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Maksimov V.P. ON CONTROL PROBLEM FOR LINEAR SYSTEM UNDER IMPULSE DISTURBANCES
A control problem for linear functional differential system with time delay of the general form is considered. The purpose of controlling is prescribed with use of a finite set of linear functionals. The system is acting under impulse disturbances which result in trajectory jumps with unknown previously instants of time and values. A construction of control actions that contains both program and jumps-positional components is proposed.
Key words: functional differential systems; control problems; impulse disturbances.
УДК 519.688
EXACT TRIANGULAR DECOMPOSITION © G.I. Malaschonok
Ключевые слова: matrix triangular decomposition; algorithm in commutative domain.
The main result which presented in this talk is existence in commutative domain R of the
matrix triangular decomposition, which has the form: A = PLDUQ, where P and Q are
permutation matrices, L and PLPT are lower triangular matrices over R, U and QT UQ are upper triangular matrices over R, D = diag(d-\d-1, ..,d-1,0,.., 0) is a diagonal matrix of rank r, di G R\{0}, i =l,...,r.
A matrix decomposition of a form
A = VwU (1)
is called the Bruhat decomposition of the matrix A, if V and U are nonsingular upper triangular
matrices and w is a matrix of permutation.
Let R be a commutative domain, F be the field of fractions over R. We want to obtain a decomposition of matrix A over domain R in the form A = VwU, where V and U are upper triangular matrices over R and w is a matrix of permutation, which is multiplied by some diagonal matrix in the field of fractions F. Moreover each nonzero element of w has the form (агаг-1)-1, where аг is some minor of order i of matrix A.
We call such triangular decomposition the Bruhat decomposition in the commutative domain
R.
Let R be a commutative domain, A = (аг,j) G Rnxn be a matrix of order n, akj be k x k minor of matrix A which disposed in the rows 1, 2,... ,k — 1,i and columns 1, 2,... ,k — 1,j for all integers i,j,k G {1,..., n}. We suppose that the row i of the matrix A is situated at the last row of the minor, and the column j of the matrix A is situated at the last column of the minor. We denote a0 = 1 and ak = ak k for all diagonal minors (1 ^ k ^ n). And we use the notation 5ij for Kronecker delta.
Let k and s be integers in the interval 0 ^ k<s ^n, Ak = (ak+:) be the matrix of minors with size (s — k) x (s — k) which has elements ak+г, i,j = k + 1,...,s — 1,s, and АП = (ajj) = = A.
Theorem! [LDU decomposition of the minors matrix].
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Let A = (aij) € Rnxn be the matrix of rank r, ai = 0 for i = k,k + 1,...,r, r ^ s ^ n, then the matrix of minors Ak is equal to the following product of three matrices:
Ak = Lk Dk Uk = (j )(5ij ak (ai-1ai )-1)(a\J). (2)
The matrix Lk = (aj j), i = k + 1.. .s, j = k + 1.. .r, is a low triangular matrix of size (s — k) x
x (r — k), the matrix Uk = (ai j), i = k + 1.. .r, j = k + 1.. .s, is an upper triangular matrix of size (r — k) x (s — k) and D^: = (Sijak(ai-1 ai)-1), i = k + 1 ...r, j = k + 1 ...r, is a diagonal matrix of size (r — k) x (r — k).
Consequencel [LDU decomposition of matrix A ].
Let A = (ai j) € Rnxn, be the matrix of rank r, r ^ n, ai = 0 for i = 1, 2,... ,r, then matrix A is equal to the following product of three matrices:
A = L°nD°n Un = (ajj )(Sij (ai-1ai)-1)(ai,j). (3)
The matrix L®n = (ajj), i = 1.. .n, j = 1.. .r, is a low triangular matrix of size n x r, the matrix Un = (ai j), i = 1.. .r, j = 1.. .n, is an upper triangular matrix of size r x n and D°n = = (Sij (ai-1ai)-1), i = 1.. .r, j = 1.. .r, is a diagonal matrix of size r x r.
Let In be the identity matrix and Pn be the matrix with second unit diagonal. Consequence2 [Bruhat decomposition of matrix A ].
Let matrix A = (aitj) have the rank r, r ^ n, and B = PnA. Let B = LDU be the LDU-decomposition of matrix B. Then V = PnLPr and U are upper triangular matrices of size n x r and r x n correspondingly and
A = V (Pr D)U (4)
is the Bruhat decomposition of matrix A.
We are interested in the block form of decomposition algorithms for LDU and Bruhat decompositions. Let us use some block matrix notations.
For any matrix A (or Aqq ) we denote by Aj j (or Apj ’ j ) the block which stands at the intersection of rows i1 + 1,... ,i2 and columns j1 + 1,..., j2 of the matrix. We denote by Ai2 the diagonal block A^’^.
LDU algorithm.
Input: (Akn, ak ), 0 ^ k<n.
Output: {Lkn, {ak+1,ak+2,..., an}, Uk, Mk, Wk },
where Dk = akdiag{akak+1,...,an-1an}-1, Mk = ak(L1nDk)-1, Wk = ak(DkUk)-1.
1. If k = n — 1, An = (an) is a matrix of the first order, then we obtain
{an, {an}, an, an-1,an-1}, D^-1 = (an)-1.
( an-1 B \
2. If k = n — 2, An-2 = r is a matrix of second order, then we obtain
n \ Y S J
( ( an-1 0 \ {an-1 an}( an-1 p N ( an-2 0 \ ( an-2 —^ \ 1
Y an J , {a ,a },\ 0 an J , V —Y an-1J , V 0 an-1 J J
(5)
where an = (an-2)
2 -1
D'l~2 = an-2diag{an-2an-1,an-1an}-1
Y
an-1 0 \ 1 ( an-1 B W an-2 0 \ ( an-2 —P
Y an ) , {a ,a },\ 0 an ) , V —Y an-1 ) , V 0 an-1
an-1 B Y S
3. If the order of the matrix An more than two (0 ^ k <n — 2 ), then we choose an integer s in the interval (k <s <n) and divide the matrix into blocks
<=(a b )■ «»
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3.1. Recursive step
{Lk, {ak+1, ak+2,..., as}, U?, Mks, W? } = LDU(A?, ak)
3.2. We compute
U = (ak )-1Mk B, L = (ak )-1CW?, (7)
An = (ak )-1as (D — Ldks U). (8)
3.3. Recursive step
3.4 Result: where
{Lsn, {as+1,as+2,..., an}, US, MS, WS} = LDU (An, as)
{Lkn, {ak+1, ak+2,..., an}, Ukn, M? , W?},
' k = ( LS 0 ) U s = ( uS u )
^ V L Lsn ) , n \ 0 Usn)
Lkn = 7S rs ),US = US Us , (9)
Mnk = Msk 0 , (10)
n \ MS L Dk M k /ak M S I’
—M^LDkMk/ak Ms
Wk =( WS —WSDkUWS/ak . ( )
Wn I 0 Ws 1 . (11)
n
Complexity.
Let y and B be constants, 3 ^ B> 2, and let M(n) = Yn13 + o(n3) be the number of multiplication operations in one n x n matrix multiplication.
Theorem 2.
The complexity of the decomposition is ~
The decomposition of the second order matrix needs in 7 multiplicative operations, so:
t(n) = 2t(n/2) + 7M (n/2), t(2) = 7.
After summation from n = 2k to 21 we obtain
7y (2°2l3^k-1) + ... + 2k-223'1) + 2k-27 = 7Yn—----------+ 7 n.
23 — 2 4
The exact triangular decomposition.
Definitionl. A decomposition of the matrix A of rank r over a commutative domain R in the product of five matrices
A = PLDUQ (12)
is called exact triangular decomposition if P and Q are permutation matrces,
L and PLPT are nonsingular lower triangular matrices over R,
U and QTUQ are nonsingular upper triangular matrices over R,
D = diag(d-1 ,d-1 ,..,d-1, 0,.., 0) is a diagonal matrix of rank r, di € R\{0}, i = 1,..r. Theorem3 [Main theorem].
Any matrix over a commutative domain has an exact triangular decomposition.
Proof of this theorem you can find in [4].
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ЛИТЕРАТУРА
1. Grigoriev D. Analogy of Bruhat decomposition for the closure of a cone of Chevalley group of a classical serie // Soviet Math. Dokl. Vol. 23. N. 2. 1981. P 393-397.
2. Malaschonok G.I. Effective Matrix Methods in Commutative Domains // Formal Power Series and Algebraic Combinatorics. Springer, Berlin. 2000. P. 506-517.
3. Malaschonok G. I. Fast Generalized Bruhat Decomposition // Computer Algebra in Scientific Computing, LNCS 6244, Springer, Berlin. 2010. P. 194-202.
4. G.I.Malaschonok Generalized Bruhat decomposition in commutative domains /in book: Computer Algebra in Scientific Computing. CASC’2013. LNCS 8136, Springer, Heidelberg, 2013, pp.231-242. Acknowledgements: This work was partially supported by the Russian Foundation for Basic Research (grant No. 12-07-00755-a).
Малашонок Г.И. ТОЧНОЕ ТРЕУГОЛЬНОЕ РАЗЛОЖЕНИЕ
Основным результатом, который представлен в этой работе, является существование в коммутативной области R матрицы треугольного разложения, которая имеет вид: A = PLDUQ, где P и Q - перестановочные матрицы, L и PLPT - нижние треугольные матрицы над R, U и QT UQ - верхние треугольные матрицы над R, D = di&gld-1, d-1,.., d-1, 0,.., 0) - диагональная матрица ранга r, di G R\{0}, i = 1, . . . , r.
Key words: треугольное разложение матрицы; алгоритм в коммутативных областях.
УДК 517.98
THE LAPLACE TRANSFORM METHOD FOR SOLVING DIFFERENTIAL EQUATIONS WITH DELAYED ARGUMENT © N.A. Malashonok
Key words: Laplace transform method; differential equations with delayed argument.
The Laplace transform method is used for differential equations with delayed argument.
There is constructed an algorithm, which is symbolic-numerical. The numerical component concerns a representation of functions, involved into the process by some kind of series.
There is a class of physical problems, which is associated with action of some kind of complementary forces - forces which are involved at various not initial time moments. Such problems frequently lead to the so called differential equations with delayed argument. Different ways of dealing with such equations exist. Applications of the Laplace transform method are well known. However there are some facts which prevent using this method in a symbolic way. Some difficulties, for example, are connected with a form of the solution of the Laplace image of the input differential equations. Only concrete kinds of equations may be solved entirely symbolically.
We restrict ourselves to the consideration of one equation, but the method works similarly with systems of equations of such type.
All functions of the argument t are supposed to satisfy the conditions for existing of their Laplace transform, and they equal zero for negative t. The points tk,tk-1 <tk, are taken in the set of t,t ^ 0. Consider an equation
n N
x(n)(t) + E E ajk x(n-j)(t — tk) = f (t), (1)
j=1 k=0
with initial conditions x(n-j)(0)= x0ll-j),j = 1,ldots,n. The function f(t) in the right-hand part is in general composite. We may consider for it the same partition points tk. The unknown
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