Group theory and controllability of partial differential control systems

Abstract

In the case of a classical control system of the form x=a(x)+b(x)u it is known that one can introduce the weak control distribution ΔWC spanned by the control Lie algebra C generated by the vector fields a,b on one side and the strong control distribution ΔSC such that ΔWC=a+ΔSC. The control system is then said to be controllable if ΔWC has maximum rank. However, in the linear case, when a(x)=Ax, b(x)=B, the well known criterion of controllability just amounts to check that ΔSC has maximum rank. The purpose of our communication is to prove that the confusion made between ΔWC and ΔSC can only be understood through group theoretical arguments and a new definition of controllability which does not depend on the state internal representation and is also valid in the frame work of partial differential control theory, that is whenever inputs and outputs are related by a (linear or nonlinear) system of partial differential equations (PDE) with any number of independent variables and any order. In this framework, the control Lie algebra is just the centralizer of the Lie algebra of the biggest group of transformations of the outputs preserving the control system for all possible inputs. A link with differential algebra is given and many explicit examples illustrate the main results.