Microsoft Word - CET--006.docx


 CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 59, 2017 

A publication of 

 

The Italian Association 
of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Zhuo Yang, Junjie Ba, Jing Pan 
Copyright © 2017, AIDIC Servizi S.r.l. 
ISBN 978-88-95608- 49-5; ISSN 2283-9216 

Trajectory Tracking for Quantum Systems with Impulsive 

Control Fields 

Wei Yang 

Shanghai Dianji University, Shanghai 201306, China 

wenshui77@126.com 

Quantum Lyapunov control is a feedback control methodology to determine control fields applied to control 

quantum systems. In this paper, with the help of the invariant principle of impulsive systems, we investigate 

the trajectory tracking for quantum systems with impulsive control fields by two Lyapunov functions based on 

the distance of quantum states and the average value of Hermitian operator which can be regarded as an 

observable of a quantum system, propose new control fields to drive the quantum systems in the form of 

sufficient conditions. One numerical simulation is presented to illustrate the effectiveness of the proposed 

control method. 

1. Introduction 

Compared to traditional solid-state systems, quantum system shows a great advantage in the information 

theory and technology, quantum computer and so on. In the last few years, because of wide variety of 

applications of quantum control theory, such as quantum chemistry, quantum information processing and 

quantum electronics etc. Considerable attention has been focused on quantum control theory, and the 

growing interest in this subject have been attributed both to theoretical and experimental breakthroughs (Dong 

et al., 2010; Wang et al., 2009; Wang et al., 2010; Mirrahimi et al., 2005; Turinici et al., 2007; Vaiano et al, 

2017) and references therein, it indicates that quantum control has become an important area of research. 

Lyapunov control was proposed as a good candidate for quantum state engineering, various Lyapunov 

functions and control methods have been studied and applied to quantum systems, such as implicit Lyapunov 

control (Zhao et al., 2012), switching control (Zhao et al., 2012), and Lyapunov functions based on state 

distance (Kuang et al., 2008; Wang et al., 2010), average value of an imaginary mechanical quantity 

(Grivopoulos et al., 2003), and state error (Kuang et al., 2008). 

Dong and Petersen introduced the switching control method to drive the system by using two controllers to 

arbitrary target state based on graph theory (Dong et al., 2011). And in (Zhao et al., 2012), Zhao, Lin and Xue 

considered another switching control method of closed quantum systems, which was via the Lyapunov 

method. Inspired by the switching control method, we developed the impulsive control method to study the 

trajectory tracking problem. As we know, impulsive dynamical systems are a special class of dynamical 

systems, which exhibit continuous evolution typically described by ordinary differential equations and 

instantaneous state jumps or impulses. Nowadays, there has been increasing interest in the analysis and 

synthesis of impulsive systems, or impulsive control systems, due to their significance both in theory and 

applications, see (Dong et al., 2008) and the references therein. 

Our aim in this paper is to improve the control effectiveness of quantum systems, and we choose the 

impulsive control method to control quantum systems, by adding an impulsive control field besides the 

continuous one. By the important theorem in (Dong et al., 2008), when one control field with given frequency, 

quantum systems governed by the Liouville equation can be described as impulsive dynamical systems. 

In this paper, based on the Lyapunov method and invariant principle of impulsive systems, our attention is 

focused on the trajectory tracking of quantum systems with impulsive control fields. In Section 2, firstly, we 

introduce the general impulsive dynamical system, then present the quantum systems with impulsive control 

fields, and introduce the invariant principle of impulsive systems. In Section 3 we give control fields to drive 

quantum systems based on the Lyapunov function, and analyze the advantages of the control method 

                               
 
 

 

 
   

                                                  
DOI: 10.3303/CET1759112

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Wei Yang, 2017, Trajectory tracking for quantum systems with impulsive control fields, Chemical Engineering 
Transactions, 59, 667-672  DOI:10.3303/CET1759112   

667



depended on impulsive control fields. We justify the effectiveness of the proposed control fields in two 

simulation experiments in Section 3. 

2. Notations and definitions 

Consider the general impulsive dynamical system described by 

1
( ( )) , )( ) , (

( ) ,

;

( ( )) .

kc k

d k

x t t

x

f x t

f tt tx

 




 


  

 (1) 

where x(t) ∈ 𝑅𝑛  denotes the system state, 𝑓𝑐 (𝑥(𝑡))  is a continuous function from 𝑅
𝑛  to 𝑅𝑛 , the set E =

{𝜏1 , 𝜏2 , L: 𝜏1 < 𝜏2 < L} ⊂ 𝑅
+ is an unbounded, closed, discrete subset of 𝑅+  which denotes the set of times 

when jumps occur, and 𝑓𝑑 : 𝑅
𝑛 → 𝑅𝑛  denotes the incremental change of the state at the time 𝜏𝑘 . In the n- 

dimensional complex space 𝐶 𝑛, we choose the most common norm ||x||: √𝑥∗𝑥, where x is represented as a 
column vector (𝑥1 , 𝑥2 , 𝐿, 𝑥𝑛)

T, and 𝑥∗ denotes its conjugate transpose. 

Denote by 𝑀𝑛(C) the space of n × n complex matrices with an inner product 𝑀𝑛(C)× 𝑀𝑛(C) →C, (a, b)=Tr(a*b), 
and the norm ||a||2=(a,a). 

We consider the following n level quantum system with two control fields, and set the Plank constant h=1: 

0 1

0

2 21
(( ) [ (t ) ( )]

( ) [ (

,

)]

) ( )

,

k

d d

fi t H H H t

i t H

t f

t

t   

 

    




 (2) 

Multiplying both sides of (2) by i , we have 

0 1 2

0

1 2
( ) [ (t ) ( )]

( )

( ) ( ) ,

,[ ( )]

k

d d

t H H H t

t H

f t t

t

f   

 

 









  (3) 

Where ρ(t), 𝜌𝑑 (𝑡) are density operators, represent the mixed state and the target state of quantum systems 
respectively, defined on an n- dimensional Hilbert space H. Density operators are positive semi-definite 

Hermitian operators with unit trace, and denote the set of density operators by Sn. Indeed,  

2 2 2
|| ( ) || ( ( )) 2 ( ) 2 ( ) 0

d d
t Tr t Tr iTr H

dt dt
H           

where ||A||2=Tr(A*A) is the Hilbert-Schmidt norm. Thus, || ρ(t)||=|| ρ(0)||, for any t>0, hence the density matrix 

evolves on a sphere decided by the initial state based on this norm: 

(0), ( ) ( ) || || (0) |( ,|| ) { | }|||
n n n

SOB t S t     . 

When the quantum system evolves freely under its own internal dynamics, i.e., there is no external field 

implemented on the system, just the free Hamiltonian 𝐻0
% is introduced. 𝐻1

%, 𝐻2
%  represent the interaction 

energy between the system and the external classical control fields 𝑓1 (𝑡)𝑎𝑛𝑑𝑓2 (𝑡) respectively, and are called 

interaction Hamiltonian. They are both n × n self-adjoint operators in the n- dimensional Hilbert space H and 

assumed to be time-independent. 

Definition 2.1: A stationary point of a Lyapunov function V(x(T)) is the point satisfies 

( )( ) 0
d d

V V x
dt dt

  . 

In order to study the trajectory tracking for quantum systems, we pay attention to the Lyapunov functions 

based on the distance of quantum states and the average value of Hermitian operator P which can be 

regarded as an observable of a quantum system, and choose the following Lyapunov functions: 

2

1

1
( ) || ( ) ( ) ||

2
d

V t t t     (4) 

2
( ) ( ( ))V t Tr P t  (5) 

In this paper, we set the first control function 𝑓1 (𝑡) is continuous, the other one 𝑓2 (𝑡) only takes effect to 

quantum systems at the impulsive points E. By the same method in [17], we obtain that quantum systems (3) 

with impulsive control fields can be described as 

668



0

0 1 1

2 2

( ) [ ( )],

( ) [ ( )],

( )

( ) , ;

( ) , ;

[ ]., ( )

k

k

d d

f t

t

t H H t t

t f t t

t H

H

t

  

  

 

 

  
 


  (6) 

Subject to quantum systems (2) or (3), we focus on finding control fields 𝑓1 (𝑡)𝑎𝑛𝑑𝑓2 (𝑡) such that the quantum 

systems with impulsive control field are driven to target states. Firstly, we introduce the invariant principle of 

impulsive systems. 

Lemma 2.1 (Shi et al., 2015): Consider the impulsive dynamical system (1), assume 𝐷𝑐 ⊂ D is a compact 
positively invariant set with respect to (5), and assume that there exists a C1 function V: 𝐷𝑐 → 𝑅 such that 

( ( )) 0, ,
c k

D tV x t x  
; 

( ( )))( ( ) ( ( )), , ;
k d k k c k

V x V x xf x tD   
  
  

 

Let { : , ( ( )) 0} { : , ( ( )))( ( ) ( ( ) })
c k c k k d k k

D V D V xG x t xx t x f Vt x    
  

        , and let M ⊂ G 

denote the largest invariant set contained in G. If X0∈ 𝐷𝑐, then x(t)→M as t → ∞. 

3. Main results 

Quantum Lyapunov control uses a feedback control methodology to determine control fields applied to control 

quantum systems. We employ two Lyapunov functions to prepare different control fields. Firstly, by the 

Lyapunov function based on quantum state distance, we have 

Theorem 1: For quantum system (5), if  

1 1 1
( ) ( [ , ])

d
t K Tf r H   , 

2 2 2
( ) ( )[( , ])( )

k d k k
f K r HT    

 
  ,  

where K1, K2>0, then quantum system (2.5) with impulsive control field can be driven to the target states. 

Proof. Choose the Lyapunov function 

2 2 2 2

1

1 1 1
|| || (( ) ) (2 )

2
2

2
( )

2
d d d d d d

TV T T rr Tr r             

When t ≠ 𝜏𝑘,  

1

2 2

0 0 0

0 0 1 1

1 1

(2 ) ( ) ( )

2 ( ) ([ ], ( )

([ , ] )

)

,

([ , ] )

( [ ])

d d d d

d d d d d d

d d

d

H T

V Tr Tr Tr

Tr H Tr H r

Tr H Tr H f H

f Tr H

     

     

   

 



 

   











  

set 
1 1 1
( ) ( [ , ])

d
t K Tf r H   , where K1>0, then 𝑉1

& ≤ 0. 

When t = 𝜏𝑘, since the quantum state ρ(t) is right continuous at the impulsive point, therefore,  

1

1

2

2 2

2

2 2

2 2

( ),

( ),

( )) ( ) ( )

(

( ( ))

( ( ))

( ( ( ) ( ) ( )[ , ])

( ( ()) ( )) ( )

(

) ( ) ( ( )[ , ])

( ( )) ( ) ( )[ , ])), ( ( )

k d k

k d k

d k d k k d k k

d k d k k d k k

k d k d k k

V

V

Tr Tr t H

Tr Tr t Tr H

V t

f

f

Tr Hf

   

   

         

         

       



 

 

  

 














 

In this situation, we choose the control field 

2 2 2
( ) ( )[( , ])( )

k d k k
f K r HT    

 
  ,  

where K2>0, then 

2 2
( ) V( ), ( ,( ) )

k k k k
V      

 
 . 

Based on different Lyapunov function, we can find different control functions to drive the quantum states. Then 

we choose the Lyapunov function based on the average value of Hermitian operator, and get 

Theorem 2: For quantum system (5), if  

669



1 1 1
( ) ([ , )] ( )t K Tr P Hf t  , 2 2 2( ) ([ , ] )( )k kK Tr P Hf   

 
  ,  

where K1, K2>0, then quantum system (2.5) with impulsive control field can be driven to the target states. 

Proof. Choose a Lyapunov function 

V2(ρ(t))=Tr(P ρ(t)). 

When t ≠ 𝜏𝑘,  

0 1 1

0 0 1 1 1 1

0 1 1

2
( ))

( ( ))

( [ ( )])

( ( ) ( ) ( ) ( ) )

( )

(

( ) ,

( ) ( )

([ , ] ( )[ , )] ( )

V t

Tr P t

Tr P H t

Tr PH t H t f t f t H

t f t

f t H

P t PH t P

Tr P H t P H







   

 



 







 



 

Since there is no relation between [P, H0] and the control component, we can set for convenience [P, H0] =0. If 

we choose simple and effective control field 

1 1 1
( ) ([ , )] ( )t K Tr P Hf t 

 

where K1>0,  

.
2

2 1 1
( ) ([ (, ] ) 0)t K Tr P HV t   . 

When t = 𝜏𝑘,, since the state ρ(t) is right continuous at the impulsive points, we have  

2

2

2 2

2 2

2 2 2

( ),

( ),

( ) ( )

( )) ( ( )

( )

( )

( ( ) [ , ])

( ( ) [ , ])

( ) ( ) [ , ])( ), ( ( )

k k

k k

k k

k k

k k k

V

V

Tr P tf

f Tr

f T

P H

Tr P t P H

V t P Hr

  

  

   

   

    

 

 

 

  















 

In this situation, we choose the control field 

2 2 2
( ) ([ , ] )( )

k k
K Tr P Hf   

 
 

,  

where K2>0, then 

2 2
( ),( ) ( ),( )

k k k k
V V     

 
 ,  

where Kj(j=1, 2) can be chose properly to adjust the control amplitude. From Lemma 2.1, the quantum system 

with impulsive control field (2.5) can be driven to the target states. 

Thus we complete the proof. 

In order to illustrate the effectiveness of the proposed method in this paper, one numerical simulation has 

been presented for a two-level quantum system and the Fourth-order Runge-Kutta method is used to solve 

with time steps size 0.04. 

Example 1: Consider the two-level quantum system with internal Hamiltonian, the first control Hamiltonian 

(Kuang et al., 2008), and the interaction Hamiltonians given as follows: 

20 1

1 0 0 0 1
, , .

0 1 0 1 0

i
H H H

i

     
       

       

Let the initial states be ρ(0) (
1

12
1

12
+

𝑖

4

1

12
−

𝑖

4
11

12

) and ρ𝑑 (0)(
1
0

0
0
) respectively. The parameters are chosen as K1=0.2, 

K2=0.05Set the quantum state ρ(0) (
𝑥(1)
𝑥(3)

𝑥(2)
𝑥(4)

), and the target state ρ𝑑 (0) (
𝑥(5)
𝑥(7)

𝑥(6)
𝑥(8)

), by the control fields  

1 1 1
( ) ([ , )] ( )t K Tr P Hf t 

 2 2 2
( ) ([ , ] )( )

k k
K Tr P Hf   

 
 

  

We have the simulation result shown in Figure 1 and Figure 2.  

670



 

Figure 1: The simulation demonstrates the control performance 

 

Figure 2: The evolution of the control field 

In Figure 1, the simulation demonstrates the control performance with impulsive control field 𝑓2𝜏𝑘
−, and the 

Lyapunov function converges to 0; Figure 2 describes the evolution of the control field 𝑓1 (𝑡), which decays to 0 

as the quantum state converges to the target state. 

4. Conclusion 

In this paper, we have introduced the Lyapunov control method to quantum systems with impulsive control 

fields, and given two effective control fields for the trajectory tracking problem. One of the theoretical results 

has been verified by a numerical simulation to illustrate the effectiveness. Compared to other control methods, 

the impulsive control method can drive the quantum state to the target state with less time and more 

accurately, benefiting from the impulsive control function at the impulsive points.  

Acknowledgments  

This work was supported by Academic Discipline Project of Shanghai Dianji University (16JCXK02) and the 

funding scheme for training young teachers in colleges and universities in Shanghai (ZZSDJ14023). 

Reference  

Cong S., Kuang S., 2007, Quantum control strategy based on state distance, Acta Auto. Sinica, 33(1), 28-31, 

DOI: 10.1360/aas-007-0028. 

Dong D., Petersen I.R., 2010, Quantum control theory and applications: a survey, IET Control Theory Appl., 4, 

2651-2671. 

671



Dong D., Petersen I.R., 2011, Controllability of quantum systems with switching control, International Journal 

of Contral., 84, 37-46. 

Dong Y.W., Sun J.T., 2008, On hybrid control of a class of stochastic non-linear Markovian switching systems, 

Automatica, 44(4), 990-995, DOI: 10.1016/j.automatica.2007.08.006 

Grivopoulos S., Bamieh B., Lyapunov-based control of quantum systems, Proceedings of 42nd IEEE 

Conferences. Dec. Contral., 1-6, 434-438. 

Kuang S., Cong S., 2008, Lyapunov control methods of closed quantum systems, Automatica, 44(1), 98-108, 

DOI: 10.1016/j.automatica.2007.05.013. 

Mirrahimi M., Rouchon P., Turinici G., 2005, Lyapunov control of bilinear Schrodinger equations, Automatica, 

41, 1987-1994.  

Shi S., Zhang Q., Yuan Z., Liu W., 2011, Hybrid impulsive control for switched singular systems, IET Ctrl. 

Theory Appl., 5(1), 103-111. 

Turinici G., 2007, Beyond bilinear controllability: applications to quantum control. Int. Series Num, Math., 155, 

293-309.  

Vaiano V., Sacco O., Sannino D., Di Capua G., Femia N., 2017, Enhanced performances of a photocatalytic 

reactor for wastewater treatment using controlled modulation of leds light, Chemical Engineering 

Transactions, 57, 553-558, DOI: 10.3303/CET1757093. 

Wang Q.F., 2009, Quantum optimal control of nuclei in the presence of perturbation in electric field, IET 

Control Theory Applications., 3(9), 1175-1182, DOI: 10.1049/iet-cta.2008.0292. 

Wang W., Wang L.C., Yi X.X., 2010, Lyapunov control on quantum open systems in decoherence-free 

subspaces, Physical Review A, 82(3), DOI: 10.1103/PhysRevA.82.034308. 

Wang X.T., Schirmer S., 2010, Analysis of Lyapunov Method for Control of Quantum States, IEEE Trans. 

Autom. Control, 55(10), 2259-2270, DOI: 10.1109/TAC.2010.2043292 

Zhao S.W., Lin H., Sun J.T., Xue Z.G., 2012, An implicit Lyapunov control for finite-dimensional closed 

quantum systems, International Journal of Robust & Nonlinear Control, 22(11), 1212-1228, DOI: 

10.1002/rnc.1748 

Zhao S.W., Lin H., Xue Z.G., 2012, Switching control of closed quantum systems via the Lyapunov method. 

Automatica, 48(8), 1833-1838, DOI: 10.1016/j.automatica.2012.05.069. 

 

672