  We consider the minimum-energy control of a car, which is modelled as a point
mass sliding on the ground in a fixed direction, and so it can be
mathematically described as the double integrator. The control variable,
representing the acceleration or the deceleration, is constrained by simple
bounds from above and below. Despite the simplicity of the problem, it is not
possible to find an analytical solution to it because of the constrained
control variable. To find a numerical solution to this problem we apply three
different projection-type methods: (i)~Dykstra's algorithm, (ii)~the
Douglas--Rachford (DR) method and (iii)~the Arag\'on Artacho--Campoy (AAC)
algorithm. To the knowledge of the authors, these kinds of (projection) methods
have not previously been applied to continuous-time optimal control problems,
which are infinite-dimensional optimization problems. The problem we study in
this article is posed in infinite-dimensional Hilbert spaces. Behaviour of the
DR and AAC algorithms are explored via numerical experiments with respect to
their parameters. An error analysis is also carried out numerically for a
particular instance of the problem for each of the algorithms.
