An algebraic framework for Infinite Dimensional Linear Systems

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An algebraic framework for Infinite Dimensional Linear Systems


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	    <date dateType="Created">Sun 1 Oct 2017</date>
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            <date dateType="Submitted">Sat 17 Feb 2018</date>
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M. FLIESS AND H. MOUNIER 1 An algebraic framework for Infinite Dimensional Linear Systems M. Fliess, H. Mounier Abstract— An algebraic theory for linear partial differential systems controlled at the boundary is presented. Some structural properties, analogous to the differential flatness of nonlinear systems, are exposed yielding an easy solution to tracking problems. Examples of the wave and Euler Bernoulli equations are treated in some detail. Index Terms— Distributed parameter systems, Delay systems, π-freeness, Flatness, controllability, Trajectory tracking, Wave equation, Euler-Bernoulli equation, Module theory. INTRODUCTION THE present document reports on recent works in the area of infinite dimensional systems modelled through delay differential or partial differential equations [1], [21], [23], [24], [25], [26], [37], [41], [42], [48], [49], [50], [52]. The adopted framework emphasizes on equation structure (rather than solutions) in order to study a given system. When dealing with linear equations, a system is associated with a module over a differential ring, this notion playing for differential equations the role played by vector spaces for linear algebraic equations. Keeping constructive notions in mind, a natural general- ization of Kalman’s controllability is π-freeness [12], [13], [21], [48], which allows the tracking of a reference trajectory in a way bearing some analogy with flat finite dimensional nonlinear systems (see [17], [18] and the references therein). This property leads to a complete parametrization of the system in terms of the so-called basic output. The obtained formulae yield every variable of the system without any integration of a differential equation. These ingredients are complemented by the use of Mikusin- ski’s operational calculus [45], [46] which avoids many of the severe analytical difficulties encountered with the classic Laplace transform (see, e.g., [8]). Another important tool is a special type of C∞ -functions, called Gevrey functions [27], yielding exact solutions to rest to rest problems for distributed parameter systems. The paper is organized as follows. We first introduce ab- stract linear systems over arbitrary commutative rings. Finite dimensional linear systems are then presented within this framework. We proceed to delay systems and relate them to M. Fliess is with the Centre de Mathématiques et de Leurs Applications, École Normale Supérieure de Cachan, 61 avenue du Président Wilson, 94235 Cachan, France, AND with the Laboratoire GAGE, École polytechnique, 91128 Palaiseau Cedex, France. E-mail: H. Mounier is with the Robotics Center (CAOR) of the École Nationale Supérieure des Mines de Paris, 60–62, Boulevard Saint-Michel, 75272 Paris Cedex 06, France, E-mail: This work was partially supported by the European Commission’s Training and Mobility of Researchers (TMR) Contract # ERBFMRX-CT970137, by the G.D.R. Medicis and by the G.D.R.-P.R.C. Automatique. the boundary control of the wave equation. The final sections are devoted to the heat and the Euler-Bernoulli equations. Acknowledgments 1: Several parts of the work presented here were done in collaboration with our colleagues and friends P. Rouchon (École des Mines de Paris, France) and J. Rudolph (Technische Universität Dresden, Germany). I. ABSTRACT LINEAR SYSTEM THEORY A. Basic definitions Any ring R is commutative, with 1 and without zero divisors. Notation The submodule spanned by a subset S of an R- module M is written [P]. An R-system Λ, or a system over R, is an R-module. Two R-systems Λ1 and Λ2 are said to be R-equivalent, or equiva- lent over R, if the R-modules Λ1 and Λ2 are isomorphic. An R-dynamics, or a dynamics over R, is an R-system Λ equipped with an input, i. e., a subset u of Λ which may be empty, such that the quotient R-module Λ/[u] is torsion. The input u is independent if the R-module [u] is free, with basis u. An output y is a subset, which may be empty, of Λ. An input-output R-system, or an input-output system over R, is an R-dynamics equipped with an output. Remarks 1: 1) Kalman’s module-theoretic setting [32] is related to the state variable description, whereas our module description encompasses all system variables without any distinction. 2) In informal terms, a system with equation a(s) x = b(s) u where s stands for d dt , the derivation operator, will be represented by a linear structure consisting in all the linear combinations of x and u and their derivatives such that the above relation is satisfied Λ = {p(s) x + q(s) u| p, q ∈ R[s], a(s) x = b(s) u} 3) In the same informal terms, an input u = (u1, . . . , um) is such that every variable z of the system satisfies a relation of the form: p(s) z = m X i=1 qi(s) ui where p, qi ∈ R[s], p 6= 0. Let A be an R-algebra et and Λ be an R-system. The A- module A ⊗R Λ is an A-system, which extends Λ. 2 M. FLIESS AND H. MOUNIER B. Relations Let Λ be an R-system. There exists an exact sequence of R-modules [59] 0 → N → F → Λ → 0 (1) where F is free. The R-module N, which is sometimes called the module of relations, should be viewed as a system of equations defining Λ. Associate to Λ a free presentation [59], i. e., the short exact sequence of R-modules F1 → F0 → Λ → 0 where F0 and F1 are free. The R-module Λ is said to be finitely generated, or of finite type, if there exists a free presentation where any basis of F0 is finite. It is said to be finitely presented if there exists a free presentation where any basis of F0 and F1 is finite. The matrix corresponding, for some given bases, to the mapping F1 → F0 is called a presentation matrix of Λ. If the ring R is Noetherian, it is known [59] that the conditions of being of finite type and of being finitely presented coincide. Then (1) may be chosen such that both F and N are of finite type. This latter case will always be verified in the sequel. Example 1: Let us determine the R-module Λ correspond- ing to a system of R-linear equations µ X κ=1 aικξκ = 0, aικ ∈ A, ι = 1, . . . , ν where ξ1, . . . , ξµ are the unknowns. Let F be the free R- module spanned by f1, . . . , fµ. Let N ⊆ F be the module of relations, i. e., the submodule spanned by Pµ κ=1 aικfκ, ι = 1, . . . , ν. Then, Λ = F/N. The ξκ’s are the residues of the fκ’s, i. e., the canonical images of the fκ’s. Take, e.g., the system given in remark 1. The free R[s]- module F is the one spanned by X and U (i.e. the set {p(s) X + q(s) U | p, q ∈ R[s]} endowed with a linear structure). The submodule N of relations is spanned by the element a(s) X −b(s) U (i.e. the set {r(s)(a(s) X −b(s) U) | r ∈ R[s]} endowed with a linear structure). Then Λ = F/N = [X, U]/[a(s) X−b(s) U], with generators x and u, the residues of X and U, and with relation a(s) x = b(s) u. C. Laplace functor 1) Mathematical preliminaries: Let K the quotient field of R. The K-vector space Λ̂ = K ⊗R Λ is called the transfer K-vector space of the R-system Λ. The functor K ⊗R •, from the category of R-modules to the category of K-vector spaces, is called the Laplace functor [14]. The (formal) Laplace transform of any λ ∈ Λ is λ̂ = 1 ⊗ λ ∈ Λ̂. Let us briefly review some basic properties. 1) The set {λi ∈ Λ | i ∈ I} is R-linearly independent if, and only if, the set {λ̂i ∈ Λ̂ | i ∈ I} is K-linearly independent. The rank of Λ, which is written rk (Λ), is, by definition, the dimension of the K-vector space Λ̂. 2) The kernel of the R-linear morphism Λ → Λ̂, λ 7→ λ̂, is the torsion submodule of Λ. The rank of an R-module is therefore zero if, and only if, it is torsion. Consider an R-module Λ spanned by ξ1, . . . , ξn which satisfy M    ξ1 . . . ξn    = 0 where M ∈ An×n . The square matrix M is said to be of maximum rank if det(M) 6= 0. The matrix M is therefore of maximum rank if, and only if, it is invertible as a matrix in Kn×n . Theorem 1: The R-module Λ is torsion if, and only if, the matrix M is of maximum rank. Proof: Λ is torsion if, and only if, Λ̂ = {0}. Note that Λ̂ is spanned by ˆ ξ1, . . . , ˆ ξn which satisfy M    ξ̂1 . . . ξ̂n    = 0 The zero solution is unique if, and only if, M ∈ Kn×n is invertible. 2) Transfer matrices: Consider an R-system, where the input u = (u1, . . . , um) and the output y = (y1, . . . , yp) are, for simplicity’s sake, finite. Since Λ/[u] is torsion, spanK(û1, . . . , ûm) = Λ̂. There exists a matrix T ∈ Kp×m , called the transfer matrix of the input-output system, such that    ŷ1 . . . ŷp    = T    û1 . . . ûm    If u is independent, û1, . . . , ûm is a basis of Λ̂. Then, T is unique. D. Input-output inversion 1) Generalities: Consider again the R-system Λ with finite input u = (u1, . . . , um) and finite output y = (y1, . . . , yp). Its output rank is, by definition, ρ = rk ([y]). The system is said to be left invertible (resp. right invertible) if ρ = rk ([u]) (resp. ρ = p). The quotient R-module Λ/[y] is the residual R-dynamics, or the zero R-dynamics. The residual dynamics is said to be trivial if Λ/[y] = {0}. Proposition 1: The R-system Λ is left invertible if, and only if, its residual R-dynamics Λ/[y] is torsion. Proof: From rk(Λ/[y]) = rk (Λ)−rk ([y]), we obtain that rk (Λ/[y]) = 0 is equivalent to rk ([y]) = rk (Λ) = rk ([u]). Remark 1: Left invertibility implies, since Λ/[y] is torsion, that y plays the role of an input. Any other system variable, i. e., any element of Λ, may be determined from the output. Right invertibility means that the components y1, . . . , yp of the output are R-linearly independent. 2) Some illustrations: Assume that the input u = (u1, . . . , um) and the output y = (y1, . . . , yp) are both finite and that u is independent, i. e., that rk([u]) = m. The next two propositions are clear. ALGEBRAIC FRAMEWORK FOR INFINITE DIMENSIONAL SYSTEMS 3 Proposition 2: An R-system with finite and independent input and finite output is left invertible (resp. right invertible) if, and only if, its transfer matrix is left invertible (resp. right invertible). The above system is said to be square if m = p. Proposition 3: Consider a square R-system with an inde- pendent input. Then, the two following properties are equiva- lent: • the system is left invertible; • the system is right invertible. An R-system with an independent input, which is both left and right invertible, is necessarily square. Such a left and right invertible system is called invertible. E. Different notions of controllability An R-system Λ is said to be R-torsion free controllable (resp. R-projective controllable, R-free controllable) if the R- module Λ is torsion free (resp. projective, free). Elementary homological algebra (see, e.g., [59]) yields the Proposition 4: R-free (resp. R-projective) controllability implies R-projective (resp. R-torsion free) controllability. Take an R-free controllable system Λ with a finite output y. This output is said to be flat, or basic, if y is a basis of Λ. The next proposition is clear. Proposition 5: Take an R-system Λ with a finite output. The output is flat if, and only if, the following two properties hold • the system is right invertible, • the residual dynamics is trivial. Then, Λ is R-free controllable. Moreover, if there is an independent input, the system is square. Remark 2: This notion corresponds, for SISO finite di- mensional systems, to a transfer function with a constant numerator. Take, e.g., a second order system in mechanics Mẍ + γẋ + κx = u where x is homogeneous to a length, u to a force, M to a mass, γ and κ to damping and stiffness coefficients respectively. This system is R[s]-free, with basis x; its transfer function is 1 Ms2 + γs + k F. π–freeness The next result [21] follows at once from [60, Proposition 2.12.17, p. 233]: Theorem and Definition 1: Let Λ be an R–system, A an R– algebra, and S a multiplicative part of A such that Λ is S−1 R– free controllable. Then, there exists an element π in S such that Λ is R[π−1 ]–free controllable. The preceding system will then be called π–free. An output being a basis of R[π−1 ]⊗R Λ is called π-flat or π-basic. Corollary 1: Let Λ be an R–torsion free controllable R– system and S a multiplicative part of R such that S−1 R is a principal ideal ring. Then, there exists π ∈ S such that Λ is R[π−1 ]–free controllable and Λ is π–free. II. FINITE DIMENSIONAL LINEAR SYSTEMS A. Modules over principal ideal rings In this section R is the principal ideal ring k[s], whose elements are of the form P finite aαsα , aα ∈ k, where k is a field, and s stands for d dt . All k[s]-modules are finitely generated and, therefore, finitely presented. The following two theorems are classic (see, e.g., [36]) Theorem 2: For a k[s]-module M, the next two conditions are equivalent: 1) M is torsion, 2) the dimension of M as a k-vector space is finite, i. e., dimkM < ∞. Theorem 3: Any k[s]-module M may be written M ≃ F ⊕ t(M) where t(M) is the torsion submodule and F a free module. Corollary 2: Any k[s]-module is free if, and only if, it is torsion free. B. State variable representation Let M be a torsion k[s]-module. The derivation s induces a k-linear endomorphism σ : M → M. The next property is clear. Proposition 6: Let M be a torsion k[s]-module. It corre- sponds to s    x1 . . . xn    = F    x1 . . . xn    (2) where F ∈ kn×n is the matrix of σ with respect to a basis x1, . . . , xn of the finite dimensional k-vector space M. Consider a k[s]-dynamics Λ with input u = (u1, . . . , um). Set n = dimk(Λ/[u]). Take in Λ a set η = (η1, . . . , ηn) whose residue in Λ/[u] is a basis. From (2) we obtain s    η1 . . . ηn    = F    η1 . . . ηn    + ν X α=1 Gαsα    u1 . . . um    (3) where F ∈ kn×n , Gα ∈ kn×m . We say that η is a generalized state, and (3) a generalized state variable representation. Let η̃ = (η̃1, . . . , η̃n) be another generalized state. As the residues of η and η̃ in Λ/[u] are bases, we obtain    η̃1 . . . η̃n    = P    η1 . . . ηn    + X finite Qγsγ    u1 . . . um    (4) where P ∈ GLk(n), Qγ ∈ kn×p . Note that the generalized state variable transformation (4) depends in general from the input and a finite number of its derivatives. Assume that ν > 1 and Jν 6= 0 in (3). Following (4) set    η1 . . . ηn    =    η1 . . . ηn    + Gνsν−1    u1 . . . um    4 M. FLIESS AND H. MOUNIER It yields s    η1 . . . ηn    = F    η1 . . . ηn    + ν−1 X α=1 Gαsα    u1 . . . um    The maximal order of derivation of u is at most ν − 1. By induction we get s    x1 . . . xn    = A    x1 . . . xn    + B    u1 . . . um    (5) where A ∈ kn×n , B ∈ kn×m . We say that (5) is a Kalman state variable representation. The n-tuple x = (x1, . . . , xn) is a Kalman state. Two Kalman states x and x̃ = (x̃1, . . . , x̃n) are related by    x̃1 . . . x̃n    = P    x1 . . . xn    (6) where P ∈ GLk(n). As a matter of fact the presence of u and of its derivatives as in (4) would yield in (5) derivatives of u of order > 1. We have proved [12] the Theorem 4: Any k[s]-dynamics admits a Kalman state vari- able representation (5). Two Kalman states are related by (6). C. Controllability 1) Definition: The three notions of free, projective and torsion free controllability over k[s] coincide. A k[s]-system Λ is therefore said to be controllable [12], if the k[s]-module Λ is free. 2) Comparison: The dynamics (5), where u is assumed to be independent, is said to be controllable à la Kalman if rk(B, AB, . . . , An−1 B) = n Theorem 5: The Kalman state variable representation (5) is controllable if, and only if, it is controllable à la Kalman. Proof: First assume that (5) is uncontrollable in the Kalman sense. Write the uncontrollable part in the Kalman decomposition: s    ξ1 . . . ξn0    = F    ξ1 . . . ξn0    where F ∈ kn0×n0 . We know that the corresponding module is torsion. Therefore the module Λ corresponding to (5) cannot be free. A similar argument shows that Λ free corresponds to a realization (5) which is controllable in the sense of Kalman. Remark 3: See [13] for the comparison with Willems’ behavioral approach [65]. III. DELAY SYSTEMS Let R be the ring k[s, e−hs , ehs ], s playing the role of d dt , and e−hs = (e−h1s , . . . , e−hrs ), the e−his being (localized) delay operators of non commensurate amplitudes; the hi’s (hi ∈ R, hi > 0) are the amplitudes of the corresponding delays. This ring may be considered as derived from a ring of polynomials in r + 1 indeterminates over the field k, knowing that s, e−h1s , . . ., e−hrs are algebraically independent. We shall do so in all subsequent sections, except for the spectral controllability (subsection III-A.2) where this ring will be viewed as a subring of the convergent power series ring k{{s}}. A (linear) delay system will be a k[s, e−hs , ehs ]-system. Note that the present definition slightly differs from previous ones [21], [49], where the advances ehis where not included. A. Controllability The resolution of Serre’s conjecture [61] due to Quillen [54] and Suslin [64] (see also [35], [68] for a detailed exposition) states that, on a polynomial ring, a projective module is free. Thus, in the present context, Quillen-Suslin’s theorem may be stated as [21]: Proposition 7: A k[s, e−hs ]–system is k[s, e−hs ]–free con- trollable if, and only if, it is k[s, e−hs ]–projective controllable. Very many notions can then be considered (through torsion freeness and freeness on the one hand, and through the varia- tion of the ground ring on the other hand). Among these, the k[s, e−hs ]-free controllability is certainly the most appealing from an algebraic viewpoint. The existence of a basis is an extremely useful feature; but this notion seems quite rare in practice (see, e.g., [48], [50]). The π-freeness retains the main advantage of freeness (existence of a basis) while being almost always satisfied in applications. Indeed, we have [21] Proposition 8: A torsion free controllable delay system Λ is π–free, where π may be chosen in k[e−hs , ehs ]. Note that a free controllable delay system is a free k[s, e−hs , ehs ]-module, which is weaker than the freeness over k[s, e−hs ]. This e−hs -freeness is quite common in practice for the special case of quasi-finite systems (see subsection III- A.3). 1) Criteria for k[s, e−hs ]–free and k[s, e−hs ]–torsion free controllability: We establish (see [21]) two criteria for k[s, e−hs ]–free and k[s, e−hs ]–torsion free controllability. The first one uses the resolution of Serre’s conjecture [54], [64], and the second one1 uses [67]. Theorem 6: A delay system Λ with presentation matrix PΛ of full generic rank β is k[s, e−hs ]–free controllable if, and only if, ∀(s, z1, . . . , zr) ∈ k̄r+1 , rk k̄ PΛ(s, z1, . . . , zr) = β where k̄ is the algebraic closure of k. This rank criterion is equivalent to the common minors of PΛ of order β having no common zero in k̄r+1 . Theorem 7: A delay system Λ is k[s, e−hs ]–torsion free controllable if, and only if, the gcd of the β ×β minors of PΛ belongs to k. 1See [69] for related results. ALGEBRAIC FRAMEWORK FOR INFINITE DIMENSIONAL SYSTEMS 5 Examples 1: 1) The system ẏ + δy = u is k[s, e−hs ]-free controllable, with basis y. 2) The system ẏ = (1 + δ)u is k[s, e−hs ]-torsion free controllable, but not k[s, e−hs ]-free controllable. 3) The system ẏ = δu is free (e.g., k[s, e−hs , ehs ]-free controllable), but not k[s, e−hs ]-free controllable. 2) Spectral controllability: The following definition of spectral controllability extends previous ones (see, e.g., [4], [57]) in our context. Definition 1: Let Λ be a delay system, with presentation matrix PΛ of full generic rank β. It is called spectrally controllable if ∀s ∈ C, rkC PΛ(s, e−hs ) = β Set Sr = k(s)[e−hs , ehs ] ∩ E, where E denotes the ring of entire functions. We have the following interpretation of spectral controllability [49] (see also [28] and [5]): Proposition 9: Let Λ be a torsion free controllable delay system. It is spectrally controllable if, and only if, it is Sr– torsion free controllable. The following result [21] gives implication relationships be- tween the notion of k[s, e−hs , ehs ]-freeness, k[s, e−hs , ehs ]- torsion freeness and spectral controllability. Proposition 10: Let Λ be a delay system. The following chain of implications is true Λ free =⇒ Λ spectrally controllable =⇒ Λ torsion free. Proof: The proof follows directly from the inclusion k[s, e−hs , ehs ] ⊂ Sr. 3) Quasi-finite systems: These systems, quite frequent in practice, are typically ones with a transfer matrix of the form H(s)e−hs with H(s) a transfer matrix without delays. A linear delay system Λ is said to be special if Λ = k[s, e−hs , ehs ] ⊗k[s] Λspec , where Λspec is a finitely generated k[s]-module. It is equivalent saying there exists a presentation matrix with entries in k[s]. Note that Λspec is not unique: it may be replaced by ehs Λspec , λspec ∈ Λspec , h ∈ spanN(h). The k[s]-linear mapping Λspec → Λ, λspec 7→ λ = 1 ⊗ λspec is injective. We will therefore consider, with a slight abuse of notations, Λspec as a subset of Λ. A dynamics Λ is said to be quasi-finite if it is special and there exists Λspec such that the control variables u belong to it. This Λspec is of course unique. It is called the fundamental dynamics, or module, and will be designated by Λspec 0 . A special input-output system is said to be quasi-finite if the next two conditions are satisfied: 1) its corresponding dynamics is quasi-finite, 2) there exists p non-negative real numbers L = (L1, . . . , Lp) such that yfut 1 = eL1s y1, . . . , yfut p = eLps yp ∈ Λ spec 0 The definition of quasi-finite input-output systems leads to the following nominal (i.e. unperturbed) state-variable repre- sentation (see [12], [19], [20]) s    xfut 1 . . . xfut n    = A    xfut 1 . . . xfut n    + B    u1 . . . um    (7)    yfut 1 . . . yfut p    =    eL1s y1 . . . eLps yp    = C    xfut 1 . . . xfut n    + D    u1 . . . um    (8) where • A ∈ kn×n , B ∈ kn×m , C ∈ kp×n , D ∈ k[s]p×m ; • n = dimk Λspec /spank[s](u)  The definition of quasi-finite input-output systems leads at once to the following property Proposition 11: The matrix T ∈ k(s, e−Ls )p×m is the transfer matrix of a nominal (i.e. perturbation-free) quasi-finite input-output system if, and only if, it is of the form T =       e−L1s 0 e−L2s ... 0 e−Lps       R where R ∈ k(s)p×m . In particular, T ∈ k(s, e−Ls ) is the nominal transfer function of a quasi-finite SISO system if, and only if, T = Re−Ls , R ∈ k(s), L ≥ 0. Such a transfer matrix, or function, will also be called quasi- finite. Example 2: The SISO unperturbed system sx = (1 − e−Ls )u, y = x, with a 1-dimensional state, is not quasi-finite, since its transfer function 1−e−Ls s is not quasi-finite. B. The wave equation 1) The torsional behavior of a flexible rod: Consider [50] the torsional behavior of a flexible rod with a torque applied to one end. A mass is attached to the other end. The system is described by the one dimensional wave equation. σ2 ∂2 q ∂τ2 (τ, z) = ∂2 q ∂z2 (τ, z) (9) ∂q ∂z (τ, 0) = −u(τ), ∂q ∂z (τ, L) = −J ∂2 q ∂τ2 (τ, L) (10) q (0, z) = q0(z), ∂q ∂τ (0, z) = q1(z) (11) Here q(τ, z) denotes the angular displacement from the unex- cited position at a point z ∈ [0, L] at time τ > 0, as shown in Figure 1; L is the length of the rod, σ the inverse of the wave propagation speed, J the inertial momentum of the mass, u(τ) the control torque and q0, q1 describe the initial angular displacement and velocity, respectively. 6 M. FLIESS AND H. MOUNIER u() J q(; z0 + dz) q(; z0) 0 z0 z0 + dz L z 1 Fig. 1. The flexible rod. 2) Delay system model: As well known, the general solu- tion of (9) may be written q(τ, z) = φ(τ + σz) + ψ(τ − σz) where φ and ψ are one variable functions. The control objec- tive will be to assign a trajectory to the angular position of the mass; the output is thus y(τ) = q(τ, L) Set t = (σ/J)τ, v(t) = (2J/σ2 )u(t) and T = σL. Easy calculations (see [50] for details) yield the following delay system (compare with [9]): ÿ(t) + ÿ(t − 2T ) + ẏ(t) − ẏ(t − 2T ) = v(t − T ) (12) One readily has v = (eT s + e−T s )ÿ + (eT s − e−T s )ẏ (13) which implies Proposition 12: The delay system with equation (12) is free, with basis y. 3) Tracking: Equation (13), yields the open loop control vd(t) = ÿd(t + T ) + ÿd(t − T ) + ẏd(t + T ) − ẏd(t − T ) The displacements of the other points of the rod can be obtained as (see [50]) qd(z, t) = 1 2 h yd(t − z + T ) + ẏd(t − z + T ) + yd(t − T + z) − ẏd(t − T + z) i IV. HEAT AND EULER-BERNOULLI EQUATIONS A. Mikusiński’s operational calculus The set of continuous functions [0, +∞) → C is a commu- tative ring C with respect to the pointwise addition + where (f + g)(t) = f(t) + g(t) and the convolution product ⋆ where (f⋆g)(t) = (g⋆f)(t) = Z t 0 f(τ)g(t−τ)dτ = Z t 0 g(τ)f(t−τ)dτ According to a famous theorem due to Titchmarsh (see [45], [46], [66]), C does not possess zero divisors, i. e., f ⋆ g = 0 ⇔ f = 0 or g = 0 The quotient field M of C is called the Mikusiński field. Any element of M is called an operator. Notations 1) A function f(t) in C is sometimes written {f(t)} when viewed as an operator in M. 2) The (convolution) product of two operators a, b ∈ M is written ab. Examples 2: 1) The neutral element 1 for the (convolu- tion) product is the Dirac operator. It is the analogue of the Dirac distribution in Schwartz’s distribution theory. The Dirac operator 1 should not be confused with the Heaviside function {1} ∈ C. 2) The inverse in M of the Heaviside function {1} is the differential operator s. It obeys to the classic rules, i. e., if f ∈ C is C1 , then {sf} = { ˙ f}+{f(0)}. The meaning of operators in the subfield C(s) of rational functions in the variable s with complex coefficients is clear. The fractional derivative √ s appears as the inverse of 1 √ s = { 1 √ 2πt }. 3) The field M contains the subring S of piecewise continuous functions R → R with left bounded supports, i. e., for any f ∈ S, there exists a constant β ∈ R, such that, for t < β, f(t) = 0. The translation operator e−hs , h ∈ R acts on f ∈ S by e−hs {f(t)} = {f(t − h)}. The inverse of e−hs is ehs . Notice that es √ −1 is not an operator, i. e., it does not belong to M. A sequence an, n > 0, of operators is said to be opera- tionally convergent [45], [46] if there exists an operator p such that the anp’s belong to C and converge almost uniformly, i. e., uniformly on any finite interval, to a function in C. A series of operators P ν>0 bν is said to be operationally convergent if the sequence Pn ν=0 bn is operationally convergent. Example 3: The operator eλ √ s , λ ∈ C, may be defined by its Taylor expansion eλ √ s = X n>0 λn s n 2 n! (14) which is operationally convergent [46]. This property does not hold for the translation operator e−hs , h ∈ R. An operational function [45] is a mapping R → M. One can define the continuity, differentiability and integrability of operational functions. B. Gevrey functions A C∞ -function ζ(t) of the real variable t is said to be Gevrey [27]2 of order, or class, µ > 1 if for any compact subset K ⊂ R and for any integer n > 0 |ζ(n) (t)| 6 Cn+1 (n!)µ t ∈ K 2This notion has now become classic among analysts (see, e.g., [30], [33], [40], [55], [58]). See, also, [29] and [46] for a slightly different setting. ALGEBRAIC FRAMEWORK FOR INFINITE DIMENSIONAL SYSTEMS 7 where C > 0 is a constant depending on ζ and K. The function φ(t) =  0 if t 6 0 e−1/td if t > 0 where d > 0, is flat3 at t = 0, i. e., φ(ν) (0) = 0 for ν > 1. It is not analytic but Gevrey of class (1 + d)/d (see, e.g., [58]). The function ηd(t) =                  0 if t < 0 Z t/T 0 exp −1/(τ(1 − τ))d  dτ Z 1 0 exp −1/(τ(1 − τ))d  dτ if t ∈ [0, T ] 1 if t > T (15) is also Gevrey of class (1 +d)/d. It is flat at t = 0 and t = 1. C. The heat equation In order to illustrate our tools consider a boundary control problem of the heat equation ∂2 w(x, t) ∂x2 = ∂w(x, t) ∂t 0 < x < 1, t > 0 (16) The initial condition is w(x, 0) = 0. Its boundary conditions are ∂w(0,t) ∂x = 0 and w(1, t) = u(t) where u(t) designates the control variable. Operational calculus is replacing (16) with the ordinary differential equation, where x is the independent variable, ŵxx(x, s) − sŵ(x, s) = 0 (17) subject to the boundary conditions ŵx(0, s) = 0, ŵ(1, s) = û (û is an operator and ŵ an operator function). The solution of (17) reads ŵ(x, s) = cosh(x √ s) cosh( √ s) û(s) (18) Replace (18) by cosh( √ s) ŵ = cosh(x √ s) û which defines a C[P, Q]-module M, where P = cosh(x √ s), Q = cosh( √ s), which is torsion free, but not free. The localized C[P, Q, (PQ)−1 ]-module C[P, Q, (PQ)−1 ] ⊗C[P,Q] M is free, of rank 1, with basis ζ̂ = ŵ(0, s): ŵ = cosh(x √ s) ζ̂ û = cosh( √ s) ζ̂ When setting w(x, t) =  X n>0 x2n (2n)! dn dtn  ζ(t) (19) u(t) =  X n>0 1 (2n)! dn dtn  ζ(t) (20) 3The word flat possesses a large variety of mathematical meanings. Think of flat coordinates, flat modules, etc. Fig. 2. A rotating Euler-Bernoulli beam with an end mass. we assume that the function ζ(t), which corresponds to the operator ζ̂, is flat at t = 0. Theorem 8: The series X n>0 x2n (2n)! ζ(n) (t) and X n>0 1 (2n)! ζ(n) (t) are absolutely convergent if, and only if, ζ is a Gevrey function of class strictly less than 2. Moreover, if ζ is flat at t = 0, then their sums are respectively equal to cosh(x √ s) {ζ} and cosh( √ s) {ζ}. Proof: The absolute convergence of the series follows at once from Stirling’s formula n! ∼ e−n nn− 1 2 √ 2π, n → +∞ The second part follows from the operational convergence of the expansion (14). D. The Euler-Bernoulli equation 1) The flexural behavior of a flexible rod: Take a flexible beam of length R, the end x = 0 of which is clamped into a motor’s axle with angle θ, the other end x = R being a mass m [24]. Suppose that its motion obeys to the Euler– Bernoulli equation (linear elasticity, weak flexion, inertia of the beam negligible with respect to the mass m, Coriolis forces negligible, i. e., θ̇ small) with the boundary conditions: ∂2 w ∂τ2 = − ∂4 w ∂4x (21) w(0, τ) = 0, ∂w(0, τ) ∂x = 0 ∂2 w(1, τ) ∂x2 = 0, ∂3 w(1, τ) ∂x3 = k1 d2 dτ2 θ(τ) + k2 ∂2 w(1, τ) ∂τ2 where w(x, t) is the deformations field with respect to the rotating axis of angle θ and where d2 θ dτ2 is the control. For notational simplicity, the quantity u(τ) = k1θ(τ)+k2w(1, τ), will play the role of an input variable. The following param- eters are the time t = R2 p ρS/(EI) τ, the length r = Rx where E, I, R, ρ and S are the usual physical quantities. 2) Operational calculus and controllability: With initial conditions w(x, 0) = 0, ∂w(x, 0) ∂τ = 0 operational calculus associates to (21) the ordinary differential equation s2 ŵ = −ŵ(4) . Its general solution reads ŵ(x, s) = aexξ √ s + be−xξ √ s + cexξ̄ √ s + de−xξ̄ √ s where ξ = exp(iπ/4) and a, b, c, d are determined from the boundary conditions. After some calculations [24], one has û = −2 h 2 + cosh( √ 2s) + cosh(i √ 2s) i ŷ 8 M. FLIESS AND H. MOUNIER ŵ(x, s) = h (cosh(ξ √ s) + cosh(¯ ξ √ s))(i sinh(xξ √ s)+ sinh(x¯ ξ √ s)) + (i sinh(ξ √ s) − sinh(¯ ξ √ s)) (− cosh(xξ √ s) + cosh(x¯ ξ √ s) i 2 √ s ξ ŷ where ŷ = π−1 (c + d) 2 and π = i sinh(ξ √ s) − sinh(¯ ξ √ s) ξs √ s Set û = γŷ and ŵ = δŷ. The next result [24] is obtained in the same manner as in Subsection IV-C. 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