ONE-SIDED AND INTERNAL CONTROLLABILITY OF SEMILINEAR WAVE EQUATIONS WITH INFINITELY ITERATED LOGARITHMS

. In two previous works we improved some earlier results of Imanuvilov, Li and Zhang, and of Zuazua on the boundary exact controllability of one-dimensional semilinear wave equations by weakening the growth assumptions on the nonlinearity. Our growth assumption is in a sense optimal. Here we adapt our method for the case of one-sided control actions. This also enables us to obtain rather general internal controllability results.

(1.1) We will obtain a boundary exact controllability result under suitable, rather weak growth assumptions on the nonlinearity f . In order to state our result, let us introduce the iterated logarithm functions log j defined by the formulas log 0 s := s and log j s := log(log j−1 s), j = 1, 2, . . . , and define the numbers e j by the equations log j e j = 1: Modifying a result in [2], we shall prove in section 2 that the formula L(x) := (1 + x 2 ) 1/2 ∞ k=1 log k (e k + x 2 ) = (1 + x 2 ) 1/2 log(e + x 2 ) log 2 (e e + x 2 ) . . .
defines an everywhere finite, even function of class C ∞ with L(0) = 1. Furthermore, L(x) is increasing for x ≥ 0, and L(x) → ∞ relatively slowly as x → ∞, so that Let us also introduce the primitive F of f defined by We shall prove the Theorem 1.1. Assume that there exists a positive number β such that If T > b − a, then for any given satisfying the final conditions Remark. This theorem improves theorem 1.3 in [1] by weakening the growth assumptions on f . Theorem 1.3 in [1] shows that the assumption (1.2) above is essentially optimal: under slightly weaker growth assumptions the system can even become ill posed.
In [1] and [2] we adapted a method of Imanuvilov for the construction of control functions. We are going to modify this approach for the study of controllability by acting at only one end-point. Consider the following problem:  For the proof we will require some new technical results. We shall then study the internal controllability problem  2. Some preliminary results. First we establish some properties of the function L mentioned in the introduction. Let us observe that (e j ) is a strictly increasing sequence of positive numbers, rapidly tending to infinity. Note that e 0 = 1 and log j e l = e l−j for all l ≥ j ≥ 0. (2.1) We have the following variant of a proposition in [2].
Furthermore, for every α > 0 and δ > 0 there exists a constant c(α, δ) > 0 such that with a suitable constant C > 0, for all real numbers a and b.
Proof. The same properties were proved in [2] for a slightly different function L.
Most of the proofs remain valid with minor changes. (In lemma 2.
Since the infinite product defining L converges locally uniformly and since its terms are analytic, L is a C ∞ function. The only part which needs a little more modification is the divergence of the integral Assume on the contrary that the integral converges. Then Performing the change of variable x = e t we obtain the equalities .
for all k ≥ 1 and t ≥ e n (because e k , e 2t ≥ 2 and t 2 ≥ 2t). Therefore we deduce from the above equalities the inequalities It follows by induction that clearly converges (because e j → ∞ very quickly) and since every e j is greater than 1, we have for all n. This contradicts (2.4).
The usefulness of the function L comes also from the following nonlinear version of Poincaré's inequality; it generalizes former results obtained by Cazenave and Haraux [3] and in [2]: Using also the above Sobolev imbedding, we conclude that with a constant S(Ω) depending only on Ω. Given δ > 0 arbitrarily, by proposition 2.1 there exists a constant c(δ) > 0 such that for all real x. Using the above estimate and denoting by |Ω| the volume of Ω, it follows that Since L ≥ 1 everywhere, in case u ≤ 1 hence we deduce the estimate (2.7) This proves (2.5) in the special case u ≤ 1. (Choose δ = ε 2 /S(Ω).) Henceforth assume that u > 1. Setting v := u/ u and applying the inequality Since u > 1 implies that we have Furthermore, since |u| > u > 1 implies that we have Substituting them into (2.8) we find that Applying (2.7) for v and using the inequality L(x) ≥ |x| hence we obtain that Choosing δ = ε 2 /(2C 2 S(Ω)) the lemma follows.
Remark. In the one-dimensional case Ω = (a, b) we may choose N * = ∞, γ = 4 and in (2.6). It follows in particular that if the length of Ω remains between two fixed positive constants, then S(Ω) may be chosen uniformly with respect to Ω. Indeed, it suffices to establish the estimate Now, for x, y ∈ Ω given arbitrarily, we have Integrating by y in Ω and then applying the Hölder inequality we obtain that Since the right-hand side does not depend on x, hence (2.9) follows.   (−d, d), (2.10) where Q is an isoscele triangle of the form for some given d > 0. Denoting by the union of its equal sides, we have the Proposition 2.4. Assume that f satisfies the growth assumption (1.2). Then for every

Finally, the trace of the solution on S belongs to H 1 (S).
Proof. Thanks to lemma 2.3 above, we may repeat the proof of theorem 1.1 and of proposition 5.1 in [1] under the present weaker growth assumption (1.2). Let us correct here a small error in [1]: in the proof of proposition 5.1 the formula (5.3) is incorrect; the correct form is the following: However, the rest of the proof of proposition 5.1 remains valid.
The function L is also useful in the study of the Cauchy-Goursat problem where Q is an isoscele trapezoid of the form for some given 0 < d < d, denotes its smaller base, and denotes its boundary without the larger base.
the problem (2.11) has a unique solution u ∈ H 1 (Q) whose traces u t (·, t), u x (·, t) are well defined in L 2 (t − d, d − t) for every 0 ≤ t ≤ d , and the function In particular, u ∈ L ∞ (Q).

Remark.
In [1] we claimed that the proposition holds true for d = d when Q becomes a triangle. There was an error in the proof when we stated that u L ∞ (Q) is bounded by a constant independent of the Lipschitz constant of f if f is globally Lipschitz continuous. Indeed, given a real number c, the solution of is given by the series However, all theorems of [1] remain valid (and the proofs become even simpler) by applying the above proposition instead of proposition 5.2 there.
Proof of the proposition. According to the preceding remark, it suffices to prove that if f is globally Lipschitz continuous, then u L ∞ (Q) is bounded by a constant independent of the Lipschitz constant of f . Although this is not true in the triangular case, it is true in the trapeziodal case. In what follows c will denote diverse constants depending on ψ H 1 (S) and η L 2 (S ) but not on t and on the Lipschitz constant of f . Put for brevity. Multiplying the differential equation in (2.11) by u t and integrating by parts we obtain the equality Applying lemma 2.3 we conclude that here and in the sequel we write · instead of · L 2 (t−d,d−t) for brevity. First of all, (2.12) yields and then using the Cauchy-Schwarz inequality, we obtain Applying lemma 2.3 it follows that u(t) is bounded. Then using, (2.1) again, we conclude as in [1] that the function is bounded, too.