Аннотация: This paper deals with problems of impulse control which allow control inputs
consisting not only of delta functions but also of their higher derivatives
(impulses of higher order). The controls are sought for in the form of
feedback strategies which leads to the application of respective generalized
dynamic programming techniques, where the role of traditional
Hamilton--Jacobi--Bellman equations is taken by respective variational
inequalities of similar structure. Further proposed are physically realizable
approximations which converge to these ideal solutions. Since the ideal
solutions allow to transfer a controllable system from one given position to
another in zero time, their approximations lead us to physically realizable
``fast'' controls with piecewise constant realizations. Such feedback control
inputs are then compared with traditional bang-bang type strategies and turn
out to be more robust. Computational schemes for related problems of
reachability and control synthesis are further described with examples of
damping oscillating systems of high order in minimal time being demonstrated.