**Some Results on Optimal Estimation and Control
for Lossy Networked Control**

**
Abstract:**
A long term goal in the theory of systems and control is to develop a
systematic methodology for the design of feedback control schemes capable of
shaping the response of complex dynamical systems, in both an equilibrium
and a nonequilibrium setting. The most classical example of a nonequilibrium
attractor for a nonlinear dynamical system is a periodic orbit. In this
talk, we present sufficient conditions for the existence of oscillations in
a nonlinear dynamical system, e.g. a closed-loop control system. Just as in
Liapunov theory, these conditions can be checked point-wise and therefore do
not require the knowledge of the trajectories of the system, in marked
contrast with existing criteria requiring the existence of cross-sections
for the dynamics. Moreover, using the recent solution of the Poincaré
Conjecture in all dimensions, we show that these same conditions are
necessary for the existence of an asymptotically stable periodic orbit.
These results are illustrated by showing the existence of an asymptotically
stable oscillation in a three dimensional, nonholonomic mathematical model
of an AC controlled rotor, controlled to turn at a constant angular velocity.
We also apply these results to show the existence of a periodic response of
a dissipative nonlinear control system, when forced with

a periodic input.

In this talk I will describe some problems
related the effect of packets