On controlled invariance in nonlinear systems (Q1087852)

Given a nonlinear control system which is locally of the form \(\dot x=X(x,u)\), where x and u are respectively n-dimensional and m-dimensional real vectors and X is a \(C^{\infty}\) map, and given an involutive distribution \(\Delta\) of dimension s, spanned by \(\{\partial /\partial x_{n-s+j}\), \(j=1,...,s\}\), the author considers the following Controlled Invariance Problem (CIP): ''Establish whether there exists a \(C^{\infty}\) feedback \(u=\phi (x)\) such that, for \(X_{\phi}=X(.,\phi (.))\), one has \([X_{\phi},\Delta](x)\subset \Delta (x)''\). The existence of such a feedback amounts to say that \(\Delta\) is a controlled invariant (CI) distribution. If, in order to achieve the desired invariance, some degrees of freedom in the control are necessarily lost, \(\Delta\) is said to be a generalized controlled invariant (GCI) distribution. Using the notion of GCI distribution, a procedure to solve the above problem, under a number of technical restrictions, is presented.