Boundedness and stability for the damped and forced single well Duffing equation

By using differential inequalities we improve some estimates of 
W.S. LOUD for the ultimate bound and asymptotic stability of the 
solutions to the Duffing equation $ u''+ c{u'} + g(u)= 
f(t)$ where $c>0$, $f $ is measurable and essentially bounded, and $g$ is continuously 
differentiable with $g'\ge b>0$.


Introduction
In this paper we consider the second order ODE u + cu + g(u) = f (t) (1) where c > 0, f ∈ L ∞ ([t 0 , +∞)) and g ∈ C 1 (R) satisfies some sign hypotheses. The typical case is More generally we shall assume that g(0) = 0 and for some b > 0 Under this condition, W.S. Loud [9,10] established that all solutions of (1) are ultimately bounded and more precisely lim t→+∞ |u(t)| ≤ min{ where f ∞ stands for f L ∞ ([t 0 ,+∞)) . The estimate (4) is rather sharp and its proof relies on a delicate geometrical argument in the phase space. The question naturally arises of a purely analytical proof of (4), which would be extendable to more complicated situations such as second order systems or even hyperbolic problems.
This paper is devoted to a partial realization of this program. However our method, unlike the geometrical approach of [9][10], introduces a distiction between the weakly damped case corresponding to the condition c ≤ 2 √ b and the strongly damped case The analytical approach provides a better estimate for u itself but, for a reason which remains obscure, we do not recover (5) in the strongly damped case.
Moreover our proof of (4) for c ≥ 2 √ b requires an additional assumption on g.
The plan of the paper is as follows: Sections 1 and 2 are devoted to obtaining an improved version of of (4) by a purely analytical method. Section 3 deals with asymptotic stability and Section 4 contains existence and uniqueness results for bounded solutions on the whole line under a smallness condition on f . These results improve a theorem of the same nature obtained in [10] by a more complicated method.

1-Ultimate bound for c small.
The main result of this section is the following Proof. First we notice that since g(u) − bu is a non-decreasing function of u, the primitive For any solution u of (1) we have for all t ∈ J d dt (u 2 + 2G(u) + cuu ) = 2(u + g(u))u + cu 2 + cuu By the above remark we have we obtain the inequality On the other hand the condition c ≤ 2 √ b yields the inequality (2u + cu) 2 = 4u 2 + c 2 u 2 + 4cuu ≤ 4(u 2 + bu 2 + cuu ) and we deduce This inequality classically implies . By letting t 0 → +∞ we find the better estimate By Condition (3) and since g(0) = 0 the following inequality is valid In particular for any ε > 0 we have for t large enough Solving this differential inequality for u 2 we deduce for t ≥ T (ε). By letting ε → 0 we obtain (1.2). For the proof of (1.3) we notice that Hence for t large enough we have Then (1.3) becomes an immediate consequence of (1.2).
and therefore (1.2) improves the estimate (4) by a factor at least 2. In addition (1.2) is optimal when g(u) = bu.
b) In the same way, condition (1.1) implies Proof. This follows immediately from the inequality valid for large t: and from the observation that along a maximizing sequence t n for G(u(t)) the quantity tends to zero.

2-Ultimate bound for c large.
In this section we keep the notation of Section 1. Our main result, in addition to (3) requires the following assumption which was not necessary for Loud [9][10] ∀s ∈ R, g(s)s ≥ 2G(s) Theorem 2.1. In addition to (3) we assume (2.1) and Then any solution of equation (1) Proof. Introducing as previously we obtain here and therefore Here the function Φ(t) is not necessarily nonegative. However introducing The differential inequality (2.4) classically yields a) It is clear that (2.3) is better than (4), however for c large (4) is almost equivalent to (2.3). Moreover (2.3) is optimal when g(u) = bu. In the limiting case and therefore (2.3) improves the estimate (4) by a factor 2.
b) Strangely enough, here we do not recover the estimate (5) on u . A weaker estimate can be obtained as follows. writing the equation (1) as

3-Asymptotic stability.
In [10] a sufficient condition is given by W.S. Loud in order for any two solutions of (1) to asymptote each other at +∞. In this section we derive such a result by a simpler method based on the precise knowledge of the bound of an associated affine problem. More precisely we shall establish the following Then assuming This result will follow as a consequence of the following lemma Then if w is a bounded solution on R of w + cw + (b + a(t))w = 0 (3.6) we have w ≡ 0.
Proof. Let C := . We rewrite (3.6) in the form and we observe that as a consequence of (3.5) we have almost everywhere in t: Now setting Hence by [7] we find, assuming w ≡ 0 w ∞ the strict inequality coming from the exact formula given in [7] since as soon as b > 0. On the other hand it is readily verified that so that we obtain w ∞ < w ∞ and this contradiction shows that w ≡ 0.
Proof of Theorem 3.1. Let u, v be two solutions of (1) satisfying (3.3) and assume that (u, v) do not asymptote at +∞. Then there is a sequence t n → +∞ such Because both solutions are bounded on the right we may assume, replacing if necessary t n by a subsequence, that u(t + t n ) and v(t+ t n ) converge uniformly on compacta of R as well as their derivatives to some limits u * and v * , the second derivatives remaining essentially bounded. Setting w = u − v we have where ζ(t) lies between u(t) and v(t), by passing to the limit we find that w * = u * −v * is a non zero bounded solution of an equation of type (3.6) where a satisfies (3.5).
This contradiction proves that u, v are asymptotic to each other. Then the equation for w shows that w and even w tends to 0 as t goes to infinity.
Corollary 3.3. In the typical case the convergence result is obtained as soon as Proof. We use again the notation Then assuming we have for some K, δ > 0 The proof of Theorem 3.6 relies on the following Then if w is any solution on J of w + cw + (b + a(t))w = 0 (3.11) we have for some K, δ > 0 By letting T → +∞ we find +∞ t w 2 (s)ds ≤ C 5 ψ (t) (3.15) and then by letting T → +∞ in (3.14) Finally the combination of (3.15) and (3.16) provides It is immediate that this implies We conclude by using This concludes the proof.
Remark 3.8. By combining Remark 3.5 and Theorem 3.6 we obtain that (3.9) is satisfied whenever Loud's condition is better than ours. It is hard to believe that (3.18) can be optimal. Together with remark 2.2, b, this suggests that the techniques of the present paper can probably be improved.

-Bounded solutions on the line.
When f is defined and bounded on R, the classical translation method of Amerio-Biroli [1,2] allows to construct bounded solutions on R. More precisely we can state the following existence result  (1) has at least one solution. u ∈ W 2,∞ (R).
Proof. Standard application of the classical translation method of Amerio-Biroli [1,2]. Using inequality (1.4) for Theorem 1.1 or (2.4) for Theorem 2.1, applied with t 0 replaced bu θ n = −n to the solution u n of (1) on J n = [−n, +∞) with u n (0) = u n (0) = 0 the methods of proof of these theorems show that u n is uniformly bounded. The result follows by passing to the limit for a suitable subsequence of u n as n tends to infinity.
The following uniqueness result is an easy consequence of Lemma 3.2. Then (1) has at most one bounded solution on R.