Advertisement

Advances in Difference Equations

, 2012:229 | Cite as

Stabilization of Lur’e-type nonlinear control systems by Lyapunov-Krasovskii functionals

  • Andrei Shatyrko
  • Josef Diblík
  • Denys Khusainov
  • Miroslava RůžičkováEmail author
Open Access
Research
Part of the following topical collections:
  1. Progress in Functional Differential and Difference Equations

Abstract

The paper deals with the stabilization problem of Lur’e-type nonlinear indirect control systems with time-delay argument. The sufficient conditions for absolute stability of the control system are established in the form of matrix algebraic inequalities and are obtained by the direct Lyapunov method.

MSC:34H15, 34K20, 93C10, 93D05.

Keywords

Lyapunov functional absolute stability time-delay argument stabilization 

1 Introduction

One of the problems of stability motion is the problem of absolute stability. The problems of absolute stability of nonlinear control systems arise in solving practical tasks. In technical control systems, the control function is the function of one variable located between two lines in the first and third quarters of the coordinate plane. The stability of the control system with a control function located in this sector is referred to, for example, in [1, 2, 3, 4, 5, 6]. Originally, the control systems of ordinary differential equations were considered. The systems with aftereffect, that better describe the real processes, become an object of study later, e.g., in [3, 4, 7, 8]. Some nonlinear systems with indirect regulation and delay argument are considered in [9, 10]. The sufficient conditions of absolute interval stability are derived in the papers [3, 6] by Lyapunov-Krasovskii functionals in the form of the sum of the quadratic form and the integral of nonlinear components of the considered system, and by the so-called S-program the coefficients of the exponential decay of solutions are calculated. But, in the case when the conditions of the theorems quoted there are not met, the linear feedback method is used to stabilize the system.

The main goal of the paper is to solve the problem of stabilization of an indirect control system. The sufficient conditions for absolute stability of the control system are obtained using Lyapunov-Krasovskii functionals which contain an exponential multiplier.

Throughout the paper we will use the following notation. Let S Open image in new window be a real symmetric square matrix. Then the symbol λ min ( S ) Open image in new window ( λ max ( S ) Open image in new window) will denote the minimal (maximal) eigenvalue of S Open image in new window. We will also use the following vector norms:
x ( t ) : = i = 1 n x i 2 ( t ) , x ( t ) τ , ξ : = t τ t e ξ ( t s ) x ( s ) 2 d s , Open image in new window

where x = ( x 1 , x 2 , , x n ) T Open image in new window and ξ is a real parameter.

The paper is organized as follows. Since for the one-dimensional process it is possible to get simple explicit criteria, Section 2 deals with the stabilization of one-dimensional processes described by two scalar equations with delay. Then the indirect control system in the general matrix form is considered in Section 3.

2 Stabilization of one-dimensional processes

Let us consider an indirect control system described by a system of two scalar equations with delay argument in the form
x ˙ ( t ) = a 1 x ( t ) + a 2 x ( t τ ) + b f ( σ ( t ) ) , Open image in new window
(1)
σ ˙ ( t ) = c x ( t ) ρ f ( σ ( t ) ) , Open image in new window
(2)
where t t 0 0 Open image in new window, x is the state function, σ is the control defined on [ t 0 , ) Open image in new window, a 1 Open image in new window, a 2 Open image in new window, b, c, τ > 0 Open image in new window, ρ > 0 Open image in new window are constants, f ( σ ) Open image in new window is a continuous nonlinear function on ℝ satisfying the so-called sector condition. It means there exist constants k 1 Open image in new window, k 2 Open image in new window, k 2 > k 1 > 0 Open image in new window such that inequalities
k 1 σ 2 f ( σ ) σ k 2 σ 2 Open image in new window
(3)

are satisfied.

Definition 1 The continuous vector function ( x , σ ) : [ t 0 τ , ) R 2 Open image in new window is said to be a solution of (1), (2) on [ t 0 , ) Open image in new window if ( x , σ ) Open image in new window is continuously differentiable on [ t 0 , ) Open image in new window and satisfies the system (1), (2) on [ t 0 , ) Open image in new window.

Definition 2 The system (1), (2) is called absolutely stable if the trivial solution ( x , σ ) = ( 0 , 0 ) Open image in new window of the system (1), (2) is globally asymptotically stable for an arbitrary function f ( σ ) Open image in new window satisfying (3).

In the investigation of absolute stability of the control systems with delay, we will use Lyapunov-Krasovskii functionals which contain, in addition to the quadratic form and the integral of the nonlinear component of the considered system, an exponential multiplier, i.e.,
V [ x ( t ) , σ ( t ) ] = h x 2 ( t ) + g t τ t e ξ ( t s ) x 2 ( s ) d s + β 0 σ ( t ) f ( s ) d s , Open image in new window
(4)
where h, g, β, ξ are positive constants, ( x , σ ) Open image in new window is a solution of (1), (2), and t t 0 Open image in new window. It is easy to see that the last term in (4) is always nonnegative due to the left-hand part of ‘sector condition’ (3). Define, using the coefficients of the functional (4), auxiliary numbers
s 11 1 = 2 a 1 h g , s 12 1 = a 2 h , s 13 1 = ( h b + 1 2 β c ) , s 22 1 = e ξ τ g , s 33 1 = β ρ Open image in new window
and a matrix
S 1 = S 1 ( g , h , β , ξ ) : = ( s 11 1 s 12 1 s 13 1 s 12 1 s 22 1 0 s 13 1 0 s 33 1 ) = ( 2 a 1 h g a 2 h h b 1 2 β c a 2 h e ξ τ g 0 h b 1 2 β c 0 β ρ ) . Open image in new window

Our first result is the theorem on absolute stability for the considered system (1), (2).

Theorem 1 Suppose that there exist constants g > 0 Open image in new window, h > 0 Open image in new window, β > 0 Open image in new window, and ξ > 0 Open image in new window such that the matrix S 1 ( g , h , β , ξ ) Open image in new window is positive definite. Then the system (1), (2) is absolutely stable.

Proof Compute the full derivative of the functional V [ x ( t ) , σ ( t ) ] Open image in new window defined by (4) along trajectories of the system (1), (2). Then
d d t V [ x ( t ) , σ ( t ) ] = 2 h x ( t ) [ a 1 x ( t ) + a 2 x ( t τ ) + b f ( σ ( t ) ) ] + g [ x 2 ( t ) e ξ τ x 2 ( t τ ) ] g ξ t τ t e ξ ( t s ) x 2 ( s ) d s + β f ( σ ( t ) ) [ c x ( t ) ρ f ( σ ( t ) ) ] = [ 2 h a 1 + g ] x 2 ( t ) + 2 h a 2 x ( t ) x ( t τ ) g e ξ τ x 2 ( t τ ) + ( 2 h b + c β ) x ( t ) f ( σ ( t ) ) β ρ f 2 ( σ ( t ) ) g ξ t τ t e ξ ( t s ) x 2 ( s ) d s = ( x ( t ) , x ( t τ ) , f ( σ ( t ) ) ) S 1 ( g , h , β , ξ ) ( x ( t ) , x ( t τ ) , f ( σ ( t ) ) ) T g ξ x ( t ) τ , ξ 2 λ min ( S ) ( x ( t ) 2 + x ( t τ ) 2 + f ( σ ( t ) ) 2 ) g ξ x ( t ) τ , ξ 2 . Open image in new window
Using (3) we get
d d t V [ x ( t ) , σ ( t ) ] λ min ( S ) ( x ( t ) 2 + x ( t τ ) 2 + k 1 2 σ 2 ( t ) ) g ξ x ( t ) τ , ξ 2 . Open image in new window
From this inequality and the estimates
h x ( t ) 2 V [ x ( t ) , σ ( t ) ] h x ( t ) 2 + g x ( t ) τ , ξ 2 + 1 2 k 2 σ 2 ( t ) , Open image in new window

where the last term can be derived using the right-hand part of ‘sector condition’ (3), we deduce the absolute stability of the system (1), (2) (we also refer to a theorem by Krasovskii in [[11], Theorem 2, p.145]). □

The crucial assumption in Theorem 1 is the assumption of positive definiteness of the matrix S 1 ( g , h , β , ξ ) Open image in new window. If we cannot find suitable constants g, h, β, and ξ to ensure positive definiteness, or such constants do not exist, Theorem 1 is not applicable. In such a case, we can modify the control function in (1) by adding a linear combination of the values of the state function at the moments t and t τ Open image in new window and we will consider a modified system
x ˙ ( t ) = a 1 x ( t ) + a 2 x ( t τ ) + b f ( σ ( t ) ) + u ( t ) , Open image in new window
(5)
σ ˙ ( t ) = c x ( t ) ρ f ( σ ( t ) ) , Open image in new window
(6)
where
u ( t ) = c 1 x ( t ) + c 2 x ( t τ ) , Open image in new window
(7)

c 1 Open image in new window and c 2 Open image in new window are suitable constants, and t t 0 0 Open image in new window.

Then we can apply the following result.

Theorem 2 Let g > 0 Open image in new window, h > 0 Open image in new window, β > 0 Open image in new window, and ξ > 0 Open image in new window be fixed. Then the system (5), (6) is absolutely stable if the constants c 1 Open image in new window, c 2 Open image in new window in the control function (7) fulfill the inequality
c 1 < 1 2 h [ s 11 1 1 s 22 1 ( s 12 1 c 2 h ) 2 1 s 33 1 ( s 13 1 ) 2 ] . Open image in new window
(8)
Proof We employ the same functional (4) and the scheme of the proof of Theorem 1. Tracing the proof of Theorem 1, we get that for the absolute stability of the system (5), (6), it is sufficient that the matrix
S 2 = S 2 ( g , h , β , ξ ) : = ( s 11 1 2 c 1 h s 12 1 c 2 h s 13 1 s 12 1 c 2 h s 22 1 0 s 13 1 0 s 33 1 ) Open image in new window
is positive definite. Applying the known positivity criterion (Sylvester criterion) [[12], p.260], [13] to the matrix S 2 Open image in new window, we require the positivity of its main diagonal minors, i.e.,
Δ 1 = s 11 1 2 c 1 h > 0 , Open image in new window
(9)
Δ 2 = ( s 11 1 2 c 1 h ) s 22 1 ( s 12 1 c 2 h ) 2 > 0 , Open image in new window
(10)
Δ 3 = ( s 11 1 2 c 1 h ) s 22 1 s 33 1 s 22 1 ( s 13 1 ) 2 s 33 1 ( s 12 1 c 2 h ) 2 > 0 . Open image in new window
(11)
The inequality (9) can be rewritten as
c 1 < 1 2 h s 11 1 . Open image in new window
(12)
By a simple modification of inequality (10), we have
2 c 1 h s 22 1 < s 11 1 s 22 1 ( s 12 1 c 2 h ) 2 . Open image in new window
Hence, taking into account that s 22 1 = e ξ τ g > 0 Open image in new window, we get a more suitable relationship,
c 1 < 1 2 h [ s 11 1 1 s 22 1 ( s 12 1 c 2 h ) 2 ] . Open image in new window
(13)
The inequality (11) can be modified into the form
2 c 1 h s 22 1 s 33 1 < s 11 1 s 22 1 s 33 1 s 22 1 ( s 13 1 ) 2 s 33 1 ( s 12 1 c 2 h ) 2 . Open image in new window
(14)
Finally, with regard to the assumptions
h > 0 , s 22 1 = e ξ τ g > 0 , s 33 1 = β ρ > 0 , Open image in new window

the inequality (14) can be written in the form (8). If this inequality holds, then obviously (12) and (13) hold as well. Moreover, it is easy to see that it is possible to find parameters c 1 Open image in new window and c 2 Open image in new window such that the inequality (8) is fulfilled. □

3 Stabilization of the indirect control systems with matrix coefficients

Our goal in this section is to extend the considerations developed in Section 2 to the study of stabilization of the indirect control systems whose coefficients are expressed in a matrix form. It means we will consider an n-dimensional process x described by the system of ( n + 1 ) Open image in new window equations,
x ˙ ( t ) = A x ( t ) + B x ( t τ ) + b f ( σ ( t ) ) , Open image in new window
(15)
σ ˙ ( t ) = c x ( t ) ρ f ( σ ( t ) ) , Open image in new window
(16)

where t t 0 0 Open image in new window, x = ( x 1 , x 2 , , x n ) T Open image in new window is the n-dimensional column vector function of the state, σ is the scalar function of the control defined on [ t 0 , ) Open image in new window, A and B are n × n Open image in new window constant matrices, b = ( b 1 , b 2 , , b n ) T Open image in new window is an n-dimensional constant column vector, c = ( c 1 , c 2 , , c n ) Open image in new window is an n-dimensional constant row vector, τ > 0 Open image in new window and ρ > 0 Open image in new window are constants, and f ( σ ) Open image in new window is a continuous nonlinear function on ℝ satisfying sector condition (3).

To investigate the system (15), (16) we use a Lyapunov-Krasovskii functional, generalizing the functional (4), in the form
V [ x ( t ) , σ ( t ) ] = x T ( t ) H x ( t ) + t τ t e ξ ( t s ) x T ( s ) G x ( s ) d s + β 0 σ ( t ) f ( s ) d s , Open image in new window
(17)

where H and G are n × n Open image in new window constant positive definite symmetric matrices, and ξ and β are positive constants.

We give a generalization of Theorem 1 to the case of the control system (15), (16). For it, we define the matrices
S 11 3 : = A T H H A G , S 12 3 : = H B , S 13 3 : = ( H b + 1 2 β c T ) , S 22 3 : = e ξ τ G Open image in new window
and
S 3 ( G , H , β , ξ ) : = ( S 11 3 S 12 3 S 13 3 , ( S 12 3 ) T S 22 3 θ ( S 13 3 ) T θ T s 33 1 ) , Open image in new window

where θ = ( θ , θ , , θ ) T Open image in new window is an n-dimensional zero column vector.

Theorem 3 Suppose that there exist positive definite symmetric matrices H, G and constants β > 0 Open image in new window, ξ > 0 Open image in new window such that the matrix S 3 ( G , H , β , ξ ) Open image in new window is positive definite. Then the system (15), (16) is absolutely stable.

Proof The scheme of the proof repeats the proof of Theorem 1. Compute the full derivative of the functional V [ x ( t ) , σ ( t ) ] Open image in new window defined by (17) along trajectories of the system (15), (16). Then
d d t V [ x ( t ) , σ ( t ) ] = x ˙ T ( t ) H x ( t ) + x T ( t ) H x ˙ ( t ) + x T ( t ) G x ( t ) e ξ τ x T ( t τ ) G x ( t τ ) ξ t τ t e ξ ( t s ) x T ( s ) G x ( s ) d s + β f ( σ ( t ) ) σ ˙ ( t ) = [ A x ( t ) + B x ( t τ ) + b f ( σ ( t ) ) ] T H x ( t ) + x T ( t ) H [ A x ( t ) + B x ( t τ ) + b f ( σ ( t ) ) ] + x T ( t ) G x ( t ) e ξ τ x T ( t τ ) G x ( t τ ) ξ t τ t e ξ ( t s ) x T ( s ) G x ( s ) d s + β f ( σ ( t ) ) [ c x ( t ) ρ f ( σ ( t ) ) ] = ( x T ( t ) , x T ( t τ ) , f ( σ ( t ) ) ) S 3 ( G , H , β , ξ ) ( x T ( t ) , x T ( t τ ) , f ( σ ( t ) ) ) T ξ t τ t e ξ ( t s ) x T ( s ) G x ( s ) d s λ min ( S ) ( x ( t ) 2 + x ( t τ ) 2 + f ( σ ( t ) ) 2 ) ξ λ min ( G ) x ( t ) τ , ξ 2 . Open image in new window
Using (3) we get
d d t V [ x ( t ) , σ ( t ) ] λ min ( S ) ( x ( t ) 2 + x ( t τ ) 2 + k 1 2 σ 2 ( t ) ) ξ λ min ( G ) x ( t ) τ , ξ 2 . Open image in new window
From this inequality and the estimates
λ min ( H ) x ( t ) 2 V [ x ( t ) , σ ( t ) ] λ max ( H ) x ( t ) 2 + λ max ( G ) x ( t ) τ , ξ 2 + 1 2 k 2 σ 2 ( t ) , Open image in new window

where the last term can be derived using the right-hand part of sector condition (3), we deduce the absolute stability of the system (15), (16) (we also refer to a theorem by Krasovskii in [[11], Theorem 2, p.145]). □

It may happen that it is not easy to find suitable positive definite symmetric matrices H, G and constants β > 0 Open image in new window, ξ > 0 Open image in new window such that the matrix S 3 ( G , H , β , ξ ) Open image in new window will be positive definite, or such matrices and constants do not exist. In such a case, we can modify the control function in the system (15), (16) by adding a linear combination of the values of the state function at the moments t and t τ Open image in new window. Therefore, instead of the system (15), (16), we will consider a modified system
x ˙ ( t ) = A x ( t ) + B x ( t τ ) + b f ( σ ( t ) ) + u ( t ) , Open image in new window
(18)
σ ˙ ( t ) = c x ( t ) ρ f ( σ ( t ) ) , Open image in new window
(19)
where
u ( t ) = C 1 x ( t ) + C 2 x ( t τ ) , Open image in new window
(20)

C 1 Open image in new window and C 2 Open image in new window are n × n Open image in new window constant matrices (the so-called control matrices), and t t 0 0 Open image in new window. Our task is to find conditions on the matrices C 1 Open image in new window, C 2 Open image in new window such that the system (18), (19) will be absolutely stable.

We will need some auxiliary results from the theory of matrices.

Lemma 1 [13]

Let A be a regular n × n Open image in new window matrix, B be an n × q Open image in new window matrix, and C be a q × q Open image in new window regular matrix. Let a Hermitian matrix S be represented as
S = ( A B B C ) . Open image in new window

Then the matrix S is positive definite if and only if the matrices A and C B A 1 B Open image in new window are positive definite.

Lemma 2 [[12], Frobenius formula]

Let A be a regular n × n Open image in new window matrix, D be a q × q Open image in new window matrix, B be an n × q Open image in new window matrix, and C be a q × n Open image in new window matrix, and the matrix
M = ( A B C D ) Open image in new window
be regular. Then the matrix R = D C A 1 B Open image in new window is regular and
M 1 = ( A 1 + A 1 B R 1 C A 1 A 1 B R 1 R 1 C A 1 R 1 ) . Open image in new window
Theorem 4 Suppose that there exist positive definite symmetric matrices H and G, control matrices C 1 Open image in new window and C 2 Open image in new window, and constants β > 0 Open image in new window and ξ > 0 Open image in new window such that
  1. (1)
    The matrices
    Δ 1 4 : = S 11 3 C 1 T H H C 1 , Open image in new window
    (21)
     
Δ 2 4 : = S 22 3 [ S 12 3 H C 2 ] T [ S 12 3 C 1 T H H C 1 ] 1 [ S 12 3 H C 2 ] Open image in new window
(22)
are positive definite.
  1. (2)
    The number
    Δ 3 4 : = β ρ ( S 13 3 ) T [ ( S 11 4 ) 1 + ( S 11 4 ) 1 S 12 4 R 1 ( S 12 4 ) T ( S 11 4 ) 1 ] S 13 3 , Open image in new window
    (23)
     
where
S 11 4 = Δ 1 4 , S 12 4 = S 12 3 H C 2 , R 1 = S 22 3 ( S 12 4 ) T ( S 11 4 ) 1 S 12 4 , Open image in new window
(24)

is positive.

Then the system (18), (19) is absolutely stable.

Proof The philosophy of the proof is the same as in the proof of Theorem 2, only the calculations will be more complicated, because now we work with the matrix case. In accordance with Theorem 3, the system (15), (16) is absolutely stable if the matrix S 3 ( G , H , β , ξ ) Open image in new window is positive definite. Define the auxiliary matrix
S 4 = S 4 ( G , H , C 1 , C 2 , β , ξ ) : = ( S 11 4 S 12 4 S 13 3 ( S 12 4 ) T S 22 3 θ ( S 13 3 ) T θ T s 33 1 ) . Open image in new window
The matrix S 4 Open image in new window plays the same role for the system (18), (19) as the matrix S 3 ( G , H , β , ξ ) Open image in new window for the system (15), (16). Therefore, the system (18), (19) is absolutely stable if the matrix S 4 ( G , H , C 1 , C 2 , β , ξ ) Open image in new window is positive definite. It follows from Lemma 1 that the matrix S 4 ( G , H , C 1 , C 2 , β , ξ ) Open image in new window is positive definite if and only if the matrix
M 4 = ( S 11 4 S 12 4 ( S 12 4 ) T S 22 3 ) Open image in new window
is positive definite and the inequality
s 33 1 > ( ( S 13 3 ) T θ T ) ( M 4 ) 1 ( S 13 3 θ ) Open image in new window
(25)

holds.

The matrix M 4 Open image in new window is positive definite (we use Lemma 1 again) if and only if the matrices
S 11 4 , S 22 3 ( S 12 4 ) T ( S 11 4 ) 1 S 12 4 Open image in new window
are positive definite. The matrix S 11 4 Open image in new window is positive definite due to (21). The matrix
S 22 3 ( S 12 4 ) T ( S 11 4 ) 1 S 12 4 = S 22 3 [ S 12 3 H C 2 ] T [ S 11 3 C 1 T H H C 1 ] 1 [ S 12 3 H C 2 ] Open image in new window

is positive definite due to (22).

We compute the inverse matrix to the matrix M 4 Open image in new window using Lemma 2. We get
( M 4 ) 1 = ( M 4 11 ( S 11 4 ) 1 S 12 4 R 1 R 1 S 12 4 ( S 11 4 ) 1 R 1 ) , Open image in new window
where
M 4 11 = ( S 11 4 ) 1 + ( S 11 4 ) 1 S 12 4 R 1 ( S 12 4 ) T ( S 11 4 ) 1 . Open image in new window
Therefore, the inequality (25) can be rewritten as
s 33 1 > ( S 13 3 ) T M 4 11 S 13 3 Open image in new window

and is valid due to (23).

Consequently, the system with control of the form (18), (19) is absolutely stable if there exist matrices C 1 Open image in new window, C 2 Open image in new window in (20) such that conditions (21)-(23) are valid. □

Remark 1 Let us recall the well-known facts that for the validity of Theorem 1, it is necessary that a 1 < 0 Open image in new window and for the validity of Theorem 3, it is necessary that all characteristic values of the matrix A have negative real parts.

Notes

Acknowledgements

The authors would like to thank the following for their support: The first and the third authors were supported by the National Scholarship Program of the Slovak Republic (SAIA), the second author was supported by Grant P201/11/0768 of the Czech Grant Agency (Prague), the fourth author was supported by the Grant Agency of the Slovak republic (VEGA 1/0090/09). The authors would like to thank the referees and the editor for helpful suggestions incorporated into this paper.

References

  1. 1.
    Aizerman MA, Gantmacher FR: Absolute Stability of Regulator Systems. Holden-Day, San Francisco; 1964.Google Scholar
  2. 2.
    Lur’e AI: Some Problems in the Theory of Automatic Control. H.M. Stationary Office, London; 1957.Google Scholar
  3. 3.
    Liao X, Yu P Series Mathematical Modelling: Theory and Applications 25. In Absolute Stability of Nonlinear Control Systems. 2nd edition. Springer, New York; 2008.CrossRefGoogle Scholar
  4. 4.
    Liao X, Wang L, Yu P Monograph Series on Nonlinear Science and Complexity. In Stability of Dynamical Systems. Elsevier, Amsterdam; 2007.CrossRefGoogle Scholar
  5. 5.
    Růžičková M, Khusainov DY, Kuzmych O: The estimation of the dynamics of indirect control switching systems. Tatra Mt. Math. Publ. 2011, 48: 197–213.MathSciNetGoogle Scholar
  6. 6.
    Yakubovich VA, Leonov GA, Gelig AK: Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities. World Scientific, Singapore; 2004.CrossRefGoogle Scholar
  7. 7.
    Khusainov DY, Shatyrko AV: Lyapunov Functions Method in Investigation of Functional-Differential Systems. Kiev National University Publ., Kiev; 1997. (In Russian)Google Scholar
  8. 8.
    Korenevskiy DG: Dynamical Systems Stability under Random Perturbation of Parameters. Algebraic Criteria. Naukova Dumka, Kiev; 1989. (In Russian)Google Scholar
  9. 9.
    Shatyrko AV, Khusainov DY: Absolute stability conditions construction of non-direct regulator systems by Lyapunov-Krasovskiy functional. Bull. Kyiv Univ., Ser. Phys. & Math. 2009, 4: 145–152. (In Ukrainian)Google Scholar
  10. 10.
    Shatyrko AV, Khusainov DY: Stability of Nonlinear Control Systems with Aftereffect. Inform.-Analit. Agency Publ., Kiev; 2012.Google Scholar
  11. 11.
    El’sgol’ts LE, Norkin SB: Introduction to the Theory of Differential Equations with Deviating Argument. 2nd edition. Nauka, Moscow; 1971. (In Russian)Google Scholar
  12. 12.
    Gantmacher FR: Theory of Matrices. Nauka, Moscow; 1988. (In Russian)Google Scholar
  13. 13.
    Horn R, Johnson C: Matrix Analysis. Cambridge University Press, Cambridge; 1985.CrossRefGoogle Scholar

Copyright information

© Shatyrko et al.; licensee Springer 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  • Andrei Shatyrko
    • 1
  • Josef Diblík
    • 2
  • Denys Khusainov
    • 1
  • Miroslava Růžičková
    • 3
    Email author
  1. 1.Kyiv National Taras Shevchenko UniversityKyivUkraine
  2. 2.Brno University of TechnologyBrnoCzech Republic
  3. 3.The University of ŽilinaŽilinaSlovakia

Personalised recommendations