ON THE THEORY OF ONE . SIDED MODELS IN SPACES WITH ARBITRARY CONES

The paper presents a way of constructing quasimonotone nonautonomous systems ensuring x-stability of the nonautonomous system. There are described extensions quasimonotone with respect to an arbitrary cone, Perron condition and invariant surface stability under perturbations U-stability on the set of non wandering points is proved to imply u-stability of quasimonotone nonlinear system and exponential u-stability on minimal attraction center provides u-stability of the total systems Examples are available.


INTRODUCTION
One-side models appeared as a means of investigation of real systems dynamical properties in connection with the application of vector Lyapunov functions the existence of which was indicated by Azbelev [1], Bellman [2], Matrosov [12], Melnikov [14].The comparison method created hereat is presented in [4,5,6,9,12,14,18, 19] and some other papers.In 1977 Lakshmikantham and Leela [7] offered a comparison method for cone-valued Lyapunov functions.The present contribution develops further the ideas of the comparison technique for the case of autonomous and non-autonomous quasimonotone extensions.
In the paper various sufficient conditions for equilibrium state stability in linear and nonlinear models are determined and a relation between the dynamical properties of these models and their behavior on limiting sets established.The concept of "extension" employed allows us to apply the fundamental results from theory of dynamical systems in the investigation of non-autonomous one-side models.

QUASIMONOTONE MODELS IN GENERAL
Let B be a compact metric space, R n be an n-dimensional Euclidean space.The Cartesian product B x R n = E with projection p" E B is a phase space for the given comparison system.
Let cone K with interior ( be given in Rn, u _> u;z __clef Ul U2 E K for each u Rn, < u 1,u 2 > = u R n" U l -< u < u2 is a cone segment, and K* is a cone conjugated with K. We consider a dynamical system H t" E E. Definition 2.1.Dynamical system H is called monotone (strictly monotone) extension if a) pHt(v) = Ht(pv) = Ft(tp) for each v = (tp,u) B x Rn; for each (V1,V1) E E such that p(v) = P(v2 The general monotone extension is generated by system (2.1) where F is a dynamical system in B, which has unique solution (F (tp0), u(t; tp0, u0)) defined for all t _> 0 and (qo, Uo) E.
Classes of functions W0(Wr) [15] that form the montone extensions have the following generators: Functions of the type of g(tp,u) = o (tp)u, where ct: B R, belong to class W 0 (extension of the monotonicity); 2 Convex linear combinations of mappings of the type of h(9,(u, z*(p)))z( 9), (,) is a scalar product, h(tp, s): B x R R is montone (strictly monotone) increasing with respect to S R for all tp B, z(p) K, and z*(tp) (*, belong to class W 0 (W r is a strictly monotone extension).
Functions of the type of g(tp,u) = A(tp)u, where A(tp) is an element of Lie algebra corresponding to Lie group of homogeneous cone K transformations, belong to class W 0.
It should be noted that the class W 0 is closed with respect to the limiting transition on compact sets in E.
Definition 2.2.Monotone extension (2.1), (2.2) is called u-stable in cone K if for any e > 0 there exists 5(e) > 0 such that 5(e)e u 0 K for any 9 M c B implies inclusion ee u(t; q0,u0) K for allt 1+, e I; ii) asymptotically u-stable in cone K if it is u-stable in the sense of Definition 2.2.i and for any (p B a neighborhood u( c R n exists such that for u o utp fl K, lim u(t; 9, Uo) = 0 takes place as t --) ,.

STABILITY AND ASYMPTOTIC BEHAVIOR
Let L be a dynamical system in E = B x R n generated by system of the type of (3.1) Ft'B B, = f((p, %), Z Rm.
Together with system (3.1),(3.2) we consider a vector-function V C [ B x Rm, K ], V((p,Z) is locally Lipschitzian on Z.
Let there exist a function Q(tp,Z): B x R m K, Q((p,0) = 0, which is non-negative relatively cone K and function g C [ B x K, R n ], g(tp,u) W0(K).
Further we designate by Z a set of nonwandering points of dynamical system F and by C/ a minimal attraction center of this system.
Theorem 3.4.Let H E .9E be a monotone extension over a compact metric space B and there exist an open in E neighborhood U of a set of nonwandering points Z belonging to dynamical system pHt(z) = Ht(p(.))= F such that for Z U and p() Z, IHt(z) pHt(z)l =# 0 as t +oo.Then for any p B there exists a neighborhood V(tp) C E such that for any Z V(tp) 13 K [3 pl(tp), IHt(z) pnt()l -9 0 takes place as t -9 +oo, Ht(z) is asymptotically stable for t Proof.Let us set u 1 a K arbitrary and consider element Ul(qO) = .(qo)u 1 in space E, .(q)> 0 and u 1 U for all qo Z c B. Since the set of nonwandering points Z is compact, there exists ,1 = const > 0 such that constant u(q) = .1 x u = u, belongs to U for all qo Z.As the set Z is invariant and by hypotheses of the theorem there exists a sequence {T0i}, i = 1, 2,... such that (u)_< Z,i 1,2,.
As the dynamical system H t" E E is continuous, there exist neighborhoods W C B of set Z in B such that estimate 0 <_ H __a.
We note that Wi+l c W i. Let D c W t::: B be a set of points qo W for which t > 0 and Ft(q) W i. Sets D are nonempty because any point from B is wandering out of W for a finite time only; Z c D and closures D are compact.k.J_ Htq0(u)> Let gl = _ _D <0, O<_tTo Set g is closed compact and as extension Ht: E -E is monotone, g l is positive invariant.Thus, trajectory Ht(7) is positive stable in the sense of Lagrange for every X E and a g.By choice of sequence T0i and monotonicity of extension Hr.E E inequality 0 <_ HmTOi(u) <_ 4.2i i = 1, 2,...

takes place for q0
, where m is a positive integer.
Suppose on the contrary.
Let Tij(q), i < j, be a wandering time of arbitrary points p e D out of Wj.Then the arguments-above imply that inequality 0 _< Ho(u) < holds for any t > m(j) Toj + Tij + m(i) T0i.
Then H is exponentially u-stable in cone K.
Proof.Due to the continuity of L 1 and its differentiability along system F+(q) trajectories, the system (3.4),(3.5) on L 1 is generated by a system of the form Ft'B B s=g(9)s,s R, ge B with continuous bounded function g" B R.
As B is compact and system (3.4),(3.5) is linear, it is enough to show that there exists a Let us set q) B arbitrary and find a sequence T ,,,, as 1 oo such that 'It T 0 o Let mo.(u) be a normed measure concentrated at q) B. For an arbitral function f" B R we define a sequence of measures by equality where Rn, (u1,vl), (u2, v2) R2, andA C[R2x R2,Rn] and is bounded.
For l.t > 0 equation (4.4b) majorizes equation (4.3b) and generates strictly monotone extension relatively cone of positive definite symmetric matrices.Therefore we can apply Lemma 3.1 and Theorem 3.5 to system (4.4b).
For system (4.3a)C+ = { ql = " + kx; q2 = -+ kx, k is an integer On the Theory of One-Sided Models in Spaces with Arbiwary Cones: Martynyuk and Obolensky 95   According to Theorem 3.6 one can observe exponential W-stability in system (4.3) if the system of inequalities (4.5) AW 0 + WoAT + IA.(/0, E) E < 0 has solutions on C/.Due to the definition of W we find that condition (4.5) is equivalent to Routh-Hurwitz condition on C/ for system (4.1)whenZ-stability is under consideration.
Example 4.2.Let in an isodromic control system (4.6a)di d--i-= "Pii + air, i = 1,2,...,n ds n (4.6b) d-'i-= a7; ai Pn+lS-if(s) i=2 the conditions 0 < p < rnin (Ok), k e [2, n + 1], f(s) > 0, for each ;e 0, frO) = 0, > 0 be fulfilled Remark 4.1.In contrast to the classical statement of the problem on stability of isodrorrfic control system here we take into account the signs before coefficients a and this allows us to separate from the general class of isodromic control systems those generating quasimonotone semi-groups.Also it should be noted that the sign of coefficients a are really different when the real automatic control systems are under consideration.
It is easily seen that condition on parameters (4.7) extends (4.8).

CONCLUDING REMARKS
The consideration of comparison system as an extension of dynamical system defined on compact manifold allows a) more detailed investigation of linear non-autonomous systems; more complete collection of dynamical properties of comparison and initial system, such as, for example, oscillating processes. [1] [2] [3] [41 [6] [7] and constants c > 0 and 0 < 8 < I exist such that lIH.(v)II lIH:(v I) 4.2.In [ 10] we established sufficient conditions for uniform asymptotic stability O for cY 0.Remark