Problem 6: Linear Quadratic Regulator (8 points). Consider the scalar control system (x(t)R) : x(t)=u(t),x(0)=0 with cost J=2101(x2(t)+u2(t))dt+21x2(1). (a) We've shown in class that the optimal input is given by the standard form u(t)=BT(t)X(t)X~1(t)x(t), where X(t) and X~(t) solve the backwards Hamiltonian linear matrix differential equation given on the first page of this test. For the system above, derive the solution (X(t),X~(t)) to this Hamiltonian system with appropriate final condition. (b) From your solution in part (a), derive the solution P(t) to the Riccati equation. (c) Finally, check that your solution from (b) solves the Riccati equation on the first page of this test with appropriate final condition