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The paper considers methods for estimating stability using Lyapunov functions, which are used for nonlinear polynomial control systems. The apparatus of the Gro¨bner basis method is used to assess the stability of a dynamical system. A description of the Gro¨bner basis method is given. To apply the method, the canonical relations of the nonlinear system are approximated by polynomials of the components of the state and control vectors. To calculate the Gro¨bner basis, the Buchberger algorithm is used, which is implemented in symbolic computation programs for solving systems of nonlinear polynomial equations. The use of the Gro¨bner basis for finding solutions of a nonlinear system of polynomial equations is considered, similar to the application of the Gauss method for solving a system of linear equations. The equilibrium states of a nonlinear polynomial system are determined as solutions of a nonlinear system of polynomial equations. An example of determining the equilibrium states of a nonlinear polynomial system using the Gro¨bner basis method is given. An example of finding the critical points of a nonlinear polynomial system using the Gro¨bner basis method and the Wolfram Mathematica application software is given. The Wolfram Mathematica program uses the function of determining the reduced Gro¨bner basis. The application of the Gro¨bner basis method for estimating the attraction domain of a nonlinear dynamic system with respect to the equilibrium point is considered. To determine the scalar potential, the vector field of the dynamic system is decomposed into gradient and vortex components. For the gradient component, the scalar potential and the Lyapunov function in polynomial form are determined by applying the homotopy operator. The use of Gro¨bner bases in the gradient method for finding the Lyapunov function of a nonlinear dynamical system is considered. The coordination of input-output signals of the system based on the construction of Gro¨bner bases is considered.

We consider planar polynomial systems of ordinary differential equations of the form $\dot x = x + P_n(x,y)$, $\dot y = y + Q_n(x,y)$, where $P_n(x,y),\ Q_n(x,y)$ are homogeneous polynomials of degree $n$. We study the algebraic and non-algebraic invariant curves of these systems with emphasis on limit cycles.

This paper study the type of integrability of differential systems with separable variables $\dot{x}=h\left (x\right )f\left (y\right )$ , $\dot{y}= g\left (y\right )$ , where $h$ , $f$ and $g$ are polynomials. We provide a criterion for the existence of generalized analytic first integrals of such differential systems. Moreover we characterize the polynomial integrability of all such systems. In the particular case $h\left (x\right )=\left (ax+b\right )^{m}$ we provide necessary and sufficient conditions in order that this subclass of systems has a generalized analytic first integral. These results extend known results from Gine et al. (Discrete Contin. Dyn. Syst. 33:4531–4547, 2013) and Llibre and Valls (Discrete Contin. Dyn. Syst., Ser. B 20:2657–2661, 2015). Such differential systems of separable variables are important due to the fact that after a blow-up change of variables any planar quasi-homogeneous polynomial differential system can be transformed into a special differential system of separable variables $\dot{x}=xf\left (y\right )$ , $\dot{y}=g\left (y\right )$ , with $f$ and $g$ polynomials.

This paper deals with properties of the algebraic variety defined as the set of zeros of a "typical" sequence of polynomials. We consider various types of "nice" varieties: set-theoretic and ideal-theoretic complete intersections, absolutely irreducible ones, and nonsingular ones. For these types, we present a nonzero "obstruction" polynomial of explicitly bounded degree in the coefficients of the sequence that vanishes if its variety is not of the type. Over finite fields, this yields bounds on the number of such sequences. We also show that most sequences (of at least two polynomials) define a degenerate variety, namely an absolutely irreducible nonsingular hypersurface in some linear projective subspace., 31 pages

International audience; This paper presents an efficient algorithm called SolveFuzzySystem, or SFS, for finding real solutions of polynomial systems whose coefficients are symmetrical L-R fuzzy numbers with bounded support for which the spread functions L and R are bijective. The real solutions of such a system are deduced from solutions of some polynomial systems with real coefficients. This algorithm is based on new results that are universal because they are independent from the spread functions. These theoretical results include the management of the fuzzy system's solutions signs. An implementation in the Fuzzy package of the free computer algebra software SageMath and a parallel version of the algorithm are described.

International audience; In this paper we introduce a general class of problems originating from gearbox vibration analysis. Based on a previous work where demodulation was formulated as a matrix approximation problem, we study the specific case applicable to amplitude and phase demodulation. This problem can be rewritten as a polynomial system. Based on algebraic methods such as linear algebra and homological algebra, we focus on the characterization of the problem and solve it in the noise-free case.

We improve on the first fall degree bound of polynomial systems that arise from a Weil descent along Semaev's summation polynomials relevant to the solution of the Elliptic Curve Discrete Logarithm Problem via Gr\"obner basis algorithms., Comment: 12 pages, final

We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C . Normal form algorithms provide an algebraic approach to solve this problem. The framework presented in Telen et al. (2018) uses truncated normal forms (TNFs) to compute the algebra structure of R / I and the solutions of I. This framework allows for the use of much more general bases than the standard monomials for R / I . This is exploited in this paper to introduce the use of two special (non-monomial) types of basis functions with nice properties. This allows us, for instance, to adapt the basis functions to the expected location of the roots of I. We also propose algorithms for efficient computation of TNFs and a generalization of the construction of TNFs in the case of non-generic zero-dimensional systems. The potential of the TNF method and usefulness of the new results are exposed by many experiments.

This paper proposes a method to solve nonlinear finite-horizon optimal control problems of discrete-time polynomial systems with polynomial terminal constraints. Algebraic equations with all variables at each time step, which are independent of variables at other time steps, are derived from the necessary conditions for optimality by eliminating variables recursively. The candidates of the optimal solution are obtained by solving these equations, and algorithms to find all of these candidates are also proposed. Because of its structure, the proposed method is suitable for nonlinear model predictive control that needs only the initial optimal control law. A simple example to illustrate the methodology and a practical example with the nonlinear model predictive control framework are provided.

International audience; Finding solutions for fuzzy polynomial systems has recently received much attention and many efforts have been made to make the available algorithms for solving such problems more and more efficient. In the present paper, Wu’s algorithm is introduced as a solution procedure to obtain fuzzy polynomial systems solutions. In this approach, the parametric form of the problem is first obtained from the computation of r -cuts of fuzzy polynomials. Wu’s algorithm is then applied in order to convert the parametric form of a fuzzy polynomial system into a finite number of characteristic sets. We then have the right relation between solutions of these sets and those of the polynomial system. The most outstanding advantage of the proposed method lies in the fact that it leads to solve triangular sets amenable to easy solution.

Numerical data structures for positive dimensional solution sets of polynomial systems are sets of generic points cut out by random planes of complimentary dimension. We may represent the linear spaces defined by those planes either by explicit linear equations or in parametric form. These descriptions are respectively called extrinsic and intrinsic representations. While intrinsic representations lower the cost of the linear algebra operations, we observe worse condition numbers. In this paper we describe the local adaptation of intrinsic coordinates to improve the numerical conditioning of sampling algebraic sets. Local intrinsic coordinates also lead to a better stepsize control. We illustrate our results with Maple experiments and computations with PHCpack on some benchmark polynomial systems., Comment: 13 pages, 2 figures, 2 algorithms, 2 tables

International audience; We study the isochronicity of centers at $O\in \mathbb{R}^2$ for systems $$\dot x=-y+A(x,y),\;\dot y=x+B(x,y),$$ where $A,\;B\in \mathbb{R}[x,y]$, which can be reduced to the Liénard type equation. When $deg(A)\leq 4$ and $deg(B) \leq 4$, using the so-called C-algorithm we found $36$ new families of isochronous centers. When the Urabe function $h=0$ we provide an explicit general formula for linearization. This paper is a direct continuation of [I. Boussaada, A.R. Chouikha, J-M. Strelcyn, Isochronicity conditions for some planar polynomial systems, To appear in Bulletins des Sciences Mathématiques, 2010] but can be read independantly.

The output frequency response function (OFRF) of Volterra systems can be described as a polynomial function of model parameters. However, the analytical determination of the OFRF is very computationally intensive, especially for higher order OFRF. To circumvent this problem, a numerical method can be adopted, provided that a series of simulation or experimental data for this polynomial function are given. In this study, it is theoretically shown that the analytical parametric relationship of OFRF up to any order can be determined accurately by using a simple Least Square method and every specific component of the output spectrum can also be determined explicitly, based on the OFRF's parametric characteristics. Practical techniques to obtain a unique and accurate solution for the Least Square method are discussed. This study provides a fundamental result for the determination of the analytical parametric relationship for this kind of system polynomial functions by using numerical methods.

We describe an algorithm to count the number of distinct real zeros of a polynomial (square) system f. The algorithm performs O(n D kappa(f)) iterations where n is the number of polynomials (as well as the dimension of the ambient space), D is a bound on the polynomials' degree, and kappa(f) is a condition number for the system. Each iteration uses an exponential number of operations. The algorithm uses finite-precision arithmetic and a polynomial bound for the precision required to ensure the returned output is correct is exhibited. This bound is a major feature of our algorithm since it is in contrast with the exponential precision required by the existing (symbolic) algorithms for counting real zeros. The algorithm parallelizes well in the sense that each iteration can be computed in parallel polynomial time with an exponential number of processors., Comment: We made minor but necessary improvements in the presentation

This paper presents a discrete-time control design methodology for input-saturating systems using a Lyapunov function with dependence on present and past states. The approach is used to bypass the usual difficulty with full polynomial Lyapunov functions of expressing the problem in a convex way. Also polynomial controllers are allowed to depend on both present and past states. Furthermore, by considering saturation limits on the control action, the information about the relationship between the present and past states is introduced via Positivstellensatz multipliers. Sum-of-squares techniques and available semi-definite programming (SDP) software are used in order to find the controller., The research work by J.L. Pitarch and A. Sala has been partially supported by the Spanish government under research project [grant number DPI2011-27845-C02-01 (MINECO)]; Generalitat Valenciana [grant number PROMETEOII/2013/004]. The work by T.M. Guerra and J. Lauber has been supported by the International Campus on Safety and Intermodality in Transportation, the European Community, Delegation Regionale a la Recherche et a la Technologie, Ministere de l'Enseignement superieur et de la Recherche, Region Nord Pas de Calais and the Centre National de la Recherche Scientifique.

International audience; Let f ∈ Q[X1,. .. , Xn] be a polynomial of degree D. We consider the problem of computing the real dimension of the real algebraic set defined by f = 0. Such a problem can be reduced to quanti-fier elimination. Hence it can be tackled with Cylindrical Algebraic Decomposition within a complexity that is doubly exponential in the number of variables. More recently, denoting by d the dimension of the real algebraic set under study, deterministic algorithms running in time D O(d(n−d)) have been proposed. However, no implementation reflecting this complexity gain has been obtained and the constant in the exponent remains unspecified. We design a probabilistic algorithm which runs in time which is essentially cubic in D d(n−d). Our algorithm takes advantage of gener-icity properties of polar varieties to avoid computationally difficult steps of quantifier elimination. We also report on a first implementation. It tackles examples that are out of reach of the state-of-the-art and its practical behavior reflects the complexity gain.

One of the main problems dealing with iterative methods for solving polynomial systemsis the initialization of the iteration. This paper provides an algorithm to initialize the search of solutions of polynomial systems.

The paper deals with the problem of finding sparse solutions to systems of polynomial equations possibly perturbed by noise. In particular, we show how these solutions can be recovered from group-sparse solutions of a derived system of linear equations. Then, two approaches are considered to find these group-sparse solutions. The first one is based on a convex relaxation resulting in a second-order cone programming formulation which can benefit from efficient reweighting techniques for sparsity enhancement. For this approach, sufficient conditions for the exact recovery of the sparsest solution to the polynomial system are derived in the noiseless setting, while stable recovery results are obtained for the noisy case. Though lacking a similar analysis, the second approach provides a more computationally efficient algorithm based on a greedy strategy adding the groups one-by-one. With respect to previous work, the proposed methods recover the sparsest solution in a very short computing time while remaining at least as accurate in terms of the probability of success. This probability is empirically analyzed to emphasize the relationship between the ability of the methods to solve the polynomial system and the sparsity of the solution., Comment: Journal of Global Optimization (2014) to appear

The main objective of this paper is to describe a class of polynomial vector fields f, whose associated dynamic system has one or more affine varieties as f-invariant and attractive sets. This result can be used for robot motion planning, thus computing robot paths, avoiding collisions with obstacles and reaching a target point.

The integrability conditions for certain systems of differential equations χ = P(χ, y), y = Q(χ, y) , where P and Q are cubic polynomials, are obtained using two distinct computational approaches. Such conditions are required in the investigation of Hilbert's 16 th problem and in the development of algorithms for the automatic solution of differential equations.

Stated in a few word, the aim of this paper is to give evidence to the following author's conjecture: ‘any’ nonlinear control system is equivalent to a (larger dimensioned) bilinear fractional differential system. The main purpose is to motivate a new approach in nonlinear control, and for this reason, in this paper, some simple examples, nevertheless yet meaningful, are given, where the above conjecture holds. Starting with a simple scalar example, in order to present the basic feature of the method, the paper is endowed with a case consisting in a two dimensional control system, which is nevertheless amenable to be readily generalized to a general state-space dimension. A subresult of this paper is interesting by itself: for classical polynomial systems, where just positive integers powers are involved, the result holds always, and the equivalent system result in an ordinary (non-fractional) bilinear system.

The problem of local linearizability of the planar linear center perturbed by cubic non- linearities in all generalities on the system parameters (14 parameters) is far from being solved. The synchronization problem (as noted in Pikovsky, A., Rosenblum, M., and Kurths, J., 2003, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series, Cambridge University Press, UK, and Blekhman, I. I., 1988, Synchronisation in Science and Technology, ASME Press Translations, New York) consists in bringing appropriate modifications on a given system to obtain a desired dynamic. The desired phase portrait along this paper contains a compact region around a singular point at the origin in which lie periodic orbits with the same period (independently from the chosen initial conditions). In this paper, starting from a five parameters non isochronous Chouikha cubic system (Chouikha, A. R., 2007, “Isochronous Centers of Lienard Type Equations and Applications,” J. Math. Anal. Appl., 331, pp. 358–376) we identify all possible monomial perturbations of degree d ∈ {2, 3} insuring local linearizability of the perturbed system. The necessary conditions are obtained by the Normal Forms method. These conditions are real algebraic equations (multivariate polynomials) in the parameters of the studied ordinary differential system. The efficient algorithm FGb (J. C. Faugère, “FGb Salsa Software,” http://fgbrs.lip6.fr) for computing the Gröbner basis is used. For the family studied in this paper, an exhaustive list of possible parameters values insuring local linearizability is established. All the found cases are already known in the literature but the contexts are different since our object is the synchronisation rather than the classification. This paper can be seen as a direct continuation of several new works concerned with the hinting of cubic isochronous centers, (in particular Bardet, M., and Boussaada, I., 2011, “Compexity Reduction of C-algorithm,” App. Math. Comp., in press; Boussaada, I., Chouikha, A. R., and Strelcyn, J.-M., 2011, “Isochronicity Conditions for some Planar Polynomial Systems,” Bull. Sci. Math, 135(1), pp. 89–112; Bardet, M., Boussaada, I., Chouikha, A. R., and Strelcyn, J.-M., 2011, “Isochronicity Conditions for some Planar Polynomial Systems,” Bull. Sci. Math, 135(2), pp. 230–249; and furthermore, it can be considered as an adaptation of a qualitative theory method to a synchronization problem.

Since analysis and simulation of biological phenomena require the availability of their fully specified models, one needs to be able to estimate unknown parameter values of the models. In this paper we deal with identifiability of parametrizations which is the property of one-to-one correspondence of parameter values and the corresponding outputs of the models. Verification of identifiability of a parametrization precedes estimation of numerical values of parameters, and thus determination of a fully specified model of a considered phenomenon. We derive necessary and sufficient conditions for the parametrizations of polynomial and rational systems to be structurally or globally identifiable. The results are applied to investigate the identifiability properties of the system modeling a chain of two enzyme-catalyzed irreversible reactions. The other examples deal with the phenomena modeled by using Michaelis–Menten kinetics and the model of a peptide chain elongation.

Colloque avec actes et comité de lecture. internationale.; International audience; Let $f_1, \ldots, f_s$ be polynomials in $\Q[X_1, \ldots, X_n]$ that generate a radical ideal and let $V$ be their complex zero-set. Suppose that $V$ is smooth and equidimensional; then we show that computing suitable sections of the polar varieties associated to generic projections of $V$ gives at least one point in each connected component of $V\cap\R^n$. We deduce an algorithm that extends that of Bank, Giusti, Heintz and Mbakop to non-compact situations. Its arithmetic complexity is polynomial in the complexity of evaluation of the input system, an intrinsic algebraic quantity and a combinatorial quantity.

In the last two decades, the homotopy continuation method has been developed into a reliable and ecient numerical algorithm for solving all isolated zeros of polynomial systems. During the last few years, a major computational breakthrough has emerged in the area. Based on the Bernshtein theory on root count, the polyhedral homotopy is established to considerably reduce the number of homotopy paths that need to be traced to nd all the isolated roots, making the method much more powerful. This article reports the most recent development of this new method along with future considerations.

Some classes of polynomial control systems are investigated and characterized in terms of their input-output behavior. The first class are those that can be realized as a set of linear state equations with a polynomial output map. The second class are those that can be realized as a set of homogeneous (w.r.t. some grading) state equations with homogeneous output map. >

An algorithm for finding all N-periodic solutions to systems of nonlinear difference equations is given. The algorithm is based on elimination theory, and in special Grobner bases. It is applicable to polynomial systems and can handle indeterminate parameters in the system equations. The system is not required to be in state space form; implicit equations are allowed. >

The purpose of this note is to re-establish, with a new and simple proof, a theorem which in fact represents the main results of [1].

In this paper, robust semi-definite programs are considered with the goal of verifying whether a particular LMI relaxation is exact. A procedure is presented showing that verifying exactness amounts to solving a polynomial system. The main contribution of the paper is a new algorithm to compute all isolated solutions of a system of polynomials. Standard techniques in computational algebra, often referred to as Stetter’s method [H.J. Stetter, Numerical Polynomial Algebra, SIAM, 2004], involve the computation of a Grobner basis of the ideal generated by the polynomials and further require joint eigenvector computations in order to arrive at the zeros of the polynomial system. Our algorithm does neither require structural knowledge on the polynomial system, nor does it rely on the computation of joint eigenvectors.

We study the isochronicity of centers at $O\in \mathbb{R}^2$ for systems $\dot x=-y+A(x,y), \dot y=x+B(x,y)$, where $A, B\in \mathbb{R}[x,y]$, which can be reduced to the Lienard type equation. Using the so-called C-algorithm we have found 27 new multiparameter isochronous centers., 31 pages