No control: u = 0
Some nearby states move away. Along x2=0 with x1>0, ẋ1=x12.
Composition-operator stabilization
From a local section to a feedback law
A local right inverse separates one hard question into two concrete objects: an associated state field whose stability can be checked directly, and a feedback law that makes the original plant reproduce that field.
The factorization
The section recovers each requested velocity.
velocity
v
α(v)=(α1(v), α2(v))
→
state + control
(α1(v), α2(v))
f
→
recovered velocity
v
Right-inverse identity f ∘ α = id
Associated state field
ẋ = α1−1(x)
⟹
Feedback pulled from the section
u(x) = α2(α1−1(x))
⟹
Closed-loop identity
f(x,u(x)) = α1−1(x)
A polynomial example
The controller replaces one unstable velocity component.
The example below is local: all initial states lie near the equilibrium at the origin. The middle and right trajectories coincide because the feedback makes the original system equal the associated stable field.
Original plant
ẋ = f(x,u) =
Motion is paused because reduced motion is enabled.
ẋ = g(x) = α1−1(x)
A stable second component is chosen while retaining the plant's first component.
f(x,u(x)) = g(x)
The feedback correction, shown in green, converts the native velocity into the chosen stable velocity.
native velocity
control correction
resultant
Choose the stable second velocity
g2(x) = −½x1 − 2x2
Make the original plant reproduce it
x1x2 + x22
+
u(x)3
=
−½x1 − 2x2
The continuous feedback
u(x) = ∛−½x1 − 2x2
−
(x1x2 + x22)
The Jacobian of g at the origin has eigenvalues (−2 ± √2)/2, both negative. Thus the displayed associated system—and therefore the controlled plant—is locally exponentially stable.