q 1 consider the first order system described byy 2y au y 0 c determin

Question

Q.1. Consider the first order system described byy = 2y + au, y(0) = c.Determine the optimal control input u(t) to minimize the functional J(y, u)=\int_{0}^{\infty}\left[y^{2}+w u^{2}\right] d t where w>0, is a weight factor for the inputs. Use both calculus of variations and dynamic programming to obtain the control law, and show that the control laws obtained from the two methods are equivalent to each other from the feedback point of view.Discuss two extreme cases where the weight factor w→0 and w→∞, and explain your findings.