5. Consider the uncontrolled system model as n = X The performance is measured over time interval (0,1) as P I=\int_{0}^{1} \frac{1}{2}\left[x^{2}+u^{2}\right] d t a. Determine the state-function of Pontryagin H. b. Determine the optimal input uº. C. Determine the equations governing the controlled system in terms of x and 2. d. Find the state transition matrix. e. Find the optimal input such that x(0) = 0 and x(1) = 2.

Fig: 1

Fig: 2

Fig: 3

Fig: 4

Fig: 5

Fig: 6

Fig: 7

Fig: 8

Fig: 9