Stabilization of nonlinear systems by similarity transformations (Q1277076)

The author considers the system: \[ \dot x=A(x)+ b(x)u \] where the pair \(A(x)\), \(b(x)\) is given and \(x\in\mathbb{R}^n\), \(u(x)= s^*(x)x\). She obtains the feedback vector \(s(x)\) to stabilize the corresponding closed loop system. For an arbitrarily chosen constant vector \(g\), a sufficient condition on the existence and an explicite form of a similarity transformation \(T(A(x), b(x),g)\) is established. This latter transforms the matrix \(A(x)\) into a Frobenius matrix, vector \(b(x)\) into \(g\), and an unknown feedback vector \(s(x)\) into the first unit vector. The stabilization of the transformed system is subject to the choice of the constant vector \(g\).