Approximating distributional behaviour of LTI differential systems using Gaussian function and its derivatives

This article is concerned with the approximation of the distributional behaviour of linear, time-invariant (LTI) systems. First, we review the different types of approximations of distributions by smooth functions and explain their significance in characterising system properties. Second, we consider the problem of changing the state of controllable LTI differential systems in a very short time. Thus, we establish an interesting relation between the time and volatility parameters of the Gaussian function and its derivatives in the approximation of distributional solutions. An algorithm is then proposed for calculating the distributional input and its smooth approximation which minimises the distance to an arbitrary target state. The optimal choice of the volatility parameter for the state transition is also derived. Finally, some complementary distance problems are also considered. The main results of this article are illustrated by numerous examples.