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We present two algebraic methods to solve the parametric optimization problem that arises in nonlinear model predictive control. We consider constrained discrete-time polynomial systems and the corresponding constrained finite-time optimal control problem. The first method is based on cylindrical algebraic decomposition. The second uses Gröbner bases and… (More)

Semidefinite characterization and computation of real radical ideals – p.1

We provide a real algebraic symbolic-numeric algorithm for computing the real variety V R (I) of an ideal I ⊆ R[x], assuming V R (I) is finite (while V C (I) could be infinite). Our approach uses sets of linear functionals on R[x], vanishing on a given set of polynomials generating I and their prolongations up to a given degree, as well as on polynomials of… (More)

For an ideal I ⊆ R[x] given by a set of generators, a new semidefi-nite characterization of its real radical I(V R (I)) is presented, provided it is zero-dimensional (even if I is not). Moreover we propose an algorithm using numerical linear algebra and semidefinite optimization techniques, to compute all (finitely many) points of the real variety V R (I)… (More)

A new technique for solving polynomial nonlinear constrained optimal control problems is presented. The problem is reformulated into a parametric optimization problem, which in turn is solved in a two step procedure. First, in a pre-computation step, the equation part of the corresponding first order optimality conditions is solved for a generic value of… (More)

In this paper, we describe new methods to compute the radical (resp. real radical) of an ideal, assuming it complex (resp. real) variety is finte. The aim is to combine approaches for solving a system of polynomial equations with dual methods which involve moment matrices and semi-definite programming. While the border basis algorithms of [17] are efficient… (More)

In this paper we propose a unified methodology for computing the set V K (I) of complex (K = C) or real (K = R) roots of an ideal I ⊆ R[x], assuming V K (I) is finite. We show how moment matrices, defined in terms of a given set of generators of the ideal I, can be used to (numerically) find not only the real variety V R (I), as shown in the authors'… (More)

Polynomial root finding is a classical and well studied problem in mathematics with practical applications in many areas. Various relevant questions can be reformulated into the task of finding all common roots of a given system of polynomials in several variables, the so-called algebraic variety or zero set. Traditionally, root finding algorithms are… (More)