January 26, 2025
A (stationary) linear system on a state space X is specified by a linear operator A\colon X\to X. If x(t) is the state of the system at time t the dynamics are determined by dx/dt = Ax(t), x(t_0) = x_0.
Exercise. Show a solution is x(t) = \exp(tA)x_0.
Hint. \exp(tA) = \sum_{n=0}^\infty t^n A^n/n!.
Exercise. If X = \bm{R}^2 and A = [0, 1;-1 0] show x(t) = (\cos t, \sin t) for x_0 = (1, 0).
Hint. Use A^2 = [-1, 0;0, 1], A^3 = [0, -1;1, 0], and A^4 = I to show \exp(tA) = [\cos t, \sin t; -\sin t, \cos t].
Note A has eigenvalues \lambda_\pm = \pm i with eigenvectors e_\pm = [\pm i, 1].
Suppose B\colon U\to X is a linear operator.