STABILITY IMPLICATIONS ON THE ASYMPTOTIC BEHAVIOR OF NONLINEAR SYSTEMS

In this paper we generalize Bownds’ Theorems (I) to the systems dY.(t) dt A(t) Y(t) and dX(t) A(t) Xt) + F(t X(t)) Moreover we also show that there dt always exists a solution X(t)-of t A(t)X + B(t) for which limt+ooSUpl IX(t) II > o (= oo) if there exists a solution Y(t) for which lira sup Iy(t) > o (= oo) t _


I. INTRODUCTION.
In this paper we shall study the stability behavior of the following systems dY(t) A(t)Y(t) 0 < t < (I.i) dt and dX(t) dt A(t)X(t) + F(t,X(t)), 0 -< t < (1.2) where A(t) is a continuous matrix on R n for all 0 < t < , F(t,X(t)) is a real valued continuous n-vector defined on [0,) X R n and X(t) and Y(t) are n-vectors.
Consider special equations of (i.i) and (1.2) y" + a(t)y 0, 0 < t < (1.3) and x" + a(t)x g(t,x,x'), 0 -< t < (1.4) where a(t) C[0,) and g(t,x,x') is continuous on [0,)R R. From some theorems of stability theory, Bownds [i] showed that (1.3) has a solution y(t) with property lim sup (ly(t) + lY'(t)I) > 0 (i 5) t_oo He also established that (1.4) has the property (1.5) provided that the zero solu- tion of (1.3) is stable and there exists a function y(t) 6L[0,) such that Ig(t,x,x') < y(t) (Ixl + Ix' I) Thus in the following section we shall extend the above results to systems (i.i) and (1.2).In section 3 we shall consider a nonhomogeneous system dX(t) dt where B(t) is a continuous vector for 0 _< t < oo.We shall prove that there always exists a solution X(t) of (i 6) for which lim sup llX(t) ll > 0(= ) if there exists t+ a solution Y(t)of (] i) for which lim sup lly(t) ll > 0(= oo) Here II II is an t+oo appropriate vector (or matrix) norm.
THEOREM 2.1. (Hartman [2, p. 60]).Suppose that, for every solution Y(t) of (i.i), the limit lim IY(t) exists and is finite.If there exists a nontrivial solution Y(t) of (i.i) for which the limit (2.1) is zero, then t t A(s)ds + as t + r t o From the above theorem we will obtain the following corollary which is a gen- Then there exists a nontrivial solution Y (t) of (I.i) for which >0 t _+o PROOF.Suppose, to the contrary, that all solutions Y(t) of (i.i) satisfy From Theorem 2.1 we obtain lim sup t t A(s)ds /-as t + This leads to a contra- r t o diction.The corollary then follows.
Throughout this paper we shall denote $(t), the fundamental matrix of (i.i) with initial condition (0) I (identity matrix).
Now we shall prove the following theorem via the Schauder-Tychonoff Theorem [2, p. 9].
THEOREM 2.2.Suppose that the null solution of (I.I) is stable and that there exists a solution Y(t) of (i.i) for which Suppose also that there exists y(t) E Ll[to') such that for some positive con- stant , Then there exists a nontrivial solution X(t) of (1.2) for which lim sup llX(t) ll > 0 t+oo PROOF.Since the null solution of (i.i) is stable, there exists a positive constant k such that (2.4) for all 0 < t < s and there exists a nontrivial solution Y(t) of (i.i) for which (2.2) holds and IIY(t) ll First, we shall show that TF c F. Taking the norm to both sides of (2.8) and using (2.3), (2.4), (2.5), and (2.6), we obtain for U 6 F IITU(t) II < IIY(t) ll + I lib(t) -l(s) f(s'U(s))llds Ll[to')' there exists T I > To so that for t e T I k y(s)ds <t (2.9) By the uniform convergence, there is an N N(I, T I) such that if n -> N, then lf(s U (s)) f(s U(s))II < i n 2kTl T < s < T I (2.10) O Then using (2.8) (2 9) (2 i0) (2 3) (2 4) and the fact that []U (t)l[ < I and n llU(t) ll-< i for T < t < oo, we obtain the following inequalities O lTUn(t)-TU(t)II I[ (t)-" t (s)f(s'Un(S))ds I (t)-l(s)f(s'U(s)dsl .T I < I t l(t)-l(s)ll lf(s'Un(S)) f(s0U(s))l]ds l(t) (s) ll lf(S,Un(S))llds + l(t) -1 T 1 (s) ll <k If(s,Un(S)) f(s,U(s)> Ids + 2k t T 1
This shows that TU converges uniformly to TU on every compact subinterval of J n o Hence T is continuous.
Third, we claim that the functions in the image set TF are equicontinuous and bounded at every point of J Since TF F, it is clear that the functions in TF o are uniformly bonded.Now we show that they are equicontinuous at eact Doint of J For each U F, the function z(t) TU(t) is a solution of the linear system o dV A(t)V + f(t,U(t)) dt Since lz(t)[i IITU(t) II 1 and lf(t,U(t))ll is uniformly bounded for U E F on dz any finite t interval, we see that is uniformly bounded on any finite interval.
Therefore, the set of all such z is equicontinuous at each point of J (see [2, p.6]).
o All of the hypotheses of the Schauder-Tychonoff Theorem are satisfied.Thus there exists a U F such that U(t) TU(t); that is, there exists a solution X( It is clear that (1.4) can be written as the form (1. where X colum(x,x').Thus we can apply Theorem 2.2 to (1.4) to obtain the fol- lowing corollary which is a generalization of Theorem 2 in [i].
COROLLARY 2.2.Suppose that the null solution of (1.3) is stable and that there exists y(t) Ll[to') such that for some positive constant Then there exists a nontrivial solution x(t) of (1.4) for which lim sup t-(Ixl + x I) > 0 PROOF.Since t A(t) 0 for Corollary 2.1, we know that there exists a sol- r ution Y(t)of (I.I) for which lim sup IIY(t)II > 0 t If we take ]IXI] ]x[ + Ix'l, then the corollary follows from Theorem 2.2.

ASYMPTOTIC BEHAVIOR FOR (1.6)
In this section we shall show that if there exists a solution Y(t) of (I.I) for which lim sup ly(t)ll > 0 (= o) then there exists a solution X(t) of (I 6) t for which lim sup ]IX(t)]i > 0 (= ) t THEOREM 3.1.Suppose that there exists a solution Y(t) of (I.i) for which Then there exists a solution X(t) of (1.6) for which lim sup :'X(t)'l > 0 PROOF.From the variation of constants formula we know that any solution X(t) of (1.6) can be written as the form below It means topics where a satellite, a rocket, or any kind of man-made object is travelling in space governed by the gravitational forces of celestial bodies and/or forces generated by propulsion systems that are available in those objects.Many topics are related to orbit determination, propagation, and orbital maneuvers related to those spacecrafts.Several other topics that are related to this subject are numerical methods, nonlinear dynamics, chaos, and control.
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