Accessibility Conditions for PDE Systems and the Adjoint System

Abstract

A group-theoretical approach is used to tackle the problem of (local) accessibility along a trajectory of systems described by partial differential equations (PDEs). In particular, a class of first-order PDE systems with boundary conditions and boundary control is applied to illustrate the key ideas of our approach. The methods mainly rely on a coordinate-free formulation of PDE systems, which also incorporates boundary conditions. The accessibility is discussed in general and an approach based on (pointwise, continuous) transformation groups and their invariants is motivated. Using an infinitesimal criterion for invariance we study special group invariants to derive (local) conditions on accessibility. It is highlighted that the basic questions lead to the investigation of a particular adjoint system. A nonlinear example demonstrates the methods and results.