Stabilization of nonlinear systems by similarity transformations

For a system x ˙ = A ( x ) + b ( x ) u , u ( x ) = s ∗ ( x ) x , x ∈ ℝ n , where the pair ( A ( x ) , b ( x ) ) is given, we obtain the feedback vector s ( x ) to stabilize the corresponding closed loop system. For an arbitrarily chosen constant vector g , a sufficient condition of the existence and an explicit form of a similarity transformation T ( A ( x ) , b ( x ) , g ) is established. The latter transforms matrix A ( x ) into the Frobenius matrix, vector b ( x ) into g , and an unknown feedback vector s ( x ) into the first unit vector. The boundaries of A ˜ ( y , g ) are determined by the boundaries of { ∂ k A ( x ) ∂ x k , ∂ k b ( x ) ∂ x k } , k = 0 , n − 1 ¯ . The stabilization of the transformed system is subject to the choice of the constant vector g .