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Title:  Feedback Stabilisation of Locally Controllable Systems 
Authors:  Isaiah, Pantelis 

Keywords:  feedback stabilisation geometric control theory 
Issue Date:  25Sep2012 
Series/Report no.:  Canadian theses 
Abstract:  Controllability and stabilisability are two fundamental properties of control systems and it is intuitively appealing to conjecture that the former should imply the latter; especially so when the state of a control system is assumed to be known at every time instant. Such an implication can, indeed, be proven for certain types of controllability and stabilisability, and certain classes of control systems. In the present thesis, we consider real analytic control systems of the form $\Sgr:\dot{x}=f(x,u)$, with $x$ in a real analytic manifold and $u$ in a separable metric space, and we show that, under mild technical assumptions, smalltime local controllability from an equilibrium $p$ of \Sgr\ implies the existence of a piecewise analytic feedback \Fscr\ that asymptotically stabilises \Sgr\ at $p$. As a corollary to this result, we show that nonlinear control systems with controllable unstable dynamics and stable uncontrollable dynamics are feedback stabilisable, extending, thus, a classical result of linear control theory.
Next, we modify the proof of the existence of \Fscr\ to show stabilisability of smalltime locally controllable systems in finite time, at the expense of obtaining a closedloop system that may not be Lyapunov stable. Having established stabilisability in finite time, we proceed to prove a converseLyapunov theorem. If \Fscr\ is a piecewise analytic feedback that stabilises a smalltime locally controllable system \mbox{$\Sgr:\dot{x}=f(x,u)$} in finite time, then the Lyapunov function we construct has the interesting property of being differentiable along every trajectory of the closedloop system obtained by ``applying" \Fscr\ to \Sgr.
We conclude this thesis with a number of open problems related to the stabilisability of nonlinear control systems, along with a number of examples from the literature that hint at potentially fruitful lines of future research in the area. 
Description:  Thesis (Ph.D, Mathematics & Statistics)  Queen's University, 20120924 10:24:22.51 
URI:  http://hdl.handle.net/1974/7506 
Appears in Collections:  Queen's Graduate Theses and Dissertations Department of Mathematics and Statistics Graduate Theses

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