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377
Flatness and defect of nonlinear systems: Introductory theory and examples
 International Journal of Control
, 1995
"... We introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous. Their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of Kalman’s controllability. The distance to flatness is ..."
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Cited by 342 (23 self)
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We introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous. Their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of Kalman’s controllability. The distance to flatness is measured by a nonnegative integer, the defect. We utilize differential algebra which suits well to the fact that, in accordance with Willems ’ standpoint, flatness and defect are best defined without distinguishing between input, state, output and other variables. Many realistic classes of examples are flat. We treat two popular ones: the crane and the car with n trailers, the motion planning of which is obtained via elementary properties of planar curves. The three nonflat examples, the simple, double and variable length pendulums, are borrowed from nonlinear physics. A high frequency control strategy is proposed such that the averaged systems become flat. ∗This work was partially supported by the G.R. “Automatique ” of the CNRS and by the D.R.E.D. of the “Ministère de l’Éducation Nationale”. 1 1
An algorithm for solving second order linear homogeneous differential equations
 J. Symbolic Comput
, 1986
"... The Galois group tells us a lot about a linear homogeneous differential equation specifically whether or not it has “closedform” solutions. Using it, we have been able to develop an algorithm for finding “closedform ” solutions. First we will compute the Galois group of some very simple equatio ..."
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Cited by 143 (1 self)
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The Galois group tells us a lot about a linear homogeneous differential equation specifically whether or not it has “closedform” solutions. Using it, we have been able to develop an algorithm for finding “closedform ” solutions. First we will compute the Galois group of some very simple equations. We then will solve a more complicated one, using the techniques of the algorithm. This example illustrates how the algorithm was discovered and the kinds of calculations used by it. 1
Symmetry reductions and exact solutions of a class of nonlinear heat equations
 Physica D
, 1993
"... Classical and nonclassical symmetries of the nonlinear heat equation ut = uxx + f(u), (1) are considered. The method of differential Gröbner bases is used both to find the conditions on f(u) under which symmetries other than the trivial spatial and temporal translational symmetries exist, and to sol ..."
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Cited by 56 (4 self)
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Classical and nonclassical symmetries of the nonlinear heat equation ut = uxx + f(u), (1) are considered. The method of differential Gröbner bases is used both to find the conditions on f(u) under which symmetries other than the trivial spatial and temporal translational symmetries exist, and to solve the determining equations for the infinitesimals. A catalogue of symmetry reductions is given including some new reductions for the linear heat equation and a catalogue of exact solutions of (1) for cubic f(u) in terms of the roots of f(u) =0. 0 Symmetry Reductions of a Nonlinear Heat Equation 1
Reduction of Systems of Nonlinear Partial Differential Equations to Simplified Involutive Forms
, 1996
"... We describe the rif algorithm which uses a finite number of differentiations and algebraic operations to simplify analytic nonlinear systems of partial differential equations to what we call reduced involutive form. This form includes the integrability conditions of the system and satisfies a consta ..."
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Cited by 55 (17 self)
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We describe the rif algorithm which uses a finite number of differentiations and algebraic operations to simplify analytic nonlinear systems of partial differential equations to what we call reduced involutive form. This form includes the integrability conditions of the system and satisfies a constant rank condition. The algorithm is useful for classifying initial value problems for determined pde systems and can yield dramatic simplifications of complex overdetermined nonlinear pde systems. Such overdetermined systems arise in analysis of physical pdes for reductions to odes using the Nonclassical Method, the search for exact solutions of Einstein's field equations and the determination of discrete symmetries of differential equations. Application of the algorithm to the associated nonlinear overdetermined system of 856 pdes arising when the Nonclassical Method is applied to a cubic nonlinear Schrodinger system yields new results. Our algorithm combines features of geometric involutiv...
On differential RotaBaxter algebras
 J. PURE APPL. ALGEBRA
, 2008
"... A RotaBaxter operator of weight λ is an abstraction of both the integral operator (when λ = 0) and the summation operator (when λ = 1). We similarly define a differential operator of weight λ that includes both the differential operator (when λ = 0) and the difference operator (when λ = 1). We fur ..."
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Cited by 53 (21 self)
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A RotaBaxter operator of weight λ is an abstraction of both the integral operator (when λ = 0) and the summation operator (when λ = 1). We similarly define a differential operator of weight λ that includes both the differential operator (when λ = 0) and the difference operator (when λ = 1). We further consider an algebraic structure with both a differential operator of weight λ and a RotaBaxter operator of weight λ that are related in the same way that the differential operator and the integral operator are related by the First Fundamental Theorem of Calculus. We construct free objects in the corresponding categories. In the commutative case, the free objects are given in terms of generalized shuffles, called mixable shuffles. In the noncommutative case, the free objects are given in terms of angularly decorated rooted forests. As a byproduct, we obtain structures of a differential algebra on decorated and undecorated planar rooted forests.
Solving Difference Equations in Finite Terms
 J. Symbolic Comput
, 1998
"... We define the notion of a liouvillian sequence and show that the solution space of a difference equation with rational function coefficients has a basis of liouvillian sequences if and only if the Galois group of the equation is solvable. Using this we give a procedure to determine the liouvillian s ..."
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Cited by 52 (3 self)
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We define the notion of a liouvillian sequence and show that the solution space of a difference equation with rational function coefficients has a basis of liouvillian sequences if and only if the Galois group of the equation is solvable. Using this we give a procedure to determine the liouvillian solutions of such a difference equation.
Algebraic analysis of linear multidimensional control systems
 IMA Jounal of Control and Information
, 1999
"... The purpose of this paper is to show how to use the modern methods of algebraic analysis in partial dierential control theory, when the input/output relations are dened by systems of partial dierential equations in the continuous case or by multishift dierence equations in the discrete case. The e ..."
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Cited by 47 (25 self)
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The purpose of this paper is to show how to use the modern methods of algebraic analysis in partial dierential control theory, when the input/output relations are dened by systems of partial dierential equations in the continuous case or by multishift dierence equations in the discrete case. The essential tool is the duality existing between the theory of dierential modules or Dmodules and the formal theory of systems of partial dierential equations. We reformulate and generalize all the formal results that can be found in the extensive literature on multidimensional systems (controllability, observability, primeness concepts, poles and zeros,...). All the results are presented through eective algorithms.
Tannakian duality for AndersonDrinfeld motives and algebraic independence of Carlitz logarithms
 Invent. Math
"... Abstract. We develop a theory of Tannakian Galois groups for tmotives and relate this to the theory of Frobenius semilinear difference equations. We show that the transcendence degree of the period matrix associated to a given tmotive is equal to the dimension of its Galois group. Using this resul ..."
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Cited by 42 (8 self)
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Abstract. We develop a theory of Tannakian Galois groups for tmotives and relate this to the theory of Frobenius semilinear difference equations. We show that the transcendence degree of the period matrix associated to a given tmotive is equal to the dimension of its Galois group. Using this result we prove that Carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent. Contents