Answer Happy

The Sun-Earth L2 point is a point beyond the Earth on the
Sun-Earth line where the gravitational forces of the Sun and the
Earth are exactly balanced by centrifugal force. A spacecraft with
a large solar sail is to be placed at an equilibrium point near L2
where the centrifugal force is augmented by the sail force. The
in-plane equations of motion (EOMs), linearized about this
equilibrium point, are
𝑥̈= 2𝑦̇ + 12.762𝑥
𝑦̈= −2𝑥̇ − 4.914𝑦 + 1.948𝑢
where u is the deviation of the solar sail angle from the
nominal (nominal solar sail position is perpendicular to the
Sun-Earth line) at the equilibrium. Here 𝑥, 𝑦 are in units of 1.51
× 106 km (the distance from the Earth to L2 point), time is in
units of 1/n, where n is Earth angular velocity about the Sun, and
the equilibrium point is 16% closer to Earth than the L2 point.
Problem 1. [30 points] Consider the open-loop system (with 𝑢 =
0). Is the system stable? [Hint: Re-write EOMs in the form 𝑋̇ = 𝐴𝑋
+ 𝐵𝑢 where 𝑋 = [ 𝑥 𝑥̇ 𝑦 𝑦̇ ] 𝑇 and compute four eigenvalues of the
matrix 𝐴. Use MATLAB command eig(A). Check their location on the
complex plane]