Approximating distributional behaviour of linear systems using Gaussian function and its derivatives

The paper is concerned with defining families of smooth functions that can be used for the approximation of impulsive types of solutions for linear systems. We review the different types of approximations of distributions in terms of smooth functions and explains their significance in the characterization of system properties where impulses were used for their characterisation. For controllable systems, we establish an interesting relation between the time t and sigma (volatility) in the approximation of distributional solutions. An algorithm is then proposed for the calculation of the coefficients of the input required to minimize the distance of our desired target state before and after approximation is proposed. The optimal choice of sigma is derived for a pre-determined time t for the state transition.