  The first part of this paper is devoted to introducing an approach to compute
the approximate minimum time function of control problems which is based on
reachable set approximation and uses arithmetic operations for convex compact
sets. In particular, in this paper the theoretical justification of the
proposed approach is restricted to a class of linear control systems. The error
estimate of the fully discrete reachable set is provided by employing the
Hausdorff distance to the continuous-time reachable set. The detailed procedure
solving the corresponding discrete set-valued problem is described. Under
standard assumptions, by means of convex analysis and knowledge of the
regularity of the true minimum time function, we estimate the error of its
approximation. Numerical examples are included in the second part.
