  The 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.
Higher-order discretization of the reachable set of the linear control problem
can balance missing regularity (e.g., Holder continuity) of the minimum time
function for smoother problems. To illustrate the error estimates and to
demonstrate differences to other numerical approaches we provide a collection
of numerical examples which either allow higher order of convergence with
respect to time discretization or where the continuity of the minimum time
function cannot be sufficiently granted, i.e., we study cases in which the
minimum time function is Holder continuous or even discontinuous.
