**Original manuscript:** 2012/09/24

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 Σ :
$$*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, small-time local controllability
from an equilibrium *p* of Σ implies the existence of a piecewise
analytic feedback *F*_{Σ} that asymptotically stabilises
Σ 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 to show stabilisability of
small-time locally controllable systems in finite time, at the expense of
obtaining a closed-loop system that may not be Lyapunov stable. Having
established stabilisability in finite time, we proceed to prove a
converse-Lyapunov theorem. If *F*_{Σ} is a piecewise
analytic feedback that stabilises a small-time locally controllable system in
finite time, then the Lyapunov function we construct has the interesting
property of being differentiable along every trajectory of the closed-loop
system obtained by "applying" *F*_{Σ} to Σ.

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.

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Last Updated: Fri Mar 15 07:59:18 2024