CONTROLABILITY OF OSCILLATING SYSTEMS
OF SECOND ORDER
Josef Diblík1, Denys Ya. Khusainov2, and Jana Lukáčová3
1
Brno University of Technology, Brno, Czech Republic
2
Kiev University, Kiev, Ukraine
3
Žilina University, Žilina, Slovak Republic
Abstract. In this paper we study the controllability of systems of second-order
ordinary differential equations describing oscillation processes. Special matrix
functions called matrix sine and matrix cosine are applied to obtain the relevant
control functions.
1 Introduction
In control theory, one of the best solved problems is the problem of controllability of
linear time-independent differential systems
x(t )  Ax(t )  bu (t )
(1)
n
where x  R  [0 )  R is continuously differentiable, A is an n  n constant
matrix, b is a constant n1 vector with the input u  R  R (see, e.g., [1,2], [4]–
[7]). As usual, a system (1) is called controllable if, for any arbitrary finite initial state
x(0)  x0 and terminal state x(t1 )  x1 with finite t1  0 , there exists a control u  (t )
such that the system
x(t )  Ax(t )  bu  (t )
has a solution x  x (t ) satisfying x (0)  x0 and x (t1 )  x1 .
In this paper we study a system of second-order differential equations describing
oscillation processes. In spite of being concerned with a particular case of the problem
of controllability for (1), our investigation brings new results. Unlike the known
investigations, ours, in addition to giving the controllability criterion, consistently
applies special matrix functions (called the matrix sine and matrix cosine functions) to
obtain formulas for the relevant control functions.
First, in part 2, auxiliary statements are given regarding the linear independence of
components of a special vector and the regularity of a special matrix, to be used in the
investigation. We also define the two above-mentioned important matrix functions:
the matrix sine and the matrix cosine. Some of their properties are analysed. In part 3
we illustrate the idea of our analysis using a scalar second-order equation
y(t )   2 y (t )  b2u (t )
(equation (13) below), describing oscillating processes. Considering an equivalent
first order system of two equations, we obtain a criterion of controllability and a
control function is developed. Finally we investigate the controllability of the system
y(t )   2 y (t )  b2u (t )
(system (27) below) and constructions of a relevant control function in part 4 where
the results given in part 3 for equation (13) are generalized.
2 Preliminaries
In this part we will prove the necessary auxiliary statements to be used in the
following investigation. The linear independence on a given interval of the
7-21
components of a vector is discussed as well as the regularity of a special matrix.
Finally, we define the matrix sine and the matrix cosine.
2.1 Independence of the components of a vector and the regularity of an
auxiliary matrix
Assume that A is a constant n  n matrix and b is a constant n1 vector. Let
 (t )  (1 (t ) 2 (t )… n (t ))T  [   ]  R n
where    ,   R ,   R be a vector defined as
 (t )  e At  b
and S n be an n  n matrix
Sn  ( b  Ab  A2b … An 1b )
Lemma 2.1 Assume
rank Sn  n
(2)
Then the components of the vector  are linearly independent on [   ] .
Proof. Using the decomposition
1
1
1
exp At  I  At  ( At ) 2  … ( At ) n 
1
2
n
where I is n  n unit matrix, we obtain
1
1
1
 (t )   exp At   b  b  Atb  ( At ) 2 b  … ( At ) n b  
1
2
n
If the components of  are linearly dependent on [   ] , then there exists a nonzero
vector  ( 1  2 … n )T such that
T
on [   ] . Assuming that
 (t ) 
T
e At b  0
(3)
T
is such a nonzero vector, we rewrite (3) as
1
1 T
1 T
T
 (t )  T b  T Atb 
( At ) 2 b  …
( At ) n b   0
(4)
1
2
n
Taking repeatedly derivatives of (4) up to order ( n  1) , we get the series (with
general terms indicated)
1
1 T 32 
1 T n 1 n 
( T  (t ))  T Ab  T A2tb 
A t b  …
A t b   0
(5)
1
2
n
1
1 T 42 
1 T n2 n 
( T  (t ))  T A2b  T A3tb 
A t b  …
A t b   0
(6)
1
2
n
…
1
1 T n 1 2 
1 T 2 n 1 n 
( T  (t ))( n 1)  T An 1b  T Antb 
A tb  
A t b 0
(7)
1
2
n
on [   ] . Identities (4)–(7) hold on [   ] if and only if the first terms of the relevant
series are equal to zero, i.e., if
T 
b  0
T
T
T
Ab  0
A2b  0
…
An 1b  0
7-22
We can view this as a system of equations to determine T . We rewrite it as
T
(8)
 ( b  Ab  A2b … An1b )  0
System (8) is homogeneous and, by our assumption, it has a nonzero solution . But
this is possible if and only if
rank ( b  Ab  A2b … An 1b )  rank S n  n
We get a contradiction with (2). Therefore,
cannot be a nonzero vector. Our
assumption is false and the conclusion of the lemma holds. 
Lemma 2.2 Let the condition (2) be valid. Then
det
  (s) (s)ds   0
t
T
0
for every t  0 .
Proof. We put   0 and   t . Since the components of  (t ) are, by Lemma 2.1,
linearly independent on [0 t ] , for any nonzero vector  ( 1  2 … n )T , there exists
at least one point t1 [0 t ] such that
 T (t1 )  0
Due to the continuity of  , there exists a neighborhood U of the point t1 such that
T (t )  0 t U  [0 t ]
Therefore
 
t
0
T
( s)
 ds 
2
T
  (s) (s)ds 
t
T
0
 0
This inequality means that the symmetric matrix
t
 (s)
0
T
(9)
(s)ds
is positively definite for every t  0 . 
Remark 2.3 Note that the matrix  ( s) T ( s) is (unlike the matrix (9)) singular for
every s  [0 t ] . Indeed, direct computation yields
 1 ( s)1 ( s) 2 ( s)1 ( s) … n ( s)1 ( s) 

1 ( s)2 ( s) 2 ( s)2 ( s) … n ( s)2 ( s) 
T

 ( s) ( s) 


…


 1 ( s)n ( s) 2 ( s)n ( s) … n ( s)n ( s) 
and
det  ( s) T ( s)  1 ( s)2 ( s)…n ( s) 
1 ( s) 2 ( s) … n ( s)
1 ( s) 2 ( s) … n ( s)
…
1 ( s) 2 ( s) … n ( s)
 0
2.2 The matrix sine and the matrix cosine
In [1, p.284] and [3, pp.123-124], the matrix functions sin t and cos t are defined
on R as uniformly convergent series
1
1
1
1
sin t  t  (t )3  (t )5   (1) k
(t ) 2 k 1  
1
3
5
(2k  1)
1
1
1
1
cos t  I  (t ) 2  (t ) 4  (t )6   (1) k
(t ) 2 k  
2
4
6
(2k )
7-23
We will call them the matrix sine and the matrix cosine of the matrix  . The symbol
I stands for an n  n unit matrix, as above. The following two lemmas are direct
consequences of the definitions of sin t and cos t . Therefore, we omit their
proofs.
Lemma 2.4 For the derivatives of sin t and cos t we have
(cos t )   sin t (sin t )   cos t
Lemma 2.5 For the integrals of sin t and cos t we have
t
 cos s ds  
0
1
sin t
t
 sin s ds  I  
0
1
cos t
It is easy, with the aid of the matrix functions, to give a solution of the second-order
linear differential system
(10)
y(t )  2 y(t )  
n
where y  R  R is an unknown vector,  is an n  n constant regular matrix and
 is an n1 null vector satisfying the initial Cauchy conditions
(11)
y(0)  y0  y(0)  y0
where y0  y0  R n . Regarding system (10), we use the term “oscillating system". This
is suggested by the scalar case represented by one second-order equation which is the
equation of a linear oscillator. A solution of (10), (11) is given by
y (t )  (cos t ) y0   1 (sin t ) y0
(12)
We observe that formula (12) describes the family of all solutions of (10) since it
covers even the case of the matrix  being singular if we define
1
1
1
1
1 (sin t )  It  2t 3  4t 5   (1)k
2 k t 2 k 1  
1
3
5
(2k  1)
3 Control of Second Order Scalar Oscillating Equations
In this part we investigate second-order scalar linear controlled equations
y(t )   2 y (t )  b2u (t )
(13)
where y  R  R is an unknown function, u  R  R is a control function,
 b2  R and b2  0 . Along with (13), we consider an initial problem
y(0)  y0  y(0)  y0
(14)
where y0  y0  R are fixed. We want to solve the problem of the existence of a
control function u (t )  u0 (t ) and find its explicit form such that the equation
y(t )   2 y (t )  b2u0 (t )
has a solution y  y0 (t ) satisfying initial data (14) and, moreover, prescribed in
advance by the terminal conditions
y(t1 )  y1 y(t1 )  y1
(15)
where the constants y1  y1  R are arbitrary and t1  0 - is an arbitrarily fixed moment
of time.
3.1 Equivalent linear system of first order and controllability criterion
Using the substitutions y  x1 , y  x2 , we rewrite (13) as a system of two first-order
equations
(16)
x1(t )  x2 (t )
7-24
2
x2(t )   x1 (t )  b2u (t )
(17)
The initial and terminal conditions for system (16), (17) related to (14) and (15) are:
x1 (0)  y0  x2 (0)  y0 x1 (t1 )  y1 x2 (t1 )  y1
We introduce an auxiliary matrix
1
 0
(18)
A 2

  0 
and vectors


 y0 
 y1 
 x (t ) 
 0
b     x(t )   1   x0     x1    
b 
 x2 (t ) 
 2
 y1 
 y0 
Then, system (16), (17) with initial and terminal conditions can be written as
x(t )  Ax(t )  bu (t )
(19)
x(0)  x0 
(20)
x(t1 )  x1
(21)
We consider the controllability of (19). Since
 0 b2 
(22)
rank (b Ab)  rank 
  2
 b2 0 
in accordance with the well-known controllability criterion, system (19) is
controllable. We summarize the investigation performed regarding equation (13).
Theorem 3.1 Equation (13) is controlled if and only if b2  0 .
3.2 Construction of a control function
Now we give a control function u0 (t ) for the problem (13)–(15).
Theorem 3.2 Let b2  0 . Then a control function u0 (t ) for the problem (13)–(15)
can be written in the form
t1
T
T
u0 (t )  bT  e A(t1 t )    e A(t1 s )bbT  e A(t1 s )  ds 
 0

1

 1

x  e At1 x0  
Proof. We will construct a suitable control function u0 (t ) for the problem (19)–(21)
since, obviously, it will be a control function of the problem (13)–(15), too. We
construct the control in the form
T
u0 (t )  bT   e A( t1 t )   
(23)
where   (1   2 )T - is a vector of unknown constants and e At is the exponential
matrix of the matrix At , i.e.
 cos t
 1 sin t 
At
e 

cos t 
  sin t
The solution to the problem (19)–(21) can be derived using the Cauchy formula
t
x(t )  e At x0   e A(t s )bu(s)ds
0
(24)
In (24) we put t  t1 and u  u0 (t ) with u0 (t ) given by (23). Then
t1
x1  e At1 x0   e A(t1 s )bbT  e A(t1 s )   ds
T
0
7-25
(25)
We rewrite (25) as
 (t1 )    x1  e At1 x0
(26)
where
t1
(t1 )   e A(t1  s )bbT  e A(t1 s )  ds
T
0
The matrix  (t1 ) is (as follows from (22)) non-singular. This is a direct consequence
of Lemma 2.1 and Lemma 2.2 after substituting 2 for n , (0 b2 )T for b , 2  2 matrix
defined by (18) for A and the matrix (b Ab) for S n . Consequently, the system of
algebraic equations (26), i.e.,
(t1 )    
with an arbitrary vector
  x1  e At1 x0
has a unique solution
   1 (t1 ) 
Finally, the control function u0 (t ) is defined as
T
u0 (t )  bT   e A(t1 t )   
T
 bT   e A(t1 t )   1 (t1 )  x1  e At1 x0 
t
T
T
bT  e A(t1 t )    1 e A(t1  s )bbT  e A(t1  s )  ds 

 


 0

1

 1

x  e At1 x0 


4 Control of Second Order Oscillating Systems
We start with investigating the controllability of a second-order oscillating system of
the form
y(t )   2 y (t )  b2u (t )
(27)
where y  R  R n is an unknown vector,  is an n  n constant regular matrix,
b2  R n , b2   is a given vector and the control u  R  R n is a given vectorfunction.
We assume that the initial and terminal conditions
y(0)  y0  y(0)  y0 y (t1 )  y1 y(t1 )  y1
(28)
for (27) are given where y0  y0 y1 and y1 are arbitrary constant vectors, and t1  0 is
an arbitrary but fixed moment of time.
We will analyse the controllability problem if there exists a control function u  u0 (t )
such that the system (27) with u  u0 (t ) , i.e., the system
y(t )   2 y (t )  b2u0 (t )
has a solution y  y0 (t )  R n satisfying the initial and terminal conditions (28). Then
we will describe a construction of the control function u0 (t ) using the matrix sine and
the matrix cosine.
4.1 Equivalent linear first order system and criterion controllability of (27)
Using substitutions y  x1 , y  x2 where x1 , x2 are new n -dimensional vectors, we
replace a system of n second-order differential equations (27) with a system of 2n
first-order differential equations
7-26
x1(t )  x2 (t )
2
x2(t )   x1 (t )  b2u (t )
The initial and terminal conditions for system (29), (30) derived from (28) are
x1 (0)  y0  x2 (0)  y0 x1 (t1 )  y1 x2 (t1 )  y1
We introduce an auxiliary 2n  2n matrix
I
 
A

2
   
and 2n -dimensional vectors


 y0 
 y1 
 x (t ) 
 
b     x(t )   1   x0     x1    
b 
 x2 (t ) 
 2
 y1 
 y0 
Then the system (29), (30) with initial and terminal conditions can be written as
x(t )  Ax(t )  bu (t )
x(0)  x0 
x(t1 )  x1
(29)
(30)
(31)
(32)
(33)
(34)
Define auxiliary matrices: an n  n matrix
S n  ( b2   2b2  ( 2 ) 2 b2 … ( 2 ) n 1 b2 )
and a 2n  2n matrix
S2 n  ( b Ab A2b… A2 n 1b )
Theorem 4.1 System (27) is controlled if and only if rank Sn  n
Proof. It is obvious that system (27) is controlled if and only if system (32) is
controlled. In accordance with the well-known control criterion for linear systems,
system (32) is controllable if and only if rank S2 n  2n
We evaluate the rank of S 2n . To do this, we compute the matrices Ak b ,
k  1 2… 2n  1 . We obtain
7-27
 
Ab   2
 
 
A2 b   2
 
I      b2 
     
   b2    
 
A3b   2
 
I       2b2 
 


    2b2    
 
A b 2
 
I    2b2    
2 


        2  b2 
 
A5b   2
 

   b 
I




  
  b


2



I     2  b2   




   2  3  

 


       b2 


   2  3b 

I   


 2 
3  


     2  b2  




 

3


I     2  b2    
        2  4  
 

     b2 
4
 
A6b   2
 
 
A7b   2
 
 
A8b   2
 
I  b2    

   
      2b2 






 




2 2 
 2 





2 2
 2

…
A
 
b 2
 
2 n 1



 (1) n 1   2  n 1b 

I  
2




n 1   


   (1) n 1   2  b2  







Then
rank S 2 n  ( b Ab A2b… A2 n 1b )
 b
2
 rank 

 b2 

 2b2

n 1
… (1) n 1   2  b2
 2  rank  b2   2b2 … (1) n 1 ( 2 ) n 1 b2
(1) n 1   2  b2 




n 1

 2b2 …

 2  rank S n 
System (32) is controlled if and only if rank S2 n  2n . Then, as it follows from the
last computation, system (27) is controlled if and only if rank Sn  n . 
4.2. Control function for the problem (27), (28)
Now we will construct a control function u0 (t ) for the problem (27), (28).
Theorem 4.2 Assume rank Sn  n . Then a control function u0 (t ) for the
problem (27), (28) can be written as
T
u0 (t )  bT  e A(t1 t )  
t1
T
T
 bT  e A(t1 t )    e A(t1  s )bbT  e A(t1  s )  ds 
 0

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1

 1

x  e At1 x0  
Proof. System (27) is controllable by Theorem 4.1. Due to the control problems (27),
(28) and (32)–(34) being equivalent, it is sufficient to construct a control function
u0 (t ) for the problem (32)–(34). It is easy to verify, that
 cos t
1 sin t 
e At  

cos t 
  sin t
We construct a control function in the form
T
u0 (t )  bT  e A(t1 t )   
(35)
where   1   2  is a 2n -dimensional constant vector and 1 ,  2 are n -dimensional
T
vectors, 1  (11  12 … 1n ) ,  2  ( 21   22 …  2n ) . Now we proceed like in part 3. From
the Cauchy formula (24) (we assume that the dimensions reflect our 2n -dimensional
case) we get, for t  t1 , u  u0 (t ) ,
t1
x(t1 )  e At1 x(0)   e A(t1 s )bbT  e A(t1 s )   ds
T
0
We define a 2n  2n matrix
t1
(t1 )   e A(t1 s )bbT  e A(t1 s )  ds
T
0
and rewrite the last relation as a system of linear 2n -th order nonhomogeneous
algebraic equations of
 (t1 )  x1  e At1 x0 
We note that the matrix  (t1 ) is non-singular. This follows directly from Lemma 2.1
and Lemma 2.2 if n is replaced by 2n , b by b , matrix A by 2n  2n matrix A
defined by (31) and matrix S n by the matrix S 2n . Then the system of algebraic
equations
(t1 )  
with
an
arbitrary
2n -dimensional
vector
   1   2  ,
T
 1  ( 11   12 …  1n ) ,
 2  ( 21   22 …  2n ) defined as
  x1  e At x0 
1
has a unique solution
   1 (t1 ) 
Finally, the control function has the form
T
u0 (t )  bT  e A(t1 t )  
t1
T
T
 bT  e A(t1 t )    e A(t1  s )bbT  e A(t1  s )  ds 
 0

1

 1

x  e At1 x0  

Remark 4.3 It is easy to see that Theorem 4.1 generalizes Theorem 3.1 and
Theorem 4.2 is a generalization of Theorem 3.2. When constructing the control
function of the form (35) we use the same methodology as that used in the scalar case.
ACKNOWLEDGEMENTS
Josef Diblík, Brno University of Technology, Brno, Czech Republic. This author was
supported by the Grant 201/08/0469 of Czech Grant Agency (Prague), and by the
7-29
Council of Czech Government MSM 0021630503, MSM 0021630519 and MSM
0021630529 (diblik@feec.vutbr.cz, diblik.j@fce.vutbr.cz).
Denys Ya. Khusainov, Kiev University, Kiev, Ukraine. This author was supported by
Slovak-Ukrainian project No SK-UA-0028-07 (khusainov@unicyb.kiev.ua).
Jana Lukáčová, Žilina University, Žilina, Slovak Republic. This author was supported
by the Grant 1/3238/06 of the Grant Agency of Slovak Republic (VEGA) and by
Slovak-Ukrainian project No SK-UA-0028-07 (jana.lukacova@fpv.uniza.sk).
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Providence, Rhode Island, 1977.
[4] R.E. KALMAN, P.L. FALB, M.A. ARBIB, Topics in Mathematical System Theory,
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[5] N.N. KRASOVSKII, Theory of Control of Motion: Linear Systems, Izdat. Nauka,
Moscow, 1968. (Russian)
[6] J. MACKI, A. STRAUSS, Introduction to Optimal Control Theory, Springer-Verlag,
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[7] J. ZABCYK, Mathematical Control Theory: An Introduction (Systems & Control:
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