Partial Differential Control Theory

Abstract

When D : ξ → η is a linear ordinary differential (OD) or partial differential (PD) operator, a "direct problem" is to find the generating compatibility conditions (CC) in the form of an operator D 1 : η → ζ such that Dξ = η implies D 1 η = 0. When D is involutive, the procedure provides successive first order involutive operators D 1 , ..., D n when the ground manifold has dimension n. Conversely, when D 1 is given, a much more difficult " inverse problem " is to look for an operator D : ξ → η having the generating CC D 1 η = 0. If this is possible, that is when the differential module defined by D 1 is " torsion-free ", that is when it does not exist any observable quantity which is a sum of derivatives of η that could be a solution of an autonomous OD or PD equation for itself, one shall say that the operator D 1 is parametrized by D. The parametrization is said to be "minimum " if the differential module defined by D does not contain any free differential submodule. The systematic use of the adjoint of a differential operator provides a constructive test with five steps. We prove and illustrate through many explicit examples the fact that a control system is controllable if and only if it can be parametrized. Accordingly, the controllability of any OD or PD control system is a " built in " property not depending on the choice of the input and output variables among the system variables. In the OD case and when D 1 is formally surjective, controllability just amounts to the injectivity of ad(D 1 ), even in the variable coefficients case, a result still not acknowledged by the control community. Among other applications, the parametrization of the Cauchy stress operator in arbitrary dimension n has attracted many famous scientists (G.B. Airy in 1863 for n = 2, J.C. Maxwell in 1863, G. Morera and E. Beltrami in 1892 for n = 3, A. Einstein in 1915 for n = 4). We prove that all these works are already explicitly using the self-adjoint Einstein operator, which cannot be parametrized. As a byproduct, they are all based on a confusion between the so-called div operator induced from the Bianchi operator D 2 and the Cauchy operator, adjoint of the Killing operator D which is parametrizing the Riemann operator D 1 for an arbitrary n. We prove that this purely mathematical result deeply questions the origin and existence of gravitational waves. Like the Michelson and Morley experiment, it is thus an open historical problem to know whether Einstein was aware of these previous works or not, as the comparison needs no comment.

A Constructive Study of the Module Structure of Rings of Partial Differential Operators

“…From a more general perspective, the constructive approach to algebraic analysis initiated in [1, 3,4,7,11,12,13,14,15,16,31,32,33,34] relies on the solvability of inhomogeneous linear systems in the ring under consideration. Within category theory, this idea was axiomatized in [2] and applied, for instance, to sheaf theory.…”

“…The corresponding system defines the infinitesimal transformations of the Lie pseudogroup formed by the contact transformations (see Example V.1.84 in [31]). We can check that ker D (R.) = Q D, where the matrix Q ∈ D 3 is defined by:…”

“…In mathematical physics, adjoint modules play an important role as shown in [31] (see also the references therein and [16]). …”

Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics"

“…We obtain the following theorem generalizing for PD control systems the well known first order Kalman form of OD control systems where the derivatives of the input do not appear ( [27], VI,1.14, p 802): When R q is involutive, the linear differential operator D :…”

“…This is particularly clear in classical control theory where the systems are classified into two categories, namely the "controllable" ones and the "uncontrollable" ones ( [14], [27]). In order to understand the problem studied by Macaulay in [M], that is roughly to determine the minimum number of solutions of a system that must be known in order to determine all the others by using derivatives and linear combinations with constant coefficients in a field k, let us start with the following motivating example:…”