Final answer:
The linearization of the nonlinear state equations around the given nominal trajectory, which is constant, leads to a linear system that is time-invariant. One checks the coefficients of the linearized system and since they do not explicitly depend on time, the system is time-invariant.
Step-by-step explanation:
To linearize the nonlinear state equations X₁(t)=u(t)[x₂(t)]² - x₁(t) and x₂(t)= - x₂(t) about a nominal trajectory, we consider a perturbation from the nominal trajectory given by x ~₁(0) = 0, x ~₂(0)=1, and u~(t)=1 for all t ≥ 0. We denote deviations from the nominal trajectory as Δx₁ and Δx₂, leading to approximated linear equations of deviation by Taylor expanding the original equations around the nominal trajectory and retaining the first-order terms.
To determine whether the linearized system is time-varying, we observe the coefficients of the linearized equations. If these coefficients are functions of time, the system is time-varying; if not, it is time-invariant. As the coefficients of the linearized equations in our case depend only on the constant nominal inputs and not on time explicitly, our linearized system should be time-invariant.