A compact form of the algebraic criterion for the delay-absolute stability of solutions of linear

A COMPACT FORM OF THE ALGEBRAIC CRITERION FOR THE DELAY-ABSOLUTE STABILITY OF SOLUTIONS OF LINEAR DIFFERENCE-DIFFERENTIAL EQUATIONS

D. G. Korenevskii and A. G. Mazko UDC 512.643

The aim of the present article is to indicate a more compact form of the criterion for the Lyapunov-asymptotic-stability of solutions of a delay-type system of stationary constant delay of the argument (absolute stability), obtained earlier by one of the authors in [i]. For systems with a single delay, the simplification of the criterion consists in the reduc- tion'of the two conditions from [i] that form the criterion, namely, the Lyapunov matrix al- gebraic equation and an unimprovable matrix inequality, to only one condition, a three-term Sylvester matrix equation, and for systems with m (m ~ 2) delays, it consists in the reduc- tion of analogous conditions to an (m + 2)-term Sylvester equation.

Moreover, in this article we give a refined formulation of a theorem of [i], concerning the case of an equation with several delays.

Let us consider the delay-type autonomous linear system of difference-differential equa- tions

dx(t)/dt =-Ax(t)+ A,x(t--~ 0 + ... + A,~x(t -- ~m), (1)

where A and A I are (n • n)-matrices, x(t) is a column-vector, and ~i are constant positive delays of the argument t, i = i ..... m.

THEOREM i. The zero solution of system (i) is Lyapunov-asymptotically-stable for arbi- trary ~i > 0 if and only if i) the matrix A is a Hurwitz matrix with a certain stability re-

serve ~ and the matrix A + ~ A~ is simply Hurwitz and 2) there exists a positive-definite i=!

solution X = X ~ > 0 of theSyivester matrix equation

rn AX + XA T § 7~X + V ..... 1 .4iXA'~ =--Y,

- ~ ~i (2)

where 7E = 71 + -.. + 7m; 7i > 0 is a collection of positive numbers such that 7E < 2a and Y is an arbitrary positive-definite symmetric (n • n)-matrix.

Proof. By the general stability theory, the zero solution x ~ 0 of the linear system (i) is asymptotically stable if there exists a positive-definite Lyapunov-Krasovskii func- tional such that its derivative with respect to time by virtue of system (i) is negative. It is also well known that this proposition admits generalizations in the class of linear systems (see, e.g., [2, p. 87]).

We will seek this quadratic functional in the form

= x (t/Hx (0 + ~ ~ I x (O/Hx (o) dO, (3) where H and ~i, i = 1, .... m, is an unknown constant positive-definite matrix and unknown positive numbers, respectively, that are subject in the sequel to determination from the con- dition of negativity of its derivative with respect to time.

The derivative of functional (3) by virtue of system (i) can be represented in the form

=--zCz, C--- ~11 Ca C4

Mathematics Institute, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41,. No. 2, pp. 278-282, February, 1989. Original article submitted February i, 1988.

250 0041-5995/89/4102-0250512.50 9 1989 Plenum Publishing Corporation Ci = -- .4"H -- ttA -- l'xH, C., = -- [HA~ ..... HA.J,

? .... 0 x (I-- "r,)

c:, = c~, c~ "~,,,,,H kx (}- ~,,,)J

(4)

The following criterion holds: C>O~=*C,>O, C~--C~Cj'Ca>O. (5) We can verify the validity of this criterion, e.g., with the help of the well-known rule of computation of block determinants [3, p. 59]. Obviously, if 7i > 0, then the inequality C~ > 0 implies that H > 0. Moreover, H > 0 <=~X = H "i > 0. On multiplication by the matrix X on the left and on the right, the last inequality in (5) reduces to the form

L(X}--R(X)= Y>O, (6)

where L(X) = -AX -XA T - 7~X and R(X)= ~]----A~XA~. Obviously, R is a positive operator with respect to the cone of nonnegative-definite (n x n)-matrices. If YZ < 2~, then the operator L is positively invertible [4]. Consequently, inequality (6) is solvable inside the cone if and only if Eq. (2) has a solution X > 0 for an arbitrary right-hand side Y > 0 [5, po 22]. In addition, functional (3) is positive, and its derivative (4) is negative. The theorem is proved.

Example I. Let us consider the second-order system with two delays

X 1 ([) ~--- -- O~IX l (t) + ~2X2 (~ -- l"2), (~] > O, "~2 > O, X~ (O = -- a~X~ (O + fiiXl (* -- "q), a2>O, %>0. (7>

We should set A=[-- % 0 '~ ,b !0 01 ~0 I~! {, p;l ~ '2 o -~j' ,~,,oj A., ,.io

o

in the vector-matrix expression (i). Using the assertion of the theorem, we obtain restric- tions on the parameters ~i and ~i, under which the zero solution of system (7) is absolutely stable~