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.
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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
In the case of measurable control functions, various definitions of generalized trajectories have been introduced in the literature by considering suitable limits of classical solutions, under key assumptions involving either the total variation of the control function or the commutativity of the vector fields. In the present paper, applying the standard Picard's iterative procedure (used for ODEs), we get a unique solution for our non-standard control system by combining a smooth mapping (an adequate composition of flows) with a solution satisfying an auxiliary affine control system in the space of parameters.
Professor Savin Treanta
Universitatea Politehnica din Bucuresti
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This page is a summary of: Local uncontrollability for affine control systems with jumps, International Journal of Control, September 2016, Taylor & Francis,
DOI: 10.1080/00207179.2016.1227090.
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