Changing the state of a linear differential system in (almost) zero time by using distributional input function

In many physical (for instance, in thermodynamics) or in more economic dynamic systems the (almost) zero time state changing is more than important. One of the most typical state changing in (almost) zero time appears whenever the financial institution managers predetermine the interest rate policy. Thus, in this paper we investigate the state changing of a linear differential system in (almost) zero time by using a linear combination of Dirac δ-function and its derivatives. Obviously, such an input is very hard to imagine physically. However, we can think of it approximately as a combination of small pulses of very high magnitude and infinitely small duration. Using linear algebra techniques and the generalized inverse theory, the input coefficients are fully determined. Finally, the whole paper ends up with the analytic presentation of an illustrative numerical example.