# Local uncontrollability for affine control systems with jumps

### What is it about?

The purpose of this paper is to formulate and prove Lie-algebraic sufficient conditions which involve the local uncontrollability of a given affine control system with jumps. In the main result of this paper, the uncontrollability conditions are not influenced when a non-standard affine control system is replaced by a standard one.

### Why is it important?

The standard form of a control system includes trajectories without any jumps. In our "non-standard" model, we consider piecewise constant control functions. It induces a finite set of jumps for each trajectory and the integral form of a trajectory uses new bounded variation controls as integration variables. In other words, the analysis of non-standard control systems includes a finite set of jumps which influence the trajectories of the control systems. Any affine control system with jumps has the property that its trajectories are contained in the kernel of a linear first order PDE. It should be noted that in our case the trajectories are piecewise smooth bounded variation functions. This explains the non-standard writing used in the present work. The main result can be easily connected with so called unreachability of an affine control system. When controllability or uncontrollability properties are involved, we need a special analysis and our paper presents the details of the uncontrollability properties connected with complete affine control systems under jumps conditions.

### Perspectives

http://dx.doi.org/10.1080/00207179.2016.1227090

The following have contributed to this page: Professor Savin Treanta