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186
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 176 (14 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. Symb. Comp
, 1986
"... In this paper wr present an algonthm for finding a "closedlorm " solution of the differential equation.1"'+.rI"+br'. where aand h are rational functions of a complex variable r. provided a "closed[orm " solution exists. The algorithm is so arranged that ii no solutio ..."
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Cited by 93 (0 self)
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In this paper wr present an algonthm for finding a "closedlorm " solution of the differential equation.1"'+.rI"+br'. where aand h are rational functions of a complex variable r. provided a "closed[orm " solution exists. The algorithm is so arranged that ii no solution is found. then ntr solutitrn can exist. In this paper we present an algorithm for finding a "closedform " solution of the dillerential equatiofl1"'+ ql +by, where a and b are rational functions of a complex variable.x. provided a "closedform " solution exists. The algorithm is so arranged that if no soluticrn is found. then no solution can exist.
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 43 (14 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...
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 43 (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.
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 33 (2 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
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 26 (6 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
DifferentialAlgebraic Dynamic Logic for DifferentialAlgebraic Programs
"... Abstract. We generalise dynamic logic to a logic for differentialalgebraic programs, i.e., discrete programs augmented with firstorder differentialalgebraic formulas as continuous evolution constraints in addition to firstorder discrete jump formulas. These programs characterise interacting discr ..."
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Cited by 23 (20 self)
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Abstract. We generalise dynamic logic to a logic for differentialalgebraic programs, i.e., discrete programs augmented with firstorder differentialalgebraic formulas as continuous evolution constraints in addition to firstorder discrete jump formulas. These programs characterise interacting discrete and continuous dynamics of hybrid systems elegantly and uniformly. For our logic, we introduce a calculus over real arithmetic with discrete induction and a new differential induction with which differentialalgebraic programs can be verified by exploiting their differential constraints algebraically without having to solve them. We develop the theory of differential induction and differential refinement and analyse their deductive power. As a case study, we present parametric tangential roundabout maneuvers in air traffic control and prove collision avoidance in our calculus.
Moving frames and singularities of prolonged group actions
 Selecta Math. (N.S
"... Abstract. The prolongation of a transformation group to jet bundles forms the geometric foundation underlying Lie’s theory of symmetry groups of differential equations, the theory of differential invariants, and the Cartan theory of moving frames. Recent developments in the moving frame theory have ..."
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Cited by 19 (13 self)
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Abstract. The prolongation of a transformation group to jet bundles forms the geometric foundation underlying Lie’s theory of symmetry groups of differential equations, the theory of differential invariants, and the Cartan theory of moving frames. Recent developments in the moving frame theory have necessitated a detailed understanding of the geometry of prolonged transformation groups. This paper begins with a basic review of moving frames, and then focuses on the study of both regular and singular prolonged group orbits. Highlights include a corrected version of the basic stabilization theorem, a discussion of “totally singular points, ” and geometric and algebraic characterizations of totally singular submanifolds, which are those that admit no moving frame. In addition to applications to the method of moving frames, the paper includes a generalized Wronskian lemma for vectorvalued functions, and methods for the solution to Lie determinant equations.