3 research outputs found
Billiard Representation for Multidimensional Quantum Cosmology near the Singularity
The degenerate Lagrangian system describing a lot of cosmological models is
considered. When certain restrictions on the parameters of the model are
imposed, the dynamics of the model near the "singularity" is reduced to a
billiard on the Lobachevsky space. The Wheeler-DeWitt equation in the
asymptotical regime is solved and a third-quantized model is suggested.Comment: 6 pages, LaTe
Ball on a beam: stabilization under saturated input control with large basin of attraction
This article is devoted to the stabilization of two underactuated planar
systems, the well-known straight beam-and-ball system and an original circular
beam-and-ball system. The feedback control for each system is designed, using
the Jordan form of its model, linearized near the unstable equilibrium. The
limits on the voltage, fed to the motor, are taken into account explicitly. The
straight beam-and-ball system has one unstable mode in the motion near the
equilibrium point. The proposed control law ensures that the basin of
attraction coincides with the controllability domain. The circular
beam-and-ball system has two unstable modes near the equilibrium point.
Therefore, this device, never considered in the past, is much more difficult to
control than the straight beam-and-ball system. The main contribution is to
propose a simple new control law, which ensures by adjusting its gain
parameters that the basin of attraction arbitrarily can approach the
controllability domain for the linear case. For both nonlinear systems,
simulation results are presented to illustrate the efficiency of the designed
nonlinear control laws and to determine the basin of attraction