Stabilization of controllable and observable systems
The author presents a nice survey about the stabilization of continuous-time smooth systems, from the definitions to his recent major contributions. Let \(\dot x= f(x, u)\) with \(f(0, 0)= 0\) and \(x\in \mathbb{R}^n\) be such that the origin is locally small time continuously reachable and that the strong accessibility rank condition holds at the origin. If \(n\geq 4\), then the origin is locally small time stabilizable with periodic continuous static state feedback. The idea of the proof consists in showing that one can drive (with a feedback) any initial condition in a neighbourhood of the origin to a vicinity of a smooth closed one-dimensional manifold \(\gamma\). Then, a new open-loop control can be chosen so that the image of a vicinity \(V\) of \(\gamma\) under the action of the flow is embedded in \(\mathbb{R}^n\backslash \{0\}\) (so that the open-loop control can be represented as a feedback) for all \(t< T\), and so that all points of \(V\) are driven to the origin at \(t= T\), provided that \(n\geq 4\) and that the variational system is controllable. For \(n= 1\), if \(f\) is analytic, using that time has the same dimension as the state, the author shows how to stabilize locally the origin with a continuous autonomous feedback. If \(f\) is affine without drift, one can stabilize the origin globally with a smooth periodic feedback. Under additional observability assumptions, the origin is locally small time stabilizable by periodic continuous dynamic output feedback. The idea of the proof involves a first step where the state is first observed using the dynamic extension, and another step where stabilization is performed using the available state. Concerning the first result, let us remark that in [the author, SIAM J. Control Optimization 33, 804-833 (1995; Zbl 0828.93054)], the feedback belongs to \(C^0(\mathbb{R}^n\times [0, T]; \mathbb{R}^m)\) of class \(C^\infty\) on \((\mathbb{R}^n)\backslash \{0\}\times [0, T]\) and vanishing on \(\{0\}\times [0, T]\). Therefore, the crossing of \(\gamma\) can occur only at the origin (because of the theorem on existence and uniqueness of solutions to differential equations) and stays there. But according to lemma 2.8 of the same source, this case can be avoided by taking a nearby control if \(n\geq 3\). This seems to answer the case \(n= 3\).
scientific article; zbMATH DE number 824189
scientific article