How do I approach this question?

Consider 0 i = C+ u B where u(.) is a 27-periodic function of time. Instead of the initial condition, we have the boundary condition x(0) = x(27) That is, we are interested in 27-periodic solutions of the linear system with 27-periodic forcing. a) Is it true that a periodic solution x(.) always exist for any given 27-periodic u(.)? [Hint: u(t) = cos(t)]. b) Consider next the adjoint system -p = A p and show that at (p (t) x (t) ) = p (t) Bu(t) c) Upon integrating both sides from 0 to 27, derive the following necessary condition that u(.) must satisfy for a 27-periodic solution x(.) to be exist: (2TT Ju(t) sin(t) dt = u(t) cos(t) dt = 0 d) Show that the condition in part (c) is also sufficient. That is, if u(.) satisfies the condition then the resulting solution x(.) is 27-periodic