484 research outputs found
Dynamics of Simple Balancing Models with State Dependent Switching Control
Time-delayed control in a balancing problem may be a nonsmooth function for a
variety of reasons. In this paper we study a simple model of the control of an
inverted pendulum by either a connected movable cart or an applied torque for
which the control is turned off when the pendulum is located within certain
regions of phase space. Without applying a small angle approximation for
deviations about the vertical position, we see structurally stable periodic
orbits which may be attracting or repelling. Due to the nonsmooth nature of the
control, these periodic orbits are born in various discontinuity-induced
bifurcations. Also we show that a coincidence of switching events can produce
complicated periodic and aperiodic solutions.Comment: 36 pages, 12 figure
On periodic solutions in the Whitney's inverted pendulum problem
In the book `What is Mathematics?' Richard Courant and Herbert Robbins
presented a solution of a Whitney's problem of an inverted pendulum on a
railway carriage moving on a straight line. Since the appearance of the book in
1941 the solution was contested by several distinguished mathematicians. The
first formal proof based on the idea of Courant and Robbins was published by
Ivan Polekhin in 2014. Polekhin also proved a theorem on the existence of a
periodic solution of the problem provided the movement of the carriage on the
line is periodic. In the present paper we slightly improve the Polekhin's
theorem by lowering the regularity class of the motion and we prove a theorem
on the existence of a periodic solution if the carriage moves periodically on
the plane
Worrisome Properties of Neural Network Controllers and Their Symbolic Representations
We raise concerns about controllers' robustness in simple reinforcement
learning benchmark problems. We focus on neural network controllers and their
low neuron and symbolic abstractions. A typical controller reaching high mean
return values still generates an abundance of persistent low-return solutions,
which is a highly undesirable property, easily exploitable by an adversary. We
find that the simpler controllers admit more persistent bad solutions. We
provide an algorithm for a systematic robustness study and prove existence of
persistent solutions and, in some cases, periodic orbits, using a
computer-assisted proof methodology.Comment: accepted to ECAI2
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