TITLE:
From Kalman to Einstein and Maxwell: The Structural Controllability Revisited
AUTHORS:
Jean-Francois Pommaret
KEYWORDS:
Differential Sequence, Differential Homological Algebra, Differential Double Duality, Control Theory, Controllability, Einstein Equations, Maxwell Equations
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.15 No.9,
September
1,
2025
ABSTRACT: In the Special Relativity paper of Einstein (1905), only a footnote provides a reference to the conformal group of space-time for the Minkowski metric
ω
. We prove that General Relativity (1915) will depend on the following cornerstone result of differential homological algebra (1990). Let
K
be a differential field and
D=K[
d
1
,⋯,
d
n
]
be the ring of differential operators with coefficients in
K
. If
M
is the differential module over
D
defined by the Killing operator
D:T→
S
2
T
*
:ξ→Ω=ℒ(
ξ
)ω
and
N
is the differential module over
D
defined by the
Cauchy=ad(
Killing
)
adjoint operator with torsion submodule
t(
N
)
, then
t(
N
)≃ex
t
D
1
(
M
)=0
and the Cauchy operator can be thus parametrized by stress functions (Airy for
n=2
, Beltrami, Maxwell, Morera for
n=3
, Einstein for
n=4
) having strictly nothing to do with
Ω
. This result is largely superseding the Kalman controllability test in classical OD control theory and is showing that controllability is a structural “built-in” property of an OD/PD control system not only depending on the choice of inputs and outputs, contrary to the engineering tradition. Indeed, as illustrated by many examples, using any control system as a way to define the above differential operators and modules, the above result amounts to prove that the system is controllable if the adjoint operator is injective. In actual practice, we invite the reader to pick up in textbooks any example depending on some parameters, treat it by the Kalman test and make any exchange between inputs and outputs to check that the controllability conditions on the parameters are still unchanged! It also points out the terrible confusion done by Einstein (1915) while following Beltrami (1892), both of them using the Einstein operator but ignoring that it was self-adjoint in the framework of differential double duality (1995). Following Weyl, we finally prove that the structure of electromagnetism and gravitation only depends on the nonlinear elations of the conformal group of space-time, showing thus that nothing is left from the mathematical foundations of both general relativity and gauge theory. These results also question the origin and existence of gravitational waves and black holes, not because of a problem of detections but because of a problem of equations.