Controllability of a system

Question

How can I show that all solutions of

$x(t)'=\pmatrix{0&-1\\ 1&0}x(t)+\pmatrix{\cos(t)\\ \sin(t)}u(t)$

are within the area $x_1sin(t)-x_2cos(t)=0$ ?

Answer 1

It holds true that $$\dot{x}_1=-x_2+\cos (t) u(t)$$ $$\dot{x}_2=x_1+\sin(t)u(t)$$ So if you carry out the calculations you have $$\frac{d}{dt}[x_1\sin t -x_2\cos t]=0$$ and therefore $$x_1(t)\sin t -x_2(t)\cos t=-x_2(0)$$