Use the Pontryagin Maximum Principle to find the optimal control u and the optimal cost J for the following problem: The state equations are . x = y, . y = z, . z = u; the cost function is J = Z 1 0 (2 + 1 2 u 2 )dt; the initial state is x(0) = 0, y(0) = 1 2 , z(0) = 1; the target is the set {x = 0} at the time t = 1; and control u is unbounded. Interpret the initial state, the target set and the optimal control strategy u if the system describes the motion of an object along a straight line, with x = position, y = velocity, z = acceleration and the control u is the time-derivative of a force acting on the object.