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.

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The present chapter contains the material taught within the module P2 of FAP 2004. The purpose of this intensive course is first to provide an introduction to "algebraic analysis". This fashionable though quite difficult domain of pure mathematics today has been pioneered by V.P. Palamodov, M. Kashiwara and B. Malgrange around 1970, after the work of D.C. Spencer on the formal theory of systems of partial differential equations. We shall then focus on its application to control theory in order to study linear control systems defined by partial differential equations with constant or variable coefficients, also called multidimensional control systems, by means of new methods from module theory and homological algebra. We shall revisit a few basic concepts and prove, in particular, that controllability, contrary to a well established engineering tradition or intuition, is an intrinsic structural property of a control system, not depending on the choice of inputs and outputs among the control variables or even on the presentation of the control system. Our exposition will be rather elementary as we shall insist on the main ideas and methods while illustrating them through explicit examples. Meanwhile, we want to stress out the fact that these new techniques bring striking results even on classical control systems of Kalman type !.

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The purpose of this paper is to present for the first time an elementary summary of a few recent results obtained through the application of the formal theory of partial differential equations and Lie pseudogroups in order to revisit the mathematical foundations of general relativity. Other engineering examples (control theory, elasticity theory, electromagnetism) will also be considered in order to illustrate the three fundamental results that we shall provide successively. 1) VESSIOT VERSUS CARTAN: The quadratic terms appearing in the "Riemann tensor" according to the "Vessiot structure equations" must not be identified with the quadratic terms appearing in the well known "Cartan structure equations" for Lie groups. In particular, "curvature + torsion" (Cartan) must not be considered as a generalization of "curvature alone" (Vessiot). 2) JANET VERSUS SPENCER: The "Ricci tensor" only depends on the nonlinear transformations (called "elations" by Cartan in 1922) that describe the "difference" existing between the Weyl group (10 parameters of the Poincaré subgroup + 1 dilatation) and the conformal group of space-time (15 parameters). It can be defined without using the indices leading to the standard contraction or trace of the Riemann tensor. Meanwhile, we shall obtain the number of components of the Riemann and Weyl tensors without any combinatoric argument on the exchange of indices. Accordingly and contrary to the "Janet sequence", the "Spencer sequence" for the conformal Killing system and its formal adjoint fully describe the Cosserat equations, Maxwell equations and Weyl equations but General Relativity is not coherent with this result. 3) ALGEBRA VERSUS GEOMETRY: Using the powerful methods of "Algebraic Analysis", that is a mixture of homological agebra and differential geometry, we shall prove that, contrary to other equations of physics (Cauchy equations, Cosserat equations, Maxwell equations), the Einstein equations cannot be "parametrized", that is the generic solution cannot be expressed by means of the derivatives of a certain number of arbitrary potential-like functions, solving therefore negatively a 1000 $ challenge proposed by J. Wheeler in 1970. Accordingly, the mathematical foundations of electromagnetism and gravitation must be revisited within this formal framework, though striking it may look like. We insist on the fact that the arguments presented are of a purely mathematical nature and are thus unavoidable.