Generation of stable limit cycles in controllable linear systems

Abstract

In this paper we prove that any controllable linear systems x ˙ = A x + B u , x ∈ ℝ n , u ∈ ℝ m , admits a polynomial feedback u= u( x) such that the closed-loop system x ˙ = A x + B u ( x ) admits an orbitally asymptotically stable limit cycle. Moreover, we prove that for any positive integer n, there exists an nth-order polynomial, autonomous, ordinary differential equation with a unique limit cycle.