  We discuss the sidewise control properties of 1-d waves. In analogy with
classical control and inverse problems for wave propagation, the problem
consists on controlling the behaviour of waves on part of the boundary of the
domain where they propagate, by means of control actions localised on a
different subset of the boundary. In contrast with classical problems, the goal
is not to control the dynamics of the waves on the interior of the domain, but
rather their boundary traces. It is therefore a goal oriented controllability
problem. We propose a duality method that reduces the problem to suitable new
observability inequalities, which consist of estimating the boundary traces of
waves on part of the boundary from boundary measurements done on another subset
of the boundary. These inequalities lead to novel questions that do not seem to
be treatable by the classical techniques employed in the field, such as
Carleman inequalities, non-harmonic Fourier series, microlocal analysis and
multipliers. We propose a genuinely 1-d solution method, based on sidewise
energy propagation estimates yielding a complete sharp solution. The obtained
observability results can be reinterpreted in terms of Fourier series. This
leads to new non-standard questions in the context of non-harmonic Fourier
series.
