5. The system state equation is x = u Find the optimal solutions x* (t) to the following problems (a) ] = √(x² +u²)dt, x(0) = 0, x(1) = 1 Hint: to obtain solution to the resulting first-order differential equations (state and costate equations), you may differentiate the state equation one more time, use the costate equation to eliminate the costate and obtain a 2nd-order differential equation only in x, which should be easily to solve. The same technique can used for the rest of the problems below) (b) ] = ƒª(x² +u²)dt, x(0) = 1, x(1) is unconstrained (c) ] = √²(x² + 2xu + u²)dt, x(0) = 1, x(2) = −3 (d) f (1+u²)dt, x(0) = 2, t is free, and the terminal constraint is 9₁ = 4x(t) + 4tf - 5 = 0