  Shape control is critical to ensure the quality of composite fuselage
assembly. In current practice, the structures are adjusted to the design shape
in terms of the $\ell_2$ loss for further assembly without considering the
existing dimensional gap between two structures. Such practice has two
limitations: (1) the design shape may not be the optimal shape in terms of a
pair of incoming fuselages with different incoming dimensions; (2) the maximum
gap is the key concern during the fuselage assembly process. This paper
proposes an optimal shape control methodology via the $\ell_\infty$ loss for
composite fuselage assembly process by considering the existing dimensional gap
between the incoming pair of fuselages. Besides, due to the limitation on the
number of available actuators in practice, we face an important problem of
finding the best locations for the actuators among many potential locations,
which makes the problem a sparse estimation problem. We are the first to solve
the optimal shape control in fuselage assembly process using the $\ell_\infty$
model under the framework of sparse estimation, where we use the $\ell_1$
penalty to control the sparsity of the resulting estimator. From statistical
point of view, this can be formulated as the $\ell_\infty$ loss based linear
regression, and under some standard assumptions, such as the restricted
eigenvalue (RE) conditions, and the light tailed noise, the non-asymptotic
estimation error of the $\ell_1$ regularized $\ell_\infty$ linear model is
derived to be the order of $O(\sigma\sqrt{\frac{S\log p}{n}})$, which meets the
upper-bound in the existing literature. Compared to the current practice, the
case study shows that our proposed method significantly reduces the maximum gap
between two fuselages after shape adjustments.
