STABILITY INVESTIGATION OF QUADRATIC SYSTEMS WITH DELAY

Systems of differential equations with quadratic right-hand sides with delay are considered in the paper. Compact matrix notation form is proposed for the systems of such type. Stability investigations are performed by Lyapunov’s second method with functions of quadratic form. Stability conditions of quadratic systems with delay, uniformly by argument deviation, and with delay depending on the system’s parameters are derived. A guaranteed radius of the ball of asymptotic stability region for zero solution is obtained.


Introduction
In this paper, sufficient conditions for asymptotic stability of a differential-delay equa- tions systems with quadratic right-hand sides are given.Compact matrix form from presentation of the quadratic differential-delay system is proposed.Lyapunov's se- cond method is taken as the main method of our investigations.There are two approaches to the application of Lyapunov's second method for differential equations.
The first one is the Lyapunov function (or direct) method.It involves a search for functions from the class of functions with a finite number of variables satisfying Lyapunov's theorems conditions.The second approach is Lyapunov's functional method stressing the point of view of functional analysis.This method is linked with a search for functionals defined on integral curve segments.Essential difficulties in applications of Lyapunov's direct method to differential equations with deviating arguments were formulated for the first time by L.E.El'sgol'ts in [5].The main diffi- culties lay in a sign estimate of Lyapunov function derivative, which are functions of 2n-variables, if nth order delay-differential systems are considered.Important steps to overcome these difficulties were made by B.S. Razumikhin [6,7].The following sufficient condition for stability was obtained.
If there exists a positive definite function V(t,x), x E E n, the total derivative of the system is a negative definite functional on curves respecting solutions and satisfy- ing the Razumikhin's condition Y(5, x( 6))<_ Y(t,x(t)), < t, t >_ to; if the set of the curves is not empty, then the system is stable.Note that this condition is not an addition for reducing the class of stability tasks.The condition allows to get an esti- mate of Lyapunov function derivatives in a more simple way.So some efficient condi- tions of stability and asymptotic stability have been given in [6] by using Razumik- hin's condition.
The rich experience with Lyapunov functions construction for ordinary differential equations explains the wide application of Lyapunov's direct method for systems with deviating arguments.

Main Results
A system of differential-delay equations with quadratic right-hand sides is considered" k(t) AlX(t + A2x(t-v)+ X(t)BlX(t + X(t-r)B2x(t + X(t-r)B3x (t-  (1)   where r > 0 is a constant delay; A1, A 2 are square n x n matrices with constant coeffi- cients; X(t) nd B, j-1,3 are rectangular n x n 2 matrices of the following block forms: T__ {B ., X(t) {Xl(t), X2(t),..., Xn(t)} Bj j, B2j,.. Bnj } where Xi(t is a square n xn matrix containing the vector xT(t)-(xl(t),x2(t), ...,xn(t)) at the ith row of the matrix and the other elements being zero, Bij are square-symmetric matrices characterizing the quadratic component of the ith-line of the system (1).That is, / 0 0 x (t) Such notation of a quadratic differential system in matrix form (1) was proposed in the articles [8,9].This form is convenient for investigation of quadratic systems of a general form.
The conclusion about stability of the zero solution of ( 1) is performed on the basis of comparison of the differential system with delay (1) with some model system.As a model system we take a linear system without delay k(t)-Ax(t), A-A 1 + A2, (2) where/, 0 <_/ <_ 1 is a certain numerical parameter selected with respect to the condi- tion of maximum stability of the system (2).If the model system (2) is asymptotical- ly stable then this property is retained by the zero solution of the system (1) in some sufficiently small neighborhood of the equilibrium position.Let the matrix A be asymptotically stable, i.e., all eigenvalues of the matrix are on the left of the imagin- ary axis.Then for an arbitrary positive definite symmetric matrix C the equation ATH+HA -C (3) has as unique solution a positive definite matrix H [2]. Stability investigation of the solution x(t) 0 is performed by Lyapunov's second (direct) method.Take the quad- ratic form v(x)-xTHx as Lyapunov function, where the positive definite matrix H is a solution of (3).For the function v(x), the following two-sided inequality is true: Amin(H) x 2 _< v(x) <_ Amax(H)I x 2. (4) Here and further on )min(" respectively $max(') are the largest respectively the smallest eigenvalues of positive definite matrices.
Let us denote a level surface of the Lyapunov function v(x) by Ova, and a region bounded by this surface as v that is Ov {(, t): v(x, t) }, v {(, t): v(, t) < }.
The following vector norms are used" 1/2 i=1 I I (t)II max { x(t + s) ).

r-r<s<O
The spectral norm is taken as a matrix norm" AI {,max(ATA)) 1/2.Now we introduce some important definitions.
Definition 1: The asymptotic stability region f of the zero solution of the system with delay is the set of initial functions x(t), -r <_ t <_ O, such that a-{x(t),-r<t<0 lim x(t) -0).
Definition 2: The set of initial function values x(t) E f located inside the ball is said to be the ball U/ of radius R of the asymptotic stability region.Theorem 1: Let there exist a parameter/, 0 <_ <_ 1 and matrix H satisfying the equation (3), such that the inequality /min(C)-21HA21(/ + (H)) > 0 holds where (H)-V/,ax(H)/,kmin(H).Then the solution x(t)-0 of the system with delay (1) is asymptotically stable for arbitrary r > O.The ball U R of the asymp- totic stability region has the radius R "min(C)-2 HA2 (/ + (H)) 3 2Amax(H) BI (H)   i=1 (6) Moreover, an arbitrary solution x(t) does not leave e-neighborhoods of the origin, that is, [x(t)[ <, t>0, if [[x(0) llr <i(), where () min{R,/(H)}. (7) Proof: Let > 0 be an arbitrary value and x(t) a solution of the system (1) such that the condition [[ x(0)[[ r < ()is satisfied, where the function ()is chosen according to (7).Then the solution x(t), -v <_ t <_ O, is located inside the region where a-Amax(H)2().We show that x(t)E v" for all t_> 0. Otherwise, there is some minimum value of S > 0 where x(t) reaches the first time the boundary Ova, x(S) gv.Consider the total derivative of the Lyapunov function v(x) along the solutions of the system (1): Exploiting the special form of the matrices X(t), X(t-v) and the above-mentioned spectral norm for matrices, we obtain Therefore, the estimate for the total derivative of the Lyapunov function is fulfilled" v(x(t)) < Amin(C) x(t) 2 -4-2{ HA21[ x(t-r) + (t)   + HI [IB x(t) 12 + B21 x(t) Ix(t-)1 / B31 Ix(t-7) 12]} x(t) According to our assumption, x(t) v a for any t < S and S" x(S) Ova, yielding   Amin(H) x(t) 2 <_ v(x(t)) < v(x(S)) <_ Amx(H) x(S) .
then the total derivative is negative definite.Thus, according to Theorem 1, the ball U R of the asymptotic stability region is the interior ball with radius R defined in (6).
Thus, the proof is completed.
For defining the largest radius R of the ball a similar remark holds true.
Stability conditions uniformly in the delay v > 0 proposed in Theorem 1 have an excessively sufficient character.They impose strong restrictions on the system (1).If the model system (2) is asymptotically stable, then the zero solution of the system (1) will be asymptotically stable without satisfying the conditions of Theorem 1.
However this is true only for sufficiently small delays v < v0, where v 0 is the admiss- ible maximum value of delay depending on the quadratic system parameters and on the choice of the matrix H.
For deduction of these conditions it is necessary to estimate the value of maximum deviation of solutions from the equilibrium position in one step.In contrary to linear systems, quadratic ones have the property of nonextensibility, i.e. in finite time their solutions may go to infinity.This property holds even for simple scalar equations.Therefore, we find conditions for the system parameters, delay T and the value of ini- tial perturbation 5, under which it is possible to estimate the maximum deviation of the solution x(t) of (1) on the interval 0 < t < r.Set" P-[I+ ([A2] Lemma 1" Let 5 > 0 and r > O, such that the inequality L exp{Lr (8) p B1-----]-+ 1 > } holds.Then for arbitrary solutions x(t) of (1) satisfying the condition 11 x(0)[[ r for 0 <_ t <_ r the following relation is fulfilled: x(t) <-L + P IB1 I(1 -exp{Lt})" Proof." Rewrite the system (1) in the integral form [Alx(s) + A2x(s--) + X(s)BlX(S + X(s-')B2x(s) + X(s-)B3x(s-')]ds.
We use these lemmata for obtaining stability conditions of the solution x(t) 0 of the system with quadratic right-hand sides and a delay depending on the system para- meters.Let fl 1, then the matrix of the model system is of tile form A A 1 + A 2.