Quantum Gravity 220.1: Bang-Bang and Switching Control
- From: OsherD <mdoctorow@xxxxxxxxx>
- Date: Sun, 23 Dec 2007 06:59:13 -0800 (PST)
From Osher Doctorow
In Bang-Bang Control in Optimal Control Engineering and Mathematics
theory, the controller switches suddenly/abruptly between two states
(see Wikipedia "Bang-Bang Control", "Hysteresis," etc.).
ArXix has 9 papers on Bang-Bang control including 1 in 2007 and 2 in
2006.
Switching Control is often closely related to Bang-Bang Control, and
has 5 papers in arXiv including 1 in 2007 and 2 in 2006, but the wider
category "Switching" in arXiv brings up 462 papers including 91 in
2007 and 94-95 in 2006. The papers have tended to increase in number
from 1994 (3 papers) and 1995 and 1996 (3 papers each) to the 90s in
2006 and 2007.
An especially interesting paper arguably is Grace Y. Lin, Y. Yu, David
Yao's "Stochastic knapsack problem revisited: switchover policies and
dynamic pricing," arXiv:0708.1146 math.PR, math.OC (optimal control).
The first 2 authors are at IBM T. J. Watson Research Center Yorktown
Heights New York, the 3rd at Columbia U. New York. Switchover
policies as defined/used by them start from accepting only orders of
the highest price and switch to including lower prices as time goes
by, the switchover time optimally decided by convey programming, and
they show that such policies are asymptotically optimal.
Stochastic knapsack problems, as in Lin, Yu, and Yao (2007), are
widely used in fields from dynamic resource allocation to admission
control in telecommunication, and in recent years has become important
in studying revenue management and dynmical/flexible pricing problems.
These types of papers arguably are often precedents for using Bang-
Bang and Switching Control in physics, and cosmology should be no
exception. The shifts from constant or decelerating to accelerating
Universe or vice versa discussed in my recent posts seem similar to
various of the scenarios of the Bang-Bang and Switching Control
papers.
Many of these papers are related to hysteresis as I indicated
earlier. Hysteresis is path-dependent memory basically, and you need
the history of the input to predict the output, unlike Markov chains
and Independent Probability-Statistics processes. Path-dependent
memory belongs to Probable Causation/Influence (PI), while Markov
chains belong to conditional probability, the two being respectively
characterized by:
1) P(A-->B) = 1 + y - x, 0 < = y < = x < = 1
2) P(B|A) = y/x, 0 < = y < = x < = 1 except that x not 0
In both (1) and (2), y = P(AB), x = P(A).
Osher Doctorow
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