abstract: In this talk we will deal with the control problem for the 1-D wave equation
(1) ω(x) ∂2t u - ∂2x u = 0
on the interval [0,1], under minimal regularity assumptions over the coefficient ω In a first time, we will show “classical” observability estimates for ω satisfying an integral Zygmund condition. In particular, this result represents an improvement to the previous one for BV coefficients. Then we will consider lower regularity hypothesis: for ω log-Lipschitz or log-Zygmund, we will prove observability estimates “with a finite loss of derivatives”. Finally, we will discuss the sharpness of our results.