Carleman linearization of differential-algebraic equations systems

Carleman linearization is a mathematical technique that transforms nonlinear dynamical systems into infinite-dimensional linear systems, enabling simplified analysis. Initially developed for ordinary differential equations (ODEs) and later extended to partial differential equations (PDEs), it has found applications in control theory, biological systems, fluid dynamics, quantum mechanics, finance, and machine learning. This paper extends Carleman linearization to differential-algebraic equation (DAE) systems by introducing auxiliary functions and projecting the resulting system onto a higher-order ODE representation. Theoretical foundations are presented along with conditions under which the transformation is valid. The method is demonstrated on synthetic DAE examples, highlighting its effectiveness even when projection from algebraic variables to state variables is nontrivial or undefined.