Differential Correspondences and Control Theory

When a differential field $K$ having $n$ commuting derivations is given together with two finitely generated differential extensions $L$ and $M$ of $K$, an important problem in differential algebra is to exhibit a common differential extension $N$ in order to define the new differential extensions $L\cap M$ and the smallest differential field $(L,M)\subset N$ containing both $L$ and $M$. Such a result allows to generalize the use of complex numbers in classical algebra. Having now two finitely generated differential modules $L$ and $M$ over the non-commutative ring ring $D=K[d_1,... ,d_n]=K[d]$ of differential operators with coefficients in $K$, we may similarly look for a differential module $N$ containing both $L$ and $M$ in order to define $L\cap M$ and $L+M$. This is {\it exactly} the situation met in linear or non-linear OD or PD control theory by selecting the inputs and the outputs among the control variables. However, in many recent books and papers, we have shown that controllability was a {\it built-in} property of a control system, not depending on the choice of inputs and outputs. The main purpose of this paper is to revisit control theory by showing the specific importance of the two previous problems and the part plaid by $N$ in both cases for the parametrization of the control system. The essential tool will be the study of {\it differential correspondences}, a modern name for what was called {\it B\"{a}cklund problem} during the last century, namely the study of elimination theory for groups of variables among systems of linear or nonlinear OD or PD equations. The difficulty is to revisit {\it differential homological algebra} by using non-commutative localization. Finally, when $M$ is a $D$-module, this paper is using for the first time the fact that the system $R=hom_K(M,K)$ is a $D$-module for the Spencer operator acting on sections.


1) INTRODUCTION
The story started in 1970 at Princeton university when the author of tis paper was a visiting student of D.C. Spencer and his colleague J. Wheeler from the nearby physics department set up a 1000 $ challenge for proving that Einstein equations could be parametrized by potential-like functions like Maxwell equations. It is only in 1995 that he found the negative solution of this chalenge, only paid back one dollar (!) by Wheeler because the relativistic community was (and still is !) convinced about the existence of such a parametrization. Accordingly, such a result can only be found today in books of control theory ( [22], [25], [44]). Presenting this result at the algebra seminar of M.-P. Malliavin in the Institut Henri Poincaréof Paris the same year, he found by chance on display in the library the translation from Japanese of the 1970 Master thesis of M. Kashiwara and discovered the usefulness of differential homological algebra that Spencer never told him about during his stay in Princeton ( [10]).
Meanwhile, U. Oberst from Innsbruck university succeeded applying these new tools to control theory, in particular for studying controllability for multi-dimensional systems with constant coefficients ( [14], [15]). However, the reader may discover on the net (www.ricam.oeaw.ac.at/oberst) how difficult it is to communicate with people familiar with analysis but not with the formal methods (jet theory, differential sequences, diagram chasing) when studying systems of ordinary differential (OD) or partial differential (PD) equations. In the meantime, the author had become aware of the new methods (tensor products of rings and fields with derivations) used by A. Bialynicki-Birula in order to study "Differential Galois Theory" ( [2], [18]) that are largely superseding the approach of E. Kolchin in classical differential algebra ( [9], [11]).
A possibility to escape from such a situation was to publish as fast as possible a book presenting for the first time in a self-contained way the non-commutative aspect of double duality for the study of systems having coefficients in a differential field K ( [21], Zbl 1079.93001). Of course, the difficulty was to use commutative algebra for the graded modules in order to study the corresponding filtred modules, a hard task indeed. A more specific application of theses new tools to mathematical physics (general relativity, gauge theory) allowed the author to justify the many doubts he already had since a long time about the origin and existence of gravitational waves and black holes, but this is out of the scope of this paper (Comparing [1] to [32] and [35] needs no comment).
In the second section we shall study the linear framework and in the third section we shall study the nonlinear framework, separating in each situation the differential geometric approach from the differential algebraic approach and providing various motivating examples. Many of the results are given without proofs that can be found in the many books ( [17][18][19][20][21][22], [24], [29], [30]) and recent papers ( [24], [27], [28], [32], [33]) that we have published. It does not seem that that the link existing between the Spencer operator and non-commutative localization of Ore domains is known. In any case, this result has never been used for applications to control theory or even mathematical physics.

2.1) LINEAR SYSTEMS
If X is a manifold of dimension n with local coordinates (x) = (x 1 , ..., x n ), we denote as usual by T = T (X) the tangent bundle of X, by T * = T * (X) the cotangent bundle, by ∧ r T * the bundle of r-forms and by S q T * the bundle of q-symmetric tensors. More generally, let E, F, . . . be vector bundles over X with local coordinates (x i , y k ), (x i , z l )), . . . for i = 1, ..., n, k = 1, ..., m, l = 1, ..., p simply denoted by (x, y), (x, z), projection π : E → X : (x, y) → (x) and changes of coordinates x = ϕ(x),ȳ = A(x)y. We shall denote by E * the vector bundle obtained by inverting the matrix A of the changes of coordinates , exactly like T * is obtained from T . We denote by ξ : X → E : (x) → (x, y = ξ(x)) a (local) section of E, Under a change of coordinates, a section transforms likeξ(ϕ(x)) = A(x)ξ(x) and the changes of the derivatives can also be obtained with more work.
We shall denote by J q (E) the q-jet bundle of E with local coordinates (x i , y k , y k i , y k ij , ...) = (x, y q ) called jet coordinates and sections ξ q : (x) → (x, ξ k (x), ξ k i (x), ξ k ij (x), ...) = (x, ξ q (x)) transforming like the sections j q (ξ) : (x) → (x, ξ k (x), ∂ i ξ k (x), ∂ ij ξ k (x), ...) = (x, j q (ξ)(x)) where both ξ q and j q (ξ) are over the section ξ of E. For any q ≥ 0, J q (E) is a vector bundle over X with projection π q while J q+r (E) is a vector bundle over J q (E) with projection π q+r q , ∀r ≥ 0. DEFINITION 1.A.1: A linear system of order q on E is a vector sub-bundle R q ⊂ J q (E) and a solution of R q is a section ξ of E such that j q (ξ) is a section of R q .

DEFINITION 1.A.2:
A system R q is said to be formally integrable when all the equations of order q + r are obtained by r prolongations only, ∀r ≥ 0 or, equivalently, when the projections π q+r+s q+r : R q+r+s → R (s) q+r ⊆ R q+r are epimorphisms ∀r, s ≥ 0.
When R q is involutive, the linear differential operator D : Introducing the set of solutions Θ ⊆ E and the Janet bundles: we obtain the canonical linear Janet sequence (Introduced in [17], p 185 + p 391): where each other operator, induced by the Spencer operator, is first order involutive and generates the compatibility conditions (CC) of the preceding one. Similarly, introducing the Spencer bundles: we obtain the canonical linear Spencer sequence also induced by the Spencer operator: In the case of analytic systems, the following theorem providing the Cartan-Kähler CK) data is well known though its link with involution is rarely quoted because it is usually presented within the framework of exterior calculus ( [17], [26]): is a linear involutive and analytic system of order q on E, there exists one analytic solution y k = f k (x) and only one such that: The monomorphism 0 → J q+1 (E) → J 1 (J q (E)) allows to identify R q+1 with its imageR 1 in J 1 (R q ) and we just need to set R q =Ē in order to obtain the first order system (Spencer form) R 1 ⊂ J 1 (Ē) which is also involutive and analytic while π 1 0 :R 1 →Ē is an epimorphism. Studying the respective symbols, we may identify g q+r andḡ r whileḡ 1 is involutive. Looking at the Janet board of multiplicative variables we haveᾱ i 1 +β i 1 =m = dim(Ē) and: We obtain therefore: is a first order linear involutive and analytic system such that π 1 0 : R 1 → E is an epimorphism, then there exists one analytic solution y k = f k (x) and only one, such that: .., f m (x) are m − β n 1 given analytic functions of x 1 , ..., x n .

2.2) DIFFERENTIAL MODULES
If A is an associative ring with unit 1 ∈ A, a subset S ⊂ A is called a multiplicative subset if 1 ∈ S, 0 / ∈ S, st ∈ S, ∀s, t ∈ S.. In the commutative case, these conditions are sufficient to localize A at S by constructing the new ring of fractions S −1 A over A. For simplicity, we shall suppose that A is an integral domain (no divisor of zero) and we shall choose S = A − {0} in order to introduce the field of fractions Q(A) = S −1 A = AS −1 . The idea is to exhibit new quantities written a s with the standard rules: The same definition can be used for any module M over A in order to introduce the module of fractions S −1 M over S −1 A with the rules: In the non-commutative case considered through all this paper, we shall meet four problems: • How to compare s −1 a with as −1 ?.
• How to decide when we shall say that s −1 a = t −1 b ?.
• How to multiply s −1 a by t −1 b ?.
• How to find a common denominator for s −1 a + t −1 b ?. LEMMA 2.2.2: If there exists a left localization of a noetherian A with respect to S, then we must have Sa ∩ As = ∅. It follows that As ∩ At ∩ S = ∅ and two fractions can be multiplied or brought to the same denominator. Finally, t(M ) is a submodule of M .
Proof: Roughly, any right fraction as −1 can be written as a left fraction t −1 b, that is we must have ta = bs. Now, if we have two fractions s −1 a and t −1 b, we can find u, v ∈ A such that us = vt ∈ S. Hence, we obtain s −1 a = (us) −1 (ua) and t −1 b = (vt) −1 vb = (us) −1 vb. As for the multiplication of fractions, we have ( Finally, given x, y ∈ t(M ), we can find s, t ∈ S such that sx = 0, ty = 0. We may thus find u, v ∈ A such that us = vt ∈ S and we get us(x + y) = usx + vty = 0 ⇒ x + y ∈ t(M ). Also, we can use ta = bs in order to obtain t(ax) = (ta)x = (bs)x = b(sx) = 0 ⇒ ax ∈ t(M ). ✷ Let K be a differential field with n commuting derivations (∂ 1 , ..., ∂ n ) and consider the ring D = K[d 1 , ..., d n ] = K[d] of differential operators with coefficients in K with n commuting formal derivatives satisfying d i a = ad i + ∂ i a in the operator sense. If P = a µ d µ ∈ D = K[d], the highest value of |µ| with a µ = 0 is called the order of the operator P and the ring D with multiplication (P, Q) −→ P • Q = P Q is filtred by the order q of the operators. We have the with the vector field ξ = ξ i (x)∂ i of differential geometry, but with ξ i ∈ K now. It follows that D = D D D is a bimodule over itself, being at the same time a left D-module by the composition P −→ QP and a right D-module by the composition P −→ P Q. We define the adjoint functor ad : D −→ D op : P = a µ d µ −→ ad(P ) = (−1) |µ| d µ a µ and we have ad(ad(P )) = P both with ad(P Q) = ad(Q)ad(P ), ∀P, Q ∈ D. Such a definition can be extended to any matrix of operators by using the transposed matrix of adjoint operators (See [20]- [22], [23], [24], [31], [33], [36], [37] for more details and applications to control theory or mathematical physics). Proof: Let U ∈ S and P ∈ D be given. In order to prove the Ore property for D, we must find V ∈ S and Q ∈ D such that V P = QU . Considering the system P y = u, U y = v, it defines a differential module M over D with the finite presentation D 2 → D → M → 0. Now, as we have only one unknown and D = Dy in this sequence, then M is a torsion module and rk D (M ) = 0. From the additivity property of the differential ranks, if there should be no compatibilty condition (CC), see the example below), then the first morphism on the left should be a monomorphism, a result leading to the contradiction 2−1+0 = 0. Accordingly we can find the operators V and Q such that P U −1 = V −1 Q. Conversely, if now V and Q are given, using the adjoint functor and the fact that ad(ad(P )) = P, ∀P ∈ D, we may obtain ad(V ) and ad(Q) such that ad(P )ad(V ) = ad(U )ad(Q) as before and thus V = ad(ad(V )) and Q = ad(ad(Q)) such that V P = QU , a result showing that x 2 ) let us consider the two first order operators U = d 2 ∈ S, P = d 1 + x 2 . Considering the formal system d 1 y + x 2 y = u, d 2 y = v, we obtain y = d 2 u − d 1 v − x 2 v and thus the involutive system with jet notations: Among the three CC that should exist, only two are non-trivial and provide the new second order (care !) involutive sytem: We obain therefore the two operator identities: leading again to the two unexpected localizations: Taking the adjoint operators, we get in particular ( In order to achieve this example and explain why such methods, up to our knowledge, have never been used for applications, it just remains to explain the equality of these two fractions in this framework. Indeed, we obtain easily the unique operator identity ( Reducing to the same denominator can be done if we use the operator identity: produced by the same last CC d 2 B = (d 1 + x 2 )A for v. We conclude this example exhibiting the corresponding long exact sequence of differential modules: Accordingly, if y = (y 1 , ..., y m ) are differential indeterminates, then D acts on y k by setting d i y k = y k i −→ d µ y k = y k µ with d i y k µ = y k µ+1i and y k 0 = y k . We may therefore use the jet coordinates in a formal way as in the previous section. Therefore, if a system of OD/PD equations is written in the form Φ τ ≡ a τ µ k y k µ = 0 with coefficients a ∈ K, we may introduce the free left differential module Dy = Dy 1 + ... + Dy m ≃ D m and consider the differential module of equations I = DΦ ⊂ Dy, both with the residual left differential module M = Dy/DΦ or D-module and we may set M = D M ∈ mod(D) if we want to specify the action of the ring of differential operators. We may introduce the formal prolongation with respect to d i by setting d i Φ τ ≡ a τ µ k y k µ+1i +(∂ i a τ µ k )y k µ in order to induce maps d i : M −→ M :ȳ k µ −→ȳ k µ+1i by residue with respect to I if we use to denote the residue Dy −→ M : y k −→ȳ k by a bar like in algebraic geometry. However, for simplicity, we shall not write down the bar when the background will indicate clearly if we are in Dy or in M . As a byproduct, the differential modules we shall consider will always be finitely generated (k = 1, ..., m < ∞) and finitely presented (τ = 1, ..., p < ∞). Equivalently, introducing the matrix of operators D = (a τ µ k d µ ) with m columns and p rows, we may introduce the morphism D p D −→ D m : (P τ ) −→ (P τ Φ τ ) over D by acting with D on the left of these row vectors while acting with D on the right of these row vectors by composition of operators with im(D) = I. The presentation of M is defined by the exact cokernel sequence D p D −→ D m p −→ M −→ 0. We notice that the presentation only depends on K, D and Φ or D, that is to say never refers to the concept of (explicit local or formal) solutions. It follows from its definition that M can be endowed with a quotient filtration obtained from that of D m which is defined by the order of the jet coordinates y q in D q y. We have therefore the inductive limit 0  [10], [21], [36], [38], [40]).
Having in mind that K is a left D-module with the action (D, K) → K : (P, a) → P (a) defined by (b, a) → ba = ab, (d i , a) → d i (a) = ∂ i a and that D is a bimodule over itself for the composition law of operators, we have only two possible constructions only depending on K, D and M : DEFINITION 2.2.6: We may define the right (care !) differential module hom D (M, D), using the bimodule structure of D D D and setting (f P )(m) = f (m)P while checking that: We are immediately facing one of the most delicate problems of this section when dealing with applications and/or effective computations, a problem not solved in the corresponding literature which has been almost entirely using fields of constants ( [14], [15], [41], [42]) though the solution is known since a long time ( [3], [20], [38]). Indeed, apart for purely mathematical reasons, the only differential modules to be met are left differential modules. By chance, one has the following theorem describing the functorial side changing procedure amounting to replace D by is formal adjoint ad(D). In the differential geometric framework, such a procedure amounts to replace an operator D : E → F by its formal adjoint ad(D) : ∧ n T * ⊗ F * → ∧ n T * ⊗ E * and one may set ad(E) = ∧ n T * ⊗ E * for any vector bundle E over X with dim(E) = dim(ad(E)) while reversing the arrows. However, as the formal adjoint of an involutive operator may not even be formally integrable, the formal adjoint of an exact (Janet, Spencer) sequence may not be an exact (Janet, Spencer) sequence at all and this is the motivation for introducing the extension modules. The simplest example when n = 1 can be found in the study of the double pendulum ( [36]). In the timevarying case with x = t and d = d t , the Kalman-type system dy = A(x)y + B(x)u is controllable if and only if the operator λ → (dλ + λA, λB) is injective, that is to say if and only if the matrix (B, (d−A)B, (d−A) 2 B, ...) has maximum rank ( Compare to [45] where the adjoint is missing).
Proof: First of all, we prove that ∧ n T * has a natural right module structure over D by introducing the basic volume n-form α = dx 1 ∧ · · · ∧ dx n = dx and defining α.P = ad(P )(1)dx, ∀P ∈ D. We have α.ξ = −∂ i ξ i dx = −div(ξ)dx = −L(ξ)α, ∀ξ = ξ i d i ∈ T where L is the classical Lie derivative on forms and obtain therefore: and check that (m ⊗ α)(ξa) = (m ⊗ α)(aξ + ξ(a)). These definition are coherent because, when d is any d i , we have div(d) = 0 and thus (m ⊗ α)d = −dm ⊗ α, a result leading to the formula: The isomorphism ad(D) ≃ D D : P → ad(P ) is also right D-linear because we have successively: These unexpected results explain why the formal adjoint cannot be avoided in non-commutative localization and is so important for applications ranging from control theory ( [21], [22], [27]) to continuum mechanics ( [5], [23]) or electromagnetism ( [31]) and even general relativity ( [33], [37]). ✷ DEFINITION 2.2.9: We define the system R = hom K (M, K) and set R q = hom K (M q , K) as the system of order q. We have the projective limit It follows that f q ∈ R q : y k µ −→ f k µ ∈ K with a τ µ k f k µ = 0 defines a section at order q and we may set f ∞ = f ∈ R for a section of R. For an arbitrary differential field K, such a definition has nothing to do with the concept of a formal power series solution (care).
Similarly to the preceding definition, we may define the left (care !) differential module hom K (D, K), using again the bimodule structure of D and setting (Qf )(P ) = f (P Q), ∀P, Q ∈ D, in particular with (ξf )(P )f (P ξ), ∀ξ ∈ T, ∀P ∈ D. However, we should have (af )(P ) = f (P a) = f (aP ) = a(f (P )), ∀a ∈, unless K is a field of constants like in most of the literature ( [14], [15], [42]). PROPOSITION 2.2.10: When M is a left D-module, then R is also a left D-module.
Proof: As D is generated by K and T as we already said, let us define: In the operator sense, it is easy to check that d i a = ad i + ∂ i a and that ξη − ηξ = [ξ, η] is the standard bracket of vector fields. Using simply ∂ in place of any ∂ i and d in place of any d i , we have: and thus recover exactly the Spencer operator of the previous section though this is not evident at all. We also get ( This operator has been first introduced, up to sign, by F.S. Macaulay as early as in 1916 but this is still not acknowledged ( [12]). For more details on the Spencer operator and its applications, the reader may look at ( [19], [24], [30], [29]). ✷ PROPOSITION 2.2.11: When M and N are left D-modules, then M ⊗ K N is also a left Dmodule.
Proof: As before, we may define: ∀ξ ∈ T, ∀m ∈ M, ∀n ∈ N and let the reader finish as an exercise. ✷ COROLLARY 2.2.12: The two structures of left D-modules obtained in these two propositions are coherent with the following adjoint isomorphism existing for any triple L, M, N ∈ mod(D): and we have successively for any ξ ∈ T : that is (ξ(g(m)))(n) = f (ξm ⊗ n) = (g(ξm))(n) and thus ξ(g(m)) = g(ξm), ∀m ∈ M . The inverse morphism can be studied similarly. ✷ Proof: As K is a field, thus a commutative ring, we have the isomorphism of left D-modules ∀n ∈ N and we may exchange M and N . As K is a left differential module for the rule (P, a) → P (a), ∀P ∈ D, ∀a ∈ K, we obtain: ✷ COROLLARY 22.14: The differential module hom K (D, K) is an injective differential module.
Proof: When 0 → M ′ → M → M " → 0 is a short exact sequence of modules and N is any module, we have only the exact sequence 0 → hom(M ", N ) → hom(M, N ) → hom(M ′ , N ) obtained by composition of morphisms. In the present situation, using the previous corollaries, we have the following commutative and exact diagram because K is a field (See [4], p 18): Chasing in this diagram, we deduce that the upper morphism is an epimorphism and hom K (D, K) is an injective module because hom(•, hom K (D, K)) transforms a short exact sequence into a short exact sequence. The reader may compare such an approach with the one used in ( [41] or [42]) in order to understand why these applications are not dealing with variable coefficients as the differential structure on R must be defined by the Spencer operator but we do not know any other reference (compare to [14]). ✷ DEFINITION 2.2.15: With any differential module M we shall associate the graded module G = gr(M ) over the polynomial ring gr(D) ≃ K[χ] by setting G = ⊕ ∞ q=0 G q with G q = M q /M q+1 and we get g q = G * q where the symbol g q is defined by the short exact sequences: We have the short exact sequences 0 −→ D q−1 −→ D q −→ S q T −→ 0 leading to gr q (D) ≃ S q T and we may set as usual T * = hom K (T, K) in a coherent way with differential geometry.
The two following definitions, which are well known in commutative algebra, are also valid (with more work) in the case of differential modules (See [21] for more details or the references ( [13], [22], [39], [40]) for an introduction to homological algebra and diagram chasing). DEFINITION 2.2.17: A differential module F is said to be free if F ≃ D r for some integer r > 0 and we shall define rk D (F ) = r. If F is the biggest free differential module contained in M , then M/F is a torsion differential module and hom D (M/F, D) = 0. In that case, we shall define the differential rank of M to be rk D (M ) = rk D (F ) = r. Accordingly, if M is defined by a linear involutive operator of order q, then rk D (M ) = α n q .

REMARK 2.2.19:
We emphasize once more that the left D-module hom K (D, K) used in the literature ([12, [14]°541], [42], [45]) is coming from the right action of D on any f ∈ hom K (D, K) through the formula (Qf )(P ) = f (P Q) and we have thus: , they are not coherent at all with the formulas of Proposition 2.2.1, namely: unless K is a field of constants, in particular because, when M = D, then P a = aP in general ( [30], p 66 for more details). Accordingly, most of the applications of differential duality to control theory must be therefore revisited with these new methods of differential homological algebra (Compare to [45]). We also claim that the use of the adjoint operator must become essential for all the applications to mathematical physics. . In order to conclude this section, we may say that the main difficulty met when passing from the differential framework to the algebraic framework is the " inversion " of arrows. Indeed, with dim(E) = m, dim(F ) = p, when an operator D : E → F is injective, that is when we have the , on the contrary, using differenial modules, we have the epimorphism D p D −→ D m → 0. The case of a formally surjective operator, like the div operator, described by the exact sequence E We are now ready for using the results of the second section on the Cartan-Kähler theorem. For such a purpose, separating the parametric jets that can be chosen arbitrarily from the principal jets that can be obtained by using the fact that the given OD or PD equations have coefficients in a differential field K, we may write the solved equations in the symbolic form y pri − c par pri y par = 0 with c ∈ K and an implicit (finite) summation in order to obtain for the sections f pri − c par pri f par = 0. Using the language of Macaulay, it follows that the so-called modular equations are E ≡ f pri a pri + f par a par = 0 with eventually an infinite number of terms in the implicit summations. Substituting, we get at once E ≡ f par (a par + c par pri a pri ) = 0. Ordering the y par as we already did and using a basis {(1, 0, ...), (0, 1, 0, ...), (0, 0, 1, 0, ...), ...} for the f par , we may select the parametric modular equations E par ≡ a par + c par pri a pri = 0.
When k is a field of constants, a polynomial P = a µ χ µ ∈ k[χ] of degree q is multiplied by a monomial χ ν with | ν |= r, we get χ ν P = a µ χ µ+ν . Hence, if 0 ≤| µ |≤ q, the "shifted " polynomial thus obtained is such that r ≤| µ + ν |≤ q + r and the difference between the maximum degree and the minimum degree of the monomials involved is always equal to q and thus fixed. When n = 1, one can exhibit a series only made with 0 or 1 like f = (1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, ...) with " zero zones " of successive increasing lengths 1, 2, 3, 4, ... and so on, separated by 1 in such a way that the contraction with the shifted polynomial is the leading term of the given polynomial and extend this procedure to n arbitrary.
Replacing χ i by d i and degree by order, we may use the results of section 2 in order to split the CK-data into m formal power series of 0 (constants), 1, ..., n variables that we shall call series of type i for i = 0, 1, ..., n. However, as the following elementary example will show, the shifting procedure cannot be applied to the variable coefficient case, namely when K is used in place of k. [14]).

WE SHALL ESCAPE FROM THIS DIFFICULTY BY MEANS OF A TRICK BASED ON A SYSTEMATIC USE OF THE SPENCER OPERATOR (Compre to
The idea will be to shift the series to the left (decreasing ordering), up to sign, instead of shifting the operator to the right (increasing ordering). For this, we notice that we want that the contraction of , that is the same series can be used but in a quite different framework. Also, in the finite dimensional case existing when the symbol g q of R q is finite type, that is when g q+r = 0 for a certain integer r ≥ 0, applying the δ-sequence inductively to g q+n+i for i = r−1, ..., 0 as in ( [17], Proposition 4.7, p 123), it is known that g q is finite type and involutive if and only if g q = 0, that is to say dim(R q ) = dim(R q−1 ). In a coherent way, we have thus obtained:ÊÊ THEOREM 2.2.20: If M is a differential module over D = K[d] defined by a first order involutive system in the m unknowns y 1 , ..., y m with no zero order equation, the differential module R = hom K (M, K) may be generated over D by a finite basis of sections containing m generators.
In the general situation, counting the number of CK data, we have α 1 q + ... + α n q = dim(g q ) and dim(R q ) = dim(g q ) + dim(R q−1 ). We obtain therefore the following result which is coherent with the number of unknowns in the Spencer form R q+1 ⊂ J 1 (R q ).
defined by an involutive system R q ⊂ J q (E), the differential module R = hom K (M, K) may be generated over D by a finite basis of sections containing dim(R q ) generators.
It is easy to check that g 2 with dim(g 2 ) = 3 is not involutive, that g 3 with dim(g 3 ) = 1 is 2-acyclic because y 123 − y 111 = 0 and that g 4 = 0 is trivially involutive. Accordingly, only the system R 4 is involutive with the 8 = 2 3 parametric jets (y, y 1 , y 2 , y 3 , y 11 , y 12 , y 13 , y 111 ) and all the three characters vanish both with all the jets of order ≥ 4 because the system is homogeneous. Let us consider the following 8 sections with all other components equal to zero: section y y 1 y 2 y 3 y 11 y 12 y 13 y 23 y 111 y 123 Taking into account the two PD equations y 23 −y 11 = 0 and y 123 −y 111 = 0, we obtain successively: Or, equivalently, working with the corresponding modular equations: and so on. It follows that all the sections can be generated by the single section f 8 and all the modular equations can be generated by the single modular equation E 8 = 0, a result absolutely not evidet at first sight but coherent with the fact that the radical of the annihilator of M is the ) an even more striking example has been provided by M. Janet in 1920 ( [6]) with the following second order system: In this case, dim(R) = 12 < ∞ but R can be generated by the single modular equation: E ≡ a 12333 + x 2 a 1333 + a 1113 = 0 because all the jets of order > 5 vanish (See [20] and [22] for more details). With now m = 2, let us consider the differential module defined by the system y 1 xx − y 1 = 0, y 2 x = 0. Setting y = y 1 − y 2 , we successively get: x , y xx = y 1 xx = y 1 , y xxx = y 1 x ⇒ y 1 = y xx , y 1 x = y x , y 1 xx = y xx , y 1 xxx = y x , ... ⇒ y 2 = y xx − y, y 2 x = 0, ... and a differential isomorphism with the module defined by the new system y xxx − y x = 0. We have seen that the sections of the second system are easily seen to be generated by the single section and so on. We obtain the corresponding board describing the maps ρ r (Φ) (Compare to [M]):Ê order y y x y xx y xxx y xxxx y xxxxx y xxxxxx ...
Let us define the sections f ′ and f " by the following board where d = d x : section y y x y xx y xxx y xxxx y xxxxx y xxxxxx ...
in order to obtain df ′ = −xf ", df " = −f ′ . Though this is not evident at first sight, the two boards are orthogonal over K in the sense that each row of one board contracts to zero with each row of the other though only the rows of the first board do contain a finite number of nonzero elements. It is absolutely essential to notice that the sections f ′ and f " have nothing to do with solutions because df ′ = 0, df " = 0 on one side and also because d 2 f ′ − xf ′ = −f " = 1 x df ′ = 0 even though d 2 f " − xf " = 0 on the other side. As a byproduct, f ′ or f " can be chosen separately as unique generating section of the inverse system over K (care) and we may write for example: , let us consider again the second order system y 1 xx − y 1 = 0, y 2 x = 0. Setting z 1 = y 1 , z 2 = y 1 x , z 3 = y 2 , we obtain the first order involutive system: It follows that the CK data for z = g(x) are {g 1 = g 1 (0), g 2 = g 2 (0), g 3 = g 3 (0)}. Using the given equations and their solved prolongations like y 1 xx − y 1 = 0, y 2 xx − y 2 = 0 and so on, we have the finite basis (care !): As dg 1 = −g 2 , dg 2 = −g 1 , dg 3 = 0, a basis with only two generators may be {g 2 , g 3 }. However: and we obtain the unique generator h (See [26] for details).

2.3) LINEAR CONTROL THEORY
The most striking aspect of the application of module/system theory to linear control theory is that it is coming from rather unexpected chases in commutative and exact diagrams looking like rather abstract at first sight. As more details and examples can be found in book form ( [21], [22]), we shall only provide below a few new results that cannot be found elsewhere. Proof: Using elementary classical homological algebra, one obtains the following commutative and exact diagram: At first, the lower southeast arrow being the composition of two epimorphisms is an epimorphism.
We have the following commutative diagram of injections: Let us start with L ′ , construct N ′ = L ′ + M and obtain L" = L ∩ N ′ . First of all, we get L ′ ⊂ L and L ′ ⊂ N ′ ⇒ L ′ ⊆ L". Now, using the left commutative square, we obtain from the previous proposition N ′ /L ′ ≃ M/(L ∩ M ). Similarly, using the right commutative square, we obtain N ′ /N " ≃ N/L and thus an isomorphism N ′ /L ′ ≃ N ′ /L". However, we have the following commutative and exact diagram: and thus L"/L ′ = 0 that is L ′ = L". Finally, starting with N ′ , we should obtain L ′ = N ′ ∩ L, then define N " = L ′ + M ⊆ N ′ and conclude as before that N ′ = N ". Replacing specialzations by injections while chasing in the following commutative and exact diagram: we have thus been able to deal only with submodules of N . In paricular, if N is torsion-free, that is t(N ) = 0, the interest of this aproach is that all the submodules are torsion-free.
✷ EXAMPLE 2.3.4: (See [21], p 736-738] for the details and diagrams) With n = 1, K = Q, let us consider the differential module defined by the OD equation We may define the input differential module M in by using u and the output differential module by using y = d 2 v. The differential module (M in + M out ) ⊂ M with a strict inclusion, is defined by the OD equation d 3 y + 3d 2 y − 4y = d 3 u − du. Localizing at the covector χ, we get the equation K is a field of constants. As we can factor by (d − 1) it follows that t(M in + M out ) is generated by as an intersection of prime differential ideals. We have proved in ( [21]) how to use these differential submodules of M both with the new differential modules M ′ in = t(M )+ M in and M ′ out = t(M )+ M out in order to study all the problems concerning poles and zeros. In the present paper, as we are only interested by controllability, we have just to study the differential submodules of the torsion-free differential module M/t(M ).

3.1) NONLINEAR SYSTEMS
If X is a manifold with local coordinates (x i ) for i = 1, ..., n = dim(X), let us consider the fibered manifold E over X with dim X (E) = m, that is a manifold with local coordinates (x i , y k ) for i = 1, ..., n and k = 1, ..., m simply denoted by (x, y), projection π : E → X : (x, y) → (x) and changes of local coordinatesx = ϕ(x),ȳ = ψ(x, y). If E and F are two fibered manifolds over X with respective local coordinates (x, y) and (x, z), we denote by E× X F the fibered product of E and F over X as the new fibered manifold over X with local coordinates (x, y, z). We denote by f : X → E : (x) → (x, y = f (x)) a global section of E, that is a map such that π • f = id X but local sections over an open set U ⊂ X may also be considered when needed. Under a change of coordinates, a section transforms likef (ϕ(x)) = ψ(x, f (x)) and, differentiating with respect to x i , we may introduce new coordinates (x i , y k , y k i ) transforming like: We shall denote by J q (E) the q-jet bundle of E with local coordinates (x i , y k , y k i , y k ij , ...) = (x, y q ) called jet coordinates and sections f q : ) where both f q and j q (f ) are over the section f of E. It will be useful to introduce a multi-index µ = (µ 1 , ..., µ n ) with length | µ |= µ 1 + ... + µ n and to set µ + 1 i = (µ 1 ..., µ i−1 , µ i + 1, µ i+1 , ..., µ n ). Also, a jet coordinate y k µ is said to be of class i if µ 1 = ... = µ i−1 = 0, µ i = 0. As the background will always be clear enough, we shall use the same notation for a vector bundle or a fibered manifold and their sets of sections [31,36]. We finally notice that J q (E) is a fibered manifold over X with projection π q while J q+r (E) is a fibered manifold over J q (E) with projection π q+r q , ∀r ≥ 0 [, , ]. DEFINITION 3.1.1: A (nonlinear) system of order q on E is a fibered submanifold R q ⊂ J q (E) and a global or local solution of R q is a section f of E over X or U ⊂ X such that j q (f ) is a section of R q over X or U ⊂ X. DEFINITION 3.1.2: When the changes of coordinates have the linear formx = ϕ(x),ȳ = A(x)y, we say that E is a vector bundle over X. Vector bundles will be denoted by capital letters E, F, . . . and will have sections denoted by ξ, η, . . . . In particular, we shall denote as usual by T = T (X) the tangent bundle of X, by T * = T * (X) the cotangent bundle, by ∧ r T * the bundle of r-forms and by S q T * the bundle of q-symmetric covariant tensors. When the changes of coordinates have the formx = ϕ(x),ȳ = A(x)y + B(x) we say that E is an affine bundle over X and we define the associated vector bundle E over X by the local coordinates (x, v) changing likex = ϕ(x),v = A(x)v.
we may introduce the vertical bundle V (E) ⊂ T (E) as a vector bundle over E with local coordinates (x, y, v) obtained by setting u = 0 and changes v l = ∂ψ l ∂y k (x, y)v k . Of course, when E is an affine bundle over X with associated vector bundle E over X, we have V (E) = E × X E. We have the short exact sequence of vector bundles over E: Accordingly, in variational calculus, the couple (f, δf ) made by a section f of E and its variation δf is nothing else but a section of V (E) while δf has no reason at all to be " small ".
For a later use, if E is a fibered manifold over X and f is a section of E, we denote by f −1 (V (E)) the reciprocal image of V (E) by f as the vector bundle over X obtained when replacing (x, y, v) by (x, f (x), v) in each chart, along with the following commutative diagram: A similar construction may also be done for any affine bundle over E. When the background is clear enough, with a slight abuse of language, we shall sometimes set E = V (E) as a vector bundle over E and call " vertical machinery " such a useful systematic notation. Looking at the transition rules of J q (E), we deduce easily the following results: PROPOSITION 3.1.4: J q (E) is an affine bundle over J q−1 (E) modeled on S q T * ⊗ E E but we shall not specify the tensor product in general.
PROPOSITION 3.1.5: There is a canonical isomorphism V (J q (E)) ≃ J q (V (E)) = J q (E) of vector bundles over J q (E) given by setting v k µ = v k ,µ at any order and a short exact sequence: of vector bundles over J q (E) allowing to establish a link with the formal theory of linear systems.
PROPOSITION 3.1.6: There is an exact sequence: is called the first prolongation of R q and we may define the subsets R q+r . In actual practice, if the system is defined by PDE Φ τ (x, y q ) = 0 the first prolongation is defined by adding the PDE d i Φ τ ≡ ∂ i Φ τ + y k µ+1i ∂Φ τ /∂y k µ = 0. accordingly, f q ∈ R q ⇔ Φ τ (x, f q (x)) = 0 and f q+1 ∈ R q+1 ⇔ ∂ i Φ τ + f k µ+1i (x)∂Φ τ /∂y k µ = 0 as identities on X or at least over an open subset U ⊂ X. Differentiating the first relation with respect to x i and substracting the second, we finally obtain: and the Spencer operator restricts to d : R q+1 → T * ⊗ R q . We set R (1) q+r = π q+r+1 q+r (R q+r+1 ).
DEFINITION 3.1.8: The symbol of R q is the family g q = R q ∩ S q T * ⊗ E of vector spaces over R q . The symbol g q+r of R q+r only depends on g q by a direct prolongation procedure. We may define the vector bundle F 0 over R q by the short exact sequence 0 → R q → J q (E) → F 0 → 0 and we have the exact induced sequence 0 → g q → S q T * ⊗ E → F 0 .
Setting a τ µ k (x, y q ) = ∂Φ τ /∂y k µ (x, y q ) whenever | µ |= q and (x, y q ) ∈ R q , we obtain: In general, neither g q nor g q+r are vector bundles over R q .
On ∧ s T * we may introduce the usual bases {dx I = dx i1 ∧ ... ∧ dx is } where we have set I = (i 1 < ... < i s ). In a purely algebraic setting, one has: PROPOSITION 3.1.9: There exists a map δ : Proof: Let us introduce the family of s-forms ω = {ω k µ = v k µ,I dx I } and set (δω) k µ = dx i ∧ ω k µ+1i . We obtain at once (δ 2 ω) k µ = dx i ∧ dx j ∧ ω k µ+1i+1j = 0 and a τ µ k (δω) k µ = dx i ∧ (a τ µ k ω k µ+1i ) = 0. ✷ The kernel of each δ in the first case is equal to the image of the preceding δ but this may no longer be true in the restricted case and we set: DEFINITION 3.1.10: Let B s q+r (g q ) ⊆ Z s q+r (g q ) and H s q+r (g q ) = Z s q+r (g q )/B s q+r (g q ) with H 1 (g q ) = H 1 q (g q ) be the coboundary space im(δ), cocycle space ker(δ) and cohomology space at ∧ s T * ⊗ g q+r of the restricted δ-sequence which only depend on g q and may not be vector bundles. The symbol g q is said to be s-acyclic if H 1 q+r = ... = H s q+r = 0, ∀r ≥ 0, involutive if it is n-acyclic and finite type if g q+r = 0 becomes trivially involutive for r large enough. In particular, if g q is involutive and finite type, then g q = 0. Finally, We have (See [17] for the diagram allowing to prove this delicate result first found by Spencer): PROPOSITION 3.1.11: If g q is 2-acyclic and g q+1 is a vector bundle over R q , then g q+r is a vector bundle over R q , ∀r ≥ 1.
LEMMA 3.1.12: If g q is involutive and g q+1 is a vector bundle over R q , then g q is also a vector bundle over R q . In this case, changing linearly the local coordinates if necessary, we may look at the maximum number β of equations that can be solved with respect to v k n...n and the intrinsic number α = m − β indicates the number of y that can be given arbitrarily.
We notice that R q+r+1 = ρ r (R q+1 ) and R q+r = ρ r (R q ) in the following commutative diagram: We finally obtain the following crucial Theorem and its Corollary (Compare to [17], p 70-75): THEOREM 3.1.13: Let R q ⊂ J q (E) be a system of order q on E such that R q+1 is a fibered submanifold of J q+1 (E). If g q is 2-acyclic and g q+1 is a vector bundle over R q , then we have R DEFINITION 3.1.14: A system R q ⊂ J q (E) is said to be formally integrable at the order q + r if π q+r+s q+r : R q+r+s → R q+r is an epimorphism of fibered manifolds for all s ≥ 1, formally integrable if π q+r+1 q+r is an epimorphism of fibered manifolds ∀r ≥ 0 and involutive if it is formally integrable with an involutive symbol g q . We have the following useful test ( [17], [43]): COROLLARY 3.1.15: Let R q ⊂ J q (E) be a system of order q on E such that R q+1 is a fibered submanifold of J q+1 (E). If g q is 2-acyclic (involutive) and if the map π q+1 q : R q+1 → R q is an epimorphism of fibered manifolds, then R q is formally integrable (involutive).
This is all what is needed in order to study nonlinear systems of ordinary differential (OD) or partial differential (PD) equations, using calligraphic letters like E for the nonlinear framework and capital letters like E = V (E) for the linear or vertical linearized framework. • The derivative of a polynomial with respect to any one of the indeterminates is a polynomial while the derivative of a rational function is a rational function, a reason sufficient for believing that the concept of derivation could be useful in algebra.

3.2) DIFFERENTIAL ALGEBRA
• The variational and linearization procedures presented in the last section and used for many applications to physics should be extended to differential algebra in order to obtain the algebraic counterpart of definition 3.1.3 and proposition 3.1.5, replacing X by a ring A or a field K.
The element da ∈ Ω A/k is called the differential of a and der k (A, M ) = hom A (Ω A/k , M ).
Proof: Let F be the free A-module made by the symbols da, a ∈ A and let N be the submodule of F generated by dα, d(a + b) − da − db, d(ab) − adb − bda for α ∈ k, a, b ∈ A. We set Ω A/k = F/N and the derivation d = d A/k : A → Ω A/k : a → da is the universal derivation allowing to define f : Another way is to take into account the limit procedure that is classically used in analysis, namely when h → 0 in order to avoid the square quatity h 2 and so on. For this, let us denote by I the kernel of the map A⊗ k A → A : a ⊗ b → ab and define Ω A/k = I/I 2 while setting da = 1 ⊗ a − a ⊗ 1, ∀a ∈ A. Using the bimodule structure of A A A while identifying a ∈ A with a ⊗ 1 ∈ A⊗ k A, it follows that d is indeed a derivation from A to the A-module Ω A/k as we have successively: and thus: where the first map is a monomorphism when f is a monomorpism. In particular, if k ⊂ K ⊂ L is a chain of field extensions, then one has the short exact sequence of vector spaces over L: A similar comment applies to y 3 and it is easy to see that the kernel of the map a/a 2 → B⊗ A Ω A/k is of the form ax 3 + by 3 , ∀a, b ∈ A.
Finally, if S is a multiplicatively closed subset of A, we may use the morphism θ S : A → S −1 A in Proposition 2.2.3, we shall study the behaviour of derivations and differentials under localization.
As δ ∈ der k (A, M ) induces a unique derivation δ ∈ der k (S −1 A, S −1 M through the known formula δ(a/s) : sδa − aδs)/s 2 , it follows that the morphism der k (S −1 A, S −1 M ) → der k (A, S −1 M ) given by δ → δ • θ S is an epimorphism. We obtain the short exact sequence: and thus the short exact sequence: Taking into account the previous standard formula, it follows that Ω S −1 A/A = 0 and we obtain:

EXAMPLE 3.2.7:
We now present in an independent manner a few OD or PD cases showing the difficulties met when studying differential ideals and ask the reader to revisit them later on while reading the main Theorems. As only a few results will be proved, the interested reader may look at [18] or [20] for more details and compare to [9] or [11].
• OD 1: If k = Q, y is a differential indeterminate and d x is a formal derivation, we may set d x y = y x , d x y x = y xx and so on in order to introduce the differential ring A = k[y, y x , y xx , ...] = k{y}. We consider the differential ideal a ⊂ A generated by the differential polynomial P = y 2 x − 4y. We have d x P = 2y x (y xx − 2) and a cannot be a prime differential ideal, . . . and so on. After no less than 4 differentiations, we let the reader discover that y 5 xxx ∈ a ⇒ y xxx ∈ rad(a) and thus a is neither prime nor perfect, that is equal to its radical, but rad(a) is perfect as it is the intersection of the prime differential ideal generated by y with the prime differential ideal generated by y 2 x − 4y and y xx − 2, both containing y xxx .
• OD 2: With the same notations, let us consider the differential ideal a ⊂ A generated by the differential polynomial P = y 2 x − 4y 3 . We have d x P = 2y x (y xx − 6y 2 ) and a cannot be prime differential ideal. Hence, we must have either y x = 0 ⇒ y = 0 or y xx − 6y 2 = 0 and so on. After 3 differentiations we obtain (y xx − 6y 2 ) 4 ∈ a ⇒ y xx − 6y 2 ∈ rad(a) and thus a is neither prime nor perfect as before but rad(a) is the prime differential ideal generated by y 2 x − 4y 3 and y xx − 6y 2 .
• P D 1: If k = Q as before, y is a differential indeterminate and (d 1 , d 2 ) are two formal derivations, let us consider the differential ideal generated by P 1 = y 22 − 1 2 (y 11 ) 2 and P 2 = y 12 − y 11 in k{y}. Using crossed derivatives and differentiating twice, we get (y 111 ) 3 ∈ a ⇒ y 111 ∈ rad(a) and thus a is again neither prime nor perfect but rad(a) is a perfect differential ideal and even a prime differential ideal p because we obtain easily from the last subsection that the resisual differential ring k{y}/p ≃ k[y, y 1 , y 2 , y 11 ] is a differential integral domain. Its quotient field is thus the differential field K = Q(k{y}/p) ≃ k(y, y 1 , y 2 , y 11 ) with the rules: d 1 y = y 1 , d 1 y 1 = y 11 , d 1 y 11 = 0, d 2 y = y 2 , d 2 y 1 = y 11 , d 2 y 11 = 0 as a way to avoid " looking for solutions ". The formal linearization is the linear system R 2 ⊂ J 2 (E) obtained in the last section where it was defined over R 2 , but not over K, by the two linear second order PDE: η 22 − y 11 η 11 = 0, η 12 − η 11 = 0 ⇒ (y 11 − 1)η 111 = 0 changing slightly the notations with η = δy and keeping the letter v only when looking at the symbols. It is at this point that the problem starts because R 2 is indeed a fibered manifold with arbitrary parametric jets (y, y 1 , y 2 , y 11 ) but R 3 = ρ 1 (R 2 ) is no longer a fibered manifold because the dimension of its symbol changes when y 11 = 1. We understand therefore that there should be a close link existing between formal integrability and the search for prime differential ideals or differential fields. The solution of this problem has been provided as early as in 1983 for studying the "Differential Galois Theory " in ( [18]). The idea is to add the third order PDE y 111 = 0 and thus consider the linearized PDE η 111 = 0 obtaining therefore a third order involutive system well defined over K with symbol g 3 = 0.
DEFINITION 3.2.8: A differential ring is a ring A with a finite number of commuting derivations (∂ 1 , ..., ∂ n ) that can be extended to derivations of the ring of quotients Q(A) as we already saw. We shall suppose from now on that A is even an integral domain and introduce the differential field K = Q(A). For example, if x 1 , ..., x n are indeterminates over Q, then Q[x] = Q[x 1 , ..., x n ] is a differential ring with quotient differential field Q(x).
If K is a differential field as above and (y 1 , ..., y m ) are indeterminates over K, we transform the polynomial ring K{y} = lim q→∞ K[y q ] into a differential ring by introducing as usual the formal derivations d i = ∂ i + y k µ+1i ∂/∂y k µ and we shall set K < y >= Q(K{y}).

DEFINITION 3.2.9:
We say that a ⊂ K{y} is a differential ideal if it is stable by the d i , that is if d i a ∈ a, ∀a ∈ a, ∀i = 1, ..., n. We shall also introduce the radical rad(a) = {a ∈ A | ∃r, a r ∈ a} ⊇ a and say that a is a perfect (or radical) differential ideal if rad(a) = a. If S is any subset of A, we shall denote by {S} the differential ideal generated by S and introduce the (non-differential) ideal ρ r (S) = {d ν a | a ∈ S, 0 ≤| ν |≤ r} in A.
LEMMA 3.2.10: If a ⊂ A is differential ideal, then rad(a) is a differential ideal containing a.
Proof: If d is one of the derivations, we have a r−1 da = 1 r da r ∈ {a r } and thus: and a ∞ = a. We have in general ρ r (a q ) ⊆ a q+r and the problem will be to know when we may have equality.
We shall say that a differential extension L = Q(K{y}/p) is a finitely generated differential extension of K and we may define the evaluation epimorphism K{y} → K{η} ⊂ L with kernel p by calling η orȳ the residue of y modulo p. If we study such a differential extension L/K, by analogy with Section 2, we shall say that R q or g q is a vector bundle over R q if one can find a certain number of maximum rank determinant D α that cannot be all zero at a generic solution of p q defined by differential polynomials P τ , that is to say, according to the Hilbert Theorem of Zeros, we may find polynomials A α , B τ ∈ K{y q } such that α A α D α + τ B τ P τ = 1. The following Lemma will be used in the next important Theorem: LEMMA 3.2.12: If p is a prime differential ideal of K{y}, then, for q sufficiently large, there is a polynomial D ∈ K[y q ] such that D / ∈ p q and : Dp q+r ⊂ rad(ρ r (p q )) ⊂ p q+r , ∀r ≥ 0 THEOREM 3.2.13: (Primality test) Let p q ⊂ K[y q ] and p q+1 ⊂ K[y q+1 ] be prime ideals such that p q+1 = ρ 1 (p q ) and p q+1 ∩ K[y q ] = p q . If the symbol g q of the algebraic variety R q defined by p q is 2-acyclic and if its first prolongation g q+1 is a vector bundle over R q , then p = ρ ∞ (p q ) is a prime differential ideal with p ∩ K[y q+r ] = ρ r (p q ), ∀r ≥ 0.
COROLLARY 3.2.14: Every perfect differential ideal of {y} can be expressed in a unique way as the non-redundant intersection of a finite number of prime differential ideals.
COROLLARY 3.2.15: (Differential basis) If r is a perfect differential ideal of K{y}, then we have r = rad(ρ ∞ (r q )) for q sufficiently large.
EXAMPLE 3.2.16: As K{y} is a polynomial ring with an infinite number of variables it is not noetherian and an ideal may not have a finite basis. With K = Q, n = 1 and d = d x , then a = {yy x , y x y xx , y xx y xxx , ...} ⇒ (y x ) 2 + yy xx ∈ a ⇒ rad(a) = {y x } is a prime differential ideal.
PROPOSITION 3.2.17: If ζ is differentially algebraic over K < η > and η is differentially algebraic over K, then ζ is differentially algebraic over K. Setting ξ = ζ − η, it follows that, if L/K is a differential extension and ξ, η ∈ L are both differentially algebraic over K, then ξ + η, ξη and d i ξ are differentially algebraic over K.
If L = Q(K{y}/p), M = Q(K{z}/q) and N = Q(K{y, z}/r) are such that p = r ∩ K{y} and q = r ∩ K{z}, we have the two towers K ⊂ L ⊂ N and K ⊂ M ⊂ N of differential extensions and we may therefore define the new tower K ⊆ L ∩ M ⊆< L, M >⊆ N . However, if only L/K and M/K are known and we look for such an N containing both L and M , we may use the universal property of tensor products an deduce the existence of a differential morphism Looking for an abstract composite differential field amounts therefore to look for a prime differential ideal in L⊗ K M which is a direct sum of integral domains (See [18] for more details). DEFINITION 3.2.18: A differential extension L of a differential field K is said to be differentially algebraic over K if every element of L is differentially algebraic over K. The set of such elements is an intermediate differential field K ′ ⊆ L, called the differential algebraic closure of K in L. If L/K is a differential extension, one can always find a maximal subset S of elements of L that are differentially transcendental over K and such that L is differentially algebraic over K < S >. Such a set is called a differential transcendence basis and the number of elements of S is called the differential transcendence degree of L/K. THEOREM 3.2.19: If L/K is a finitely generated differential extension, then any intermediate differential field K ′ between K and L is also finitely generated over K. Comparing the differential geometric approach to nonlinear algebraic systems with the differential algebraic approach just presented while setting δy q = η q , we obtain: COROLLARY 3.2.21: When L/K is a finitely generated differential extension, then Ω L/K is a differential module over the differential ring L⊗ K K[d] = L[d] of differential operators with coefficients in L. The linearized "system " R = hom L (Ω L/K , L) is thus a (left) differential module for the Spencer operator like in the linear framework.
It is not evident to grasp these results in order to apply them to control theory or mathematical physics for two reasons. The first is that the formal theory of nonlinear systems has not been accepted by differential geometers because of the homological background based on the so-called "vertical machinery" and the systematic use of the Spencer δ-cohomology. The recent study of the Schwarzschild and Kerr metrics (Compare [35] to [1]) is providing a good example of such a poor situation. The second is the fact that, when K is a true differential field and M is differential module defined over the noncommutative ring K[d 1 , . . . , d n ] = K[d] of differential operators with coefficients in K, then the "system" R = hom K (M, K) is still not used today because its differential structure highly depends on the Spencer operator which has never been introduced in physics. As a good example, we may quote the fact that the Cosserat couple stress equations are just described by the formal adjoint of the linear Spencer operator ( [19]).

3.3) NONLINEAR CONTROL THEORY
As we have already explained in ( [18]), the generalized " Bäcklund problem " is nothing else than the study of nonlinear differential correspondences in the theory of differential elimination. We shall provide, below and successively, a differential geometric definition followed by a differential algebraic definition and all the problem will be to establish a link between them.
When X is a manifold of dimension n, let us consider two fibered manifolds over X, namely E with lcal coordinates (x, y) and F with local coordinates (x, z). The fibered roduct E× X F is a fibered manifold over X with ocal coordinates (x, y, z) and we have the canonical identification: with local coordinates (x, y q , z q ). For most applications, we shall suppose that E = X × Y and F = X × Z. DEFINITION 3.3.1: Let R q ⊂ J q (E× X F ) be a nonlinear system of order q on E× X F called a differential correspondence between (y, z). When r → ∞, we may consider the resolvent systems P q+r ⊂ J q+r (E) for y and Q q+r ⊂ J q+r (F ) for z, induced by the canonical projections of E× X F onto E and F respectively.
Roughly, finding P amounts to eliminate z while finding Q amounts to eliminate y and we shall only consider te first problem as the second will be similar.
• In the linear case, pushing y on the left and z on the right, we are left with the search of the CC for y or the CC for z that may be quite difficult. One of the best examples has been provided by M. Janet with the second order system (See [20] or [22] for details): where y = f (x) can be given arbitrarily for getting z = g(x) while z = g(x) must satisfy one CC of order 3 and one CC of order 6.

THEOREM OF THE RESOLVENT SYSTEMS 3.3.2:
In general, one may find two integers r, s ≥ 0 such that R (s) q+r is formally integrable (involutive) with formally integrable (involutive) projections P (s) q+r ⊂ J q+r (E) and Q (s) q+r ⊂ J q+r (F ). Moreover, r and s can be (tentatively) found by a finite algorithm preserving the symmetry existing between E and F .
Proof: First of all, we know that, in general, one can find the two integers r, s ≥ 0 in such a way that R (s) q+r is formally integrable (involutive). Hence, using the commutative and exact diagram: we may suppose, without any loss of generality, that R q is formally integrable (involutive). Now, chasing in the commutative diagram: we obtain therefore P q+r+s ⊆ ρ s (P q+r ) = J s (P q+r ) ∩ J q+r+s (E) ⊂ J s (J q+r (E)), ∀r, s ≥ 0. Then, chasing in the commutative diagram: we notice that π q+r+s q+r : P q+r+s −→ P q+r is an epimorphism ∀r, s ≥ 0. Finally, chasing in the commutative and exact diagram: we deduce that each P q+r is formaly integrable at each q + r, ∀r ≥ 0, though not always formally integrable as we shall see on examples. Looking at the symbol h of P, we have h q+r+s ⊆ ρ s (h q+r ) over P q+r+s . According to standard Noetherian arguments, such a situation is stabilizing for r and s large enough but such an approach is not constructive in general.
For this reason, we shall prefer to use a different approach which is closer to the one met in the case of linear differential correspondences. For this, if z = g(x) is an arbitrary section of F , we shall consider the new system for y defined by A q = j q (f ) −1 (R q ) over K < g >. Such a system, which is in general neither involutive nor even formally integrable as we shall see on examples, may also be not even compatible as it may not provide a fibered manifold but this way may give informations on the order of the OD or PD equations that should be satisfied by z. A similar procedure could be used by setting y = f (x) and introducing B q = j q (f ) −1 (R q ) in order to obtain a system for z over K < f >. ✷ Let us now turn to the differential algebraic counterpart.

DEFINITION 3.3.3:
If K is a differential field and we have a differential algebraic correspondence defined by a prime differential ideal r ⊂ K{y, z}, we may define the resolvent system for y by the resolvent differential ideal p = r ∩ K{y} and the resolvent system for z by the resolvent differential ideal q = r ∩ K{z}.

LEMMA 3.3.4:
The resolvent ideal for y is the prime differential resolvent ideal p = r ∩ K{y} for which one can find a differential basis. Similarly, the prime differential resolvent ideal for z is q = r ∩ K{z}.
Proof: We have the commutative and exact diagram: First of all, B is an integral domain because r is a prime differential ideal. It follows from a chase that the induced morphism A → B is a monomorphism and A ≃ im(A) ⊂ B is thus also an integral domain, a result showing that p is a prime differential ideal. It is essential to notice that projections of ideals cannot be used in the nonlinear framework. Hence, the idea is to reduce the study of differential algebraic correspondences to the study of purely algebraic correspondences. ✷ We end this last section with a few basic motivating examples showing the importance of the non-commutative localization of integral domains for explicit computations and applications. We hope therefore that these examples could be used as test examples for future applications of computer algebra (Compare to [16]). EXAMPLE 3.3.5: With m = 1, n = 2, q = 2, K = Q(x 1 , x 2 ) while using local coordinates (x 1 , x 2 , y) for the fibered manifold E let us consider anew the nice example presented by J. Johnson in ( [8]), namely the nonlinear system R 2 ∈ J 2 (E) defined by the two algebraic PD equations : We let the reader prove successively as an exercise that: 2 is adding 6x 2 yy 2 = 0 and thus y 1 = 0. R (4) 2 is adding (y) 2 = 0. Accordingly, the prime ideal p 2 ∈ K[y, y 1 , y 2 , y 11 , y 12 , y 22 ] generated by the two given differential polynomias (P 1 , P 2 ) is such that y ∈ rad(ρ 4 (p 2 )) ⇒ rad(ρ ∞ (p 2 )) = (y), a result not evident at first sight and leading to the trivial differential extension L = K. The linearization procedure is even less evident. Indeed, starting with the linearized second order system: we let the reader prove that we successively get: 2 is adding yη = 0 but one cannot conclude. Such an example is proving that, in general, one must start from a formally integrable or even involutive system in order to be able to define the module of Kähler differentials for the differential extension L/K. EXAMPLE 3.3.6: (Burgers) With n = 2, m = 2, local coordinates (x 1 , x 2 , y, z) and differential field K = Q, let us consider the algebraic first order involutive system R 1 defined by two differential algebraic PD equations: These two differential polynomials generate a prime differential ideal r ⊂ K{y, z} and provide thus a differential extension N/K. Indeed, K[y, y 1 , y 2 ; z, z 1 , z 2 ]/r 1 ≃ K[y, y 1 ; z, z 1 ] is an integral domain and r 1 is a prime ideal. Then, using one prolongation, we may introduce the following second order system R 2 = ρ 1 (R 1 ): and use the Janet tabular to prove that it is a nonlinear involutive system. It follows that K[y, ..., y 22 ; z, ..., z 22 ]/r 2 ≃ K[y, y 1 , y 11 ; z, z 1 , z 11 ] is an integral domain and r 2 is a prime ideal. Thanks to Theorem 3.2.13, we obtain thus finally K{y, z}/r ≃ K[y, y 1 , y 11 , ...; z, z 1 , z 11 , ...] which is also an integral domain. It follows hat p = r ∩ K{y} and q = r ∩ K{z} are prime differetial ideals. Taking now any section y = f (x), we obtain the system B 1 = j 1 (f ) −1 (R 1 ) for z: and its first prolongation B 2 = j 2 (f ) −1 (R 2 ) for z: First of all, this is a fibered manifold if and only if f is solution of the second order system P 2 defined by the single second order PD equation: which is the resolvent system for y generating the prime differential ideal p ⊂ K{y} allowing to define a differential extension L = Q(K{y}/p) of K and we have p = r ∩ K{y} ⇒ L ⊂ N . We are thus left with the first order (nonlinear) system for z: which is easily seen to be involutive for any solution y = f (x) of P 2 . Taking finally any section z = g(x), we obtain the system A 1 = j 1 (g) −1 (R 1 ): 1 of its first prolongation A 2 = j 2 (g) −1 (R 2 ): is compatible if and only if g is solution of the second order system Q 2 defined by the single second order PD equation obtained after substitution of y = ∂ 2 g(x): which is the resolvent system for z generating the prime differential ideal q = r ∩ K{z} allowing to define a differential extension M = Q(K{z}/q) of K and we have q = r ∩ K{z} ⇒ M ⊂ N .
We are thus left with the only zero order (linear) equation for y, namely: for any solution z = g(x) of Q 2 . The differential correspondence that must be used is thus R 2 . Both L, M and N are differential algebraic extensions of K of zero differential transcendence degree. EXAMPLE 3.3.7: (Korteweg-de Vries) With the same notations, we let the reader provide the details of the following similar example with the second order nonliear differential correspondence R 2 : by exhibiting the nonlinear formally integrable involutive system R 2 .

4) CONCLUSION
The author of this paper got his PhD thesis under the supervising of Prof. A. Lichnerowicz and has been collaborating with him till his death in 1998. At the end of his life, he became more and more convinced that the variational origin of mathematical physics (elasticity, electromagnetism, general relativity) through the corresponding Euler-Lagrange equations was a kind of "screen " hiding a more important concept allowing to describe the "duality" existing between "fields " and "inductions ". After the author discovered in 1995 the impossibility to parametrize the Einstein operator along a challenge proposed by J. Wheeler in 1970, he did notice that, in control theory, "a control system is controllable if and only if it is parametrizable " and that the "screen " is just the "differential double duality " involved in differential homological algebra through the use of the "extension modules " (See Zbl 1079.93001). Hence it remained to study the systematic use of the formal adjoint in the noncommutative situation met when linearizing nonlinear systems of OD or PD equations. Among the best useful examples, one has the following differential sequence, indicating below the fiber dimensions of the vector bundles involved with F 0 = S 2 T * : 0 → Θ → n → n(n + 1)/2 → n 2 (n 2 − 1)/12 → n 2 (n 2 − 1)(n − 2)/24 → ...
where Θ is the sheaf of Killing vector fields for the Euclidean metric when n = 3 in elasticity or the Minkowski metric when n = 4 in general relativity. Defining the adjoint operators: Cauchy = ad(Killing), Beltrami = ad(Riemann), Lanczos = ad(Bianchi) one discovers that Lanczos was in fact dreaming to construct the adjoint differential sequence: where ad(E) = ∧ n T * ⊗ E * for any vector bundle E where E * is obtained from E by inverting the transition rules when changing local coordinates. Accordingly, the only true problem left was to prove that each operator is indeed parametrized by the preceding one in both sequences, a highly non evident fact. Similarly, we have the Poincaré sequence for the exterior derivative: n → n(n − 1)/2 → n(n − 1)(n − 2)/6 with dA = F ⇒ dF = 0 for electromagnetism when n = 4 where A is the EM potential and F the EM field describing the first Maxwell operator and its parametrization. The adjoint sequence: is used for the EM induction and second Maxwell operator both with its parametrization by the so-called pseudo-potential. In both situations, there is no need to appeal to variational calculus which is only used for exhibiting the respective constitutive laws. We hope that the many tricky examples presented in this paper will be used later on as test-examples for computer algebra.