Global Bilinearization and Controllability of Control-Affine Nonlinear Systems: A Koopman Spectral Approach

This paper considers the problem of global bilinearization of the drift and control vector fields of a control-affine system. While there are linearization techniques like Carleman linearization for embedding a finite-dimensional non-linear system into an infinite-dimensional space, they depend on the analytic property of the vector fields and work only on polynomial space. The proposed method utilizes the Koopman Canonical Transform to transform the dynamics and ensures bilinearity from the projection of the Koopman operator associated with the control vector fields on the eigenspace of the drift Koopman operator. The resulting bilinear system is then subjected to controllability analysis using the Myhill semigroup method and Lie algebraic structures. The results are supported by a numerical example.