Time averaging for functional differential equations
Hindawi Publishing Corporation
Journal of Applied Mathematics
We present a result on the averaging for functional differential equations on finite time intervals. The result is formulated in both classical mathematics and nonstandard analysis; its proof uses some methods of nonstandard analysis.
1. Introduction
The idea of the method of averaging is to determine conditions in which
solutions of an autonomous dynamical system can be used to
approximate solutions of a more complicated time varying dynamical system.
The method of averaging has become one of the most important tool ever
developed for nonlinear time varying systems. Applications have been
found in celestial mechanics, noise control, nonlinear oscillations,
stability analysis, bifurcation theory and vibrational control, among many
other fields. Although averaging of ordinary differential equations is
considered a mature field?the reader may consult [
1, 5, 8, 22, 24
] for
more references and information on the subject (see also [13,
14])?averaging of functional differential equations has only recently been
developed (see [
6, 7, 9, 11, 12, 15, 16, 19
]).
This paper aims to present a result on the averaging for functional
differential equations of the form
x? (t) = f
t
?
(1.1)
on finite time intervals. The result is not new (see [10] and the references
therein). However, by means of nonstandard analysis methods, we
propose a new proof where all the analysis is achieved in Rd (it is not the
case in [10]) which makes it more simple.
The paper is organized as follows. Section 2 contains the notation and
conditions required to state and prove our main result as well as the
main result itself. The proof of this result is given in Section 4.2. To avoid
complicating the proof unnecessarily, several subsidiary lemmas have
been placed in Section 4.1.
The main result is formulated in both classical mathematics and
nonstandard analysis. Its proof makes use of Robinson?s nonstandard analysis
(NSA) [
21
]. We will work in the axiomatic form IST (for internal set
theory) of nonstandard analysis, given by Nelson [
20
]. For that, Section 3.1
is devoted to a short description of IST. Then, in Section 3.2, we present
the nonstandard translate (Theorem 3.6) in the language of IST of our
main result (Theorem 2.2). We recall that IST is a conservative extension of
ordinary mathematics. This means that any statement of ordinary
mathematics which is a theorem of IST was already a theorem of ordinary
mathematics, so there is no need to translate the proof.
2. Notation, conditions, and main result
Let r ? 0 be a given constant. Throughout this paper C0 = C([?r, 0], Rd)
will denote the Banach space of all continuous functions from [?r, 0] into
Rd with the norm ? = sup{|?(?)| : ?r ? ? ? 0}, where | ? | is a norm
of Rd. Let t0 ? R and T > t0. If x(t) is a continuous function defined on
[t0 ? r, T ] and t ? [t0, T ], then xt ? C0 is defined by xt(?) = x(t + ?) for
? ? [?r, 0].
The hypotheses, which are denoted by the letter H, are listed as
follows.
(H1) The functional f : R ?C0 ? Rd in (1.1) is continuous.
(H2) The functional f is Lipschitzian in u ? C0, that is, there exists
some constant k such that
f ?, u1 ? f ?, u2
? k u1 ? u2 , ?? ? R, ?u1, u2 ? C0.
(H3) For all u ? C0, there exists a limit
f 0(u) := lim
T?? T 0
1 T
(resp., the solution of (1.1)) such that yt0 = ? (resp., xt0 = ?) is denoted by
y = y(?; t0, ?) (resp., x = x(?; t0, ?)) and J (resp., I) will denote its maximal
interval of definition.
Remark 2.1. Existence and uniqueness of solutions of (2.3) will be
justified a posteriori. Indeed, we will show in Lemma 4.1 below that the
function f 0 is k-Lipschitz so that existence and uniqueness are
guaranteed.
Under the above assumptions, we will state the main result of this
paper which gives nearness of solutions of (1.1) and (2.3) on finite time
intervals.
Theorem 2.2. Let assumptions (H1), (H2), and (H3) hold. Let ? ? C0 and t0 ?
R. Let x be the solution of (1.1) and y the solution of (2.3) with xt0 = yt0 = ?.
Then for any ? > 0 and T > t0, T ? J, there exists ?0 = ?0(?, T ) > 0 such that,
for ? ? (0, ?0], x is defined at least on [t0, T ] and |x(t) ? y(t)| < ? on t ? [t0, T ].
3. Nonstandard main result
3.1. Internal set theory
In IST we adjoin to ordinary mathematics (say ZFC) a new undefined
unary predicate standard (st). The axioms of IST are the usual axioms of
ZFC plus three others which govern the use of the new predicate. Hence,
all theorems of ZFC remain valid in IST. What is new in IST is an addition,
not a change. We call a formula of IST external in the case where it
involves the new predicate st; otherwise, we call it internal. Thus internal
formulas are the formulas of ZFC. The theory IST is a conservative
extension of ZFC, that is, every internal theorem of IST is a theorem of ZFC.
Some of the theorems which are proved in IST are external and can be
reformulated so that they become internal. Indeed, there is a reduction
a (...truncated)