A first-order system is given by

with the initial condition x(0) = .v0. We want to design a feedback controller
u = -kx
such that the performance index

is minimized.
(a) Let A = 1. Develop a formula for / in terms of k, valid for any x0, and use an m-file to plot J / x20 versus k. From the plot, determine the approximate value of k = k min that minimizes J / X2/Q (b) Verify the result in part (a) analytically, (c) Using the procedure developed in part (a), obtain a plot of &min versus A, where kmin is the gain that minimizes the performance index.

(x1) + Au'(1)) dt