Problem 4: Linear Maps (4 points). Consider a linear state equation with specified forcing function

x(t) = A(t)x(t) + f(t)

and specifified two-point boundary conditions on x(t):

Hox(t0) + Hfx(tf ) = h

Here, Ho and Hf are n n matrices, h is an n 1 vector, and tf > t0.

(a) Under what condition(s) does there exist a solution x(t) to the state equation that satisfies the boundary conditions?

(b) Under what condition(s) does there exist a unique solution satisfying the boundary conditions?