Consider the nonlinear system: $$ \begin{align} \dot{x}_1 &= x_2 - x^2_1 \tag{1} \\ \dot{x}_2 &= -2x_1 - x_2 + x^3_1 + u \tag{2} \end{align} $$ (a) Determine a constant external control $u^*$ so that an equilibrium point has $x_1 = 2$.
(b) Find all the equilibrium points of the system with $u=u^*$.
(c) Determine a linearized model about each equilibrium point of the system.
(d) Determine which of these equilibrium points is locally stable.
For part(a), the equilibrium point(s) can be determined as follows:
$$ \begin{align} 0 &= x_2 - x^2_1 \tag{3} \\ 0 &= -2x_1 - x_2 + x^3_1 + u \tag{4} \end{align} $$
From Eq(3), we get $x_2=x^2_1$ and substitute it in Eq(4), we get $$ \begin{align} 0 &= -2x_1 - x^2_1 + x^3_1 + u \\ 0 &= x_1(-2 - x_1 + x^2_1) + u \\ 0 &= x_1(x_1-2)(x_1+1) + u \tag{5} \end{align} $$ For an equilibrium point to have $x_1=2$, the control $u$ must be zero in Eq(5). While the question doesn't restrict $u\neq 0$ but I feel the question requires a constant value other than the null. Any suggestions about this?