design ( ) as where ∈ , > 0.
Next, let us analyze the convergence to the target state. For the control system governed by (20) in the non-degenerate case, one can deduce thaṫ5( ) = 0 holds for all ≥ 0 , 0 ∈ is equivalent to which implies that the system will converge to a state which commutes with 0 . Thus based on the LaSalle invariance principle, the convergence of the control system governed by (20) can be depicted by Theorem 5.
Theorem 5 (see [37]). Consider the control system governed by (20) with the control field ( ) designed in (23). If (i) the internal Hamiltonian is strongly regular, that is, From Theorem 5, one can see that if the control system satisfies the conditions (i)-(iv), the control system will converge from any initial state to a state that commutes with the 6 The Scientific World Journal internal Hamiltonian, which may not be the target state. Next, what we need to do is to make the control system converge to the target state.
Denote the state in 4 as 4 ; then [ 4 , 0 ] = 0 holds which implies that 4 and 0 have the same eigenstates. Since the evolution of ( ) is unitary, ( ) for ≥ 0 are isospectral. So the eigenvalues of 4 are a permutation of the eigenvalues of 0 . Thus the set 4 has countable elements. If the initial state is generic, that is, the eigenvalues of the initial state are mutually different, the set 4 will have ! elements. For the target state which commutes with the internal Hamiltonian, that is, [ , 0 ] = 0, in order to make the system converge to the target state which commutes with 0 from any initial state 0 , Kuang and Cong proposed a restriction as where other represents any other state in the set 4 except the target state .
In such a way, if there exists a to make the restriction (26) hold, any state trajectory of the system will converge to the target state which commutes with 0 from any initial state 0 .
For how to make conditions (iii)-(iv) to be satisfied, please read Section 2.1.3. One can deduce the design principle of such that 5 ( ) < 5 ( other ), the result can be depicted by Proposition 6. It is difficult to design such that (26) holds for any initial state 0 and any target state which satisfies [ , 0 ] = 0. One possible method is to introduce a series of implicit function perturbations into the control laws, this method will be presented in Section 3.3.
For the target state which commutes with the internal Hamiltonian, the design principle of is as follows: > 0 for any ̸ = ∈ {1, 2, . . . , }; (ii) Design according to Proposition 6; (iii) Equation (16) In conclusion, if the control system satisfies the conditions (i)-(iv) in Theorem 5 and there exists a to make the restriction (26) hold, any state trajectory of the system will converge to the target mixed state which commutes with the internal Hamiltonian from any initial state 0 .

Target Superposition
Equation (27) implies that the system will converge to the set 5 = span{| 1 ⟩, | 2 ⟩, . . . , | ⟩, | ⟩; ∈ {1, . . . , }, ̸ = 1, . . . , }, which means that the set 5 contains infinite elements. Therefore this control method cannot ensure the system converge to the target superposition state by adding the restriction defined by (15). But this control method can ensure that the system will converge to the superposition of the eigenstates corresponding to target state. In fact, once one element of changes, all the populations of the levels will change accordingly. Therefore when there are not very many eigenstates corresponding to target state, the system maybe converge to the target state by regulating the eigenvalues of . Otherwise, the system maybe cannot converge to the target state. Consider the extreme situation in which the target state is the superposition of all the eigenstates. According to the design principle of P, we should design = , where is a real number, and is the unit matrix. Obviously, in this special case, the control method proposed in Section 2.1.3 will become invalid.
From Section 2.2, one can see that, for the control system governed by (20) Because the set 6 has infinite elements, this control method also cannot ensure the system will converge to the target state by adding the restriction defined by (26). But when the target state does not have so many nonzero off-diagonal elements, the system maybe converges to the target state by regulating the diagonal elements of . When there are many nonzero off-diagonal elements in the target state, the degree of freedom of may be not enough.
From the above analyses, we can conclude that by means of using the control methods proposed in Sections 2.1 and 2.2, the control system may but cannot be ensured to converge to the target superposition state. This problem can be solved by means of designing the control laws in the interaction picture of the control system.
Consider the -level control system in the interaction picture as Choose the Lyapunov function based on the average value of an imaginary mechanical quantity defined by (21). The first order time derivative of the Lyapunov function (21) can be obtained aṡ5 The Scientific World Journal 7 To ensurė5 ≤ 0, one can design ( ) as where > 0 and = ( ), ( = 1, . . . , ) are monotonic increasing functions through the coordinate origin of plane − . The control system governed by (28) is a nonautonomous system; thus the LaSalle invariance principle cannot be used to analyze the convergence. One can use the improved Barbalat lemma which can be used for the nonautonomous system. According to the improved Barbalat lemma, the convergence of the control system governed by (28) can be depicted by Theorem 7.
Theorem 7. Consider the control system governed by (28) with the control field ( ) designed in (30). If (i) the internal Hamiltonian is strongly regular; (ii) the control Hamiltonians are full connected, then any state trajectory will converge to the limit set For the case that is chosen as a diagonal matrix, the limit set is reduced to R 1 ≡ { : [ , ] = 0}. For the case that is chosen as a nondiagonal matrix, if rank̃( ⃗ ) = 2 − holds, then the limit set R 1 is regular; namely, R 1 ≡ { : [ , ] = 0}, wherẽ( ⃗ ) is the first 2 − rows of ( ⃗ ), ( ⃗ ) is the real ( 2 − 1) * ( 2 − 1) matrix corresponding to the Bloch representation of , and is a linear map from Hermitian or anti-Hermitian matrices into ( ).
[ , ] = 0 means that and 0 have the same eigenstates. For the target state = | ⟩⟨ | being a superposition state, in order to make the target state contain in R 1 ≡ { : [ , ] = 0}, P can be designed as where | 1 ⟩ = | ⟩ and ⟨ | ⟩ = 0, for ̸ = . Then some deduction shows that = | ⟩⟨ |( = 1). In order to make the system converge to the target superposition state from any initial pure state, one can design such that where oth represents any other state in the set R 1 except the target state . For the satisfaction of 5 ( ) < 5 ( oth ), one can design based on Proposition 6. In order to make (32) holds, one must design , ( = 1, . . . , ) such that However, for any initial pure state the target superposition state, there may not exist a such that (33) holds.
In conclusion, the design principle of for the case that the target state = | ⟩⟨ | being a superposition state is (31), (33), and Proposition 6. We can also conclude that if the control system satisfies the conditions (i) and (ii) in Theorem 7 and there exists a to make (33) hold, any state trajectory of the system will converge to the target superposition state from any initial pure state 0 .
Remark 8. One can also solve the problem of the convergence of the control system governed by (1) to the target superposition state by means of designing the control laws in the interaction picture of the control system. to make [̂, 0 ] = 0 hold. Correspondingly, the control system governed by (28) after this unitary transformation will become

Target Mixed State Which Does Not
wherê= † ,̂( ) = † ( ) . Then control laws can be designed according to Section 2.3. The research results show that the designed control laws and Theorem 7 are also valid with every physical quantity changing accordingly. In order to makêcontain in R 2 ≡ {̂: [̂, ] = 0}, needs to be designed such that For sake of the convergence from any initial mixed state 0 to any target mixed state which does not commute with the internal Hamiltonian 0 , needs to be designed such that wherêo th represents any other state in the set R 2 ≡ {̂: In such a way, if there exists a to make (35) hold, any state trajectory of the system will converge to the target mixed state which does not commute with the internal Hamiltonian from any initial mixed state 0 .
Next, let us analyze how to make the restriction defined by (35) hold. Some deductions show that if one designs based on Proposition 6 with changing intôaccordingly, then 8 The Scientific World Journal to make the whole restriction defined by (35) hold, , ( = 1, . . . , ) needs to be designed such that where is the ( , )th element of . In conclusion, the design principle of for the case that the target state is a mixed state which does not commute with 0 is > 0 for any ̸ = ∈ {1, 2, . . . , }; (ii) Proposition 6 with changing intô; (iii) Equations (36) and (16).
We can conclude from the above analysis that for the case that the target state is a mixed state which does not commute with 0 , if the control system is strongly regular and full connected, and one can seek an imaginary mechanical quantity to make (36) hold, then the control system can converge from any initial mixed state to the target mixed state which does not commute with the internal Hamiltonian.

Degenerate Cases
The convergence conditions (i) and (ii) proposed in the Theorem 1 through Theorem 7 are relevant to the internal Hamiltonian 0 and the control Hamiltonians , ( = 1, . . . , ). They are system parameters which cannot be changed. And in practice, many actual systems do not satisfy these convergence conditions. These systems are called in the degenerate cases. In order to solve the convergence of the control systems in degenerate cases, the existing method is to introduce a series of implicit function perturbations into the control laws and choose a Lyapunov function which is an implicit function [37][38][39][40].
In this section, we also divide the target state into four categories as: target eigenstate, the target mixed state which commute with the internal Hamiltonian, target superposition state, and the target mixed state which does not commute with the internal Hamiltonian. For these four target state categories, respectively, research results for the degenerate cases are summarized and analyzed.

Target Eigenstate.
In this Section, the convergence of the control system based on the Lyapunov control method in the degenerate cases to any target eigenstate from any initial pure state will be summarized and analyzed.

Implicit Lyapunov Control Based on State Distance.
In order to solve the convergence of the control systems in the degenerate cases, several researchers introduced a series of implicit function perturbations into the control laws [37][38][39][40]. After a series of perturbations ( ) introducing into the control laws, the dynamical equation (1) becomeṡ (37) where ( ) + V ( ) = ( ), ( = 1, . . . , ) are the total control laws.
The existing design method of ( ) is to design it to be a monotonically increasing functional of ( ), that is, where the function (⋅) satisfies (0) = 0, ( ) > 0 and ( ) > 0 for every > 0 and is the independent variable of the function (⋅).
The Scientific World Journal 9 The right side of (39) contains | , 1 ( ),..., ( ) ⟩ which is a functional of the perturbations ( ), ( = 1, . . . , ). One can see that the relation between and the time is defined by equations and cannot be expressed by an explicit expression, so is the implicit function of the time . From (38) and (39), one can also see that the Lyapunov function 6 is the implicit function of the time . The existence of ( ) can be depicted by Lemma 9.

Implicit Lyapunov Control Based on the State Error.
The basic idea of this method is similar to that of the state distance. In the Lyapunov control based on the state error, a global phase control item is added into the control system to facilitate the design of the control laws. Thus the control system can be depicted bẏ where ( ) = ( ) + V ( ), ( = 1, . . . , ) and is the control laws which need to design.
The existence of ( ) can be depicted by Lemma 11.
Based on the LaSalle invariance principle, the convergence of the control system governed by (44) can be depicted by Theorem 12.
Theorem 12 (see [38,39]). Consider the control system (43) with control fields ( ) designed in (45) and Lemma 11,V ( )  By designing appropriate perturbations ( ), conditions (i) and (ii) in Theorem 12 can be satisfied in most cases. From Theorem 12, one can see that if one design appropriate control laws ( ) = ( ) + V ( ), ( = 1, . . . , ) and to make the conditions (i) and (ii) in Theorem 12 hold, the control system depicted by (43) can converge to the equivalent state of the target eigenstate | ⟩ from any initial pure state.

Implicit Lyapunov Control Based on Average Value of an Imaginary Mechanical Quantity.
For the implicit Lyapunov control based on the average value of an imaginary mechanical quantity, one can consider the control system depicted by (37). In this control method, the introduced implicit function perturbations ( ) mainly have two tasks. One task is to solve the convergent problem of the control system in the degenerate cases. The basic idea is similar to that of the state distance. Another one is to choose a simpler restriction (| ⟩) < (| other ⟩), which can be satisfied for any initial state and any target state by designing the imaginary mechanical quantity. In order to ensure the system converge to the target state by adding (| ⟩) < (| other ⟩), we can design all the perturbations ( ) = 0 hold for = 1, . . . , only at | ⟩, that is, (1) (| ⟩) = 0, ( = 1, . . . , ), and (2) for | ⟩ ̸ = | ⟩, there exists at least one such that (| ⟩) ̸ = 0. For sake of completing these two tasks, we can design ( ) as a monotonically increasing functional of ( ): where ≥ 0, and for = 1, . . . , , there exists at least a > 0, and the function (⋅) satisfies (0) = 0, ( ) > 0 and ( ) > 0 for every > 0, is the independent variable of the function (⋅).
The specific Lyapunov function based on the average value of an imaginary mechanical quantity can be selected as: where 1 ,..., = ( 1 ( ), . . . , ( )) is a functional of ( ) and positive definite.
The existence of ( ) can be depicted by Lemma 13.
Based on LaSalle's invariance principle, the convergence of the control system governed by (37) can be depicted by Theorem 14. From Theorem 14, one can see that if the target state | ⟩ is an eigenstate, | ⟩ is contained in 9 . In order to make the system converge to the target eigenstate from any initial pure state | 0 ⟩, on the one hand, needs to be designed to make hold, where | other ⟩ represents any other state in the set 9 except the target state. On the other hand, because / > 0,̇≤ 0, ≥ 0 holds, when V ( ) = 0, ( ) = ̸ = 0 holds for some time, we can design = − , (0 < ≪ ) to make the state trajectory evolve but not stay in 9 until | ⟩ is reached.
Next, let us analyze how to realize the convergence conditions in Theorem 14 and restriction defined by (54). By designing appropriate ( ), conditions (i) and (ii) in Theorem 14 can be satisfied in most cases. In order to make condition (iii) be satisfied, the eigenvalues of can be designed constant, denoted by 1 , 2 , . . . , , and can be designed as To make condition (iv) satisfied, ̸ = (∀ ̸ = ; 1 ≤ , ≤ ) can be designed. The results on how to make (54) hold can be depicted by Theorem 15.
We can conclude from the above analyses that by using the implicit Lyapunov control based on the imaginary mechanical quantity, if one designs appropriate control laws ( ) = ( )+V ( ), ( = 1, . . . , ) to make the conditions (i)-(iv) in Theorem 15 and (54) hold, the control system governed by (37) in the degenerate cases can converge from any initial pure state to the target eigenstate. The design principle of the imaginary mechanical quantity is Theorem 15 and (55).

Target Superposition
State. The method proposed in Section 3.1 cannot guarantee the control system governed by (37) or (43) in the degenerate cases converge from any initial pure state to the target superposition state. In order to solve this problem, one can introduce a series of constant disturbances into the control laws. Thus the mechanical equation (37) becomeṡ And the mechanical equation (43) in Section 3.1.2 becomeṡ where ∈ .
The basic idea of solving the convergence to the target superposition state is to design to make the target state | ⟩ be an eigenstate of 0 = 0 + ∑ =1 ⋅ 0 can be viewed as the new internal Hamiltonian of the control system. If the number of the control Hamiltonians is large enough, by designing appropriate , ( 0 + ∑ =1 )| ⟩ = | ⟩ can be satisfied in most cases, where is the eigenvalue of 0 + ∑ =1 corresponding to | ⟩. Then one can design the control laws and analyze the convergence according to the method for the target eigen state cases. Research results show that every conclusion in Section 3.1 also holds with changing 0 into 0 .

Target Mixed State Which Commutes with the Internal Hamiltonian.
Consider the -level closed quantum system governed by the following quantum Liouville equation: where ( ) + V ( ) = ( ) are the total control laws. The design ideas are similar to those of Section 3.1. Consider the Lyapunov function based on the imaginary mechanical quantity as The existence of ( ) can be depicted by Lemma 16. Based on the LaSalle invariance principle, the convergence of the control system can be depicted by Theorem 17. For the target state which commutes with the internal Hamiltonian; that is, [ , 0 ] = 0, is contained in 10 . In order to make the system converge to the target state which commutes with 0 from any initial stat; on the one hand, needs to be designed to make 9 ( ) < 9 ( other ) hold, where other represents any other state in the set 10 except the target state. On the other hand, was designed as = − , (0 < ≪ ) when V ( ) = 0, ( ) = ̸ = 0 holds for some time to make the state trajectory evolve but not stay in 10 until is reached.
For the satisfaction of conditions (i)-(iv) in Theorem 17, one can follow that of Section 3.1.3. In order to make (62) hold, one can design based on Proposition 6. The total design principle of is (55) and Proposition 6.
We can conclude from the above analyses that for the target state which commutes with the internal Hamiltonian, by using the implicit Lyapunov control method based on the imaginary mechanical quantity, if one designs appropriate control laws ( ) = ( ) + V ( ), ( = 1, . . . , ) to make the conditions (i)-(iv) in Theorem 17 and (62) hold, the control system governed by (58) in the degenerate cases can converge from any initial state to the target state which commutes with the internal Hamiltonian, which contains the target eigenstate and the target mixed state which commutes with the internal Hamiltonian.

Target Mixed State Which Does Not Commute with
Internal Hamiltonian. For the target mixed state which does not commute with the internal Hamiltonian, the design idea is similar to Section 3.2. The difference is to design to make [ , 0 ] = 0. If the number of the control Haimltonians is large enough, by designing appropriate , [ , 0 ] = 0 can be satisfied in most cases. Then one can design the control laws and analyze the convergence according to the method mentioned in Section 3.3. Research results show that every conclusion in Section 3.3 also holds with changing 0 into 0 .

Conclusion
In this paper, for the non-degenerate and degenerate cases, the existing quantum Lyapunov control based on the state distance, state error, and average value of an imaginary mechanical quantity for the control systems have been summarized and analyzed. For the target state being the eigenstate, the mixed state which commutes with the internal Hamiltonian, the superposition state, and the mixed state which does not commute with the internal Hamiltonian, respectively, the design methods of the control laws have been summarized; the convergence to the target state has been summarized and analyzed. Research results show that the Lyapunov-based quantum control method can make the control system converge from any initial state to the target state in both non-degenerate and degenerate cases. After ten years of development, the quantum control theory based on the Lyapunov stability theorem has been established.