Carleman linearization of nonlinear systems and its finite-section approximations (Q6642456)

The authors consider Carleman linearization of the nonlinear dynamical system\N\[\N\dot{\boldsymbol{x}} = \boldsymbol{f}(t, \boldsymbol{x}(t)), \quad t \geq t_0,\N\]\Nwhere \(\boldsymbol{x}(t_0) = \boldsymbol{x}_0 \neq \boldsymbol{0}\), with the origin \(\boldsymbol{0}\) as equilibrium and where \(\boldsymbol{x}\in \mathbb{R}^n\) is the state of the system and \(\boldsymbol{f}(t, \boldsymbol{x})\) an analytics function about \(\boldsymbol{x}\) on a neighborhood of the equilibrium \(\boldsymbol{0}\).\N\NExponential convergence of the finite section scheme over a short time interval and also over a whole time is also investigated. The Carleman linearization of several benchmark systems are discussed to validate and illustrate the theoretical findings.