Local solvability of degenerate, overdetermined first-order partial differential equations. A control theoretic perspective (Q1903056)

Existence of classical solutions of degenerate overdetermined systems of first order PDE is considered. The author considers systems for the determination of scalar functions \(\alpha: M\to \mathbb{R}\) \((M\subseteq \mathbb{R}^n\) is open) of the form \[ X_i \alpha= c_i(x),\quad i= 1,\dots, d,\quad x\in M \] and for the determination of vector functions \(\beta: M\to \mathbb{R}^r\) of the form \[ X_i \beta= C_i(x)\beta,\quad i= 1,\dots, d,\quad x\in M. \] Here the vector fields \(X_i\) are allowed to become linearly dependent at certain points of the ambient manifold. The method of the proofs is a combination of tools from nonlinear control theory involving invariant manifolds and foliations and ideas arising from the method of characteristics.