Semi-algebraic Sets Research Articles

Polynomial Level-Set Method for Polynomial System Reachable Set Estimation

In this paper, we present a polynomial level-set method for advecting a semi-algebraic set for polynomial systems. This method uses the sub-level representation of sets. The problem of flowing these sets under the advection map of a dynamical system is converted to a semi-definite program, which is then used to compute the coefficients of the polynomials. The method presented in this paper does not require either the sets being positively invariant or star-shaped. Hence, the proposed algorithm can describe the behavior of system states both inside and outside the domain of attraction and can also be used to describe more complex shapes of sets. We further address the related problems of constraining the degree of the polynomials. Various numerical examples are presented to show the effectiveness of advection approach.

Borel measures with a density on a compact semi-algebraic set

Let \({\mathbf{K} \subset \mathbb{R}^n}\) be a compact basic semi-algebraic set. We provide a necessary and sufficient condition (with no a priori bounding parameter) for a real sequence y = (yα), \({\alpha \in \mathbb{N}^n}\) , to have a finite representing Borel measure absolutely continuous w.r.t. the Lebesgue measure on K, and with a density in \({\cap_{p \geq 1} L_p(\mathbf{K})}\) . With an additional condition involving a bounding parameter, the condition is necessary and sufficient for the existence of a density in L∞(K). Moreover, nonexistence of such a density can be detected by solving finitely many of a hierarchy of semidefinite programs. In particular, if the semidefinite program at step d of the hierarchy has no solution, then the sequence cannot have a representing measure on K with a density in Lp(K) for any p ≥ 2d.

UNMANNED VEHICLE OBSTACLE DETECTION AND AVOIDANCE USING DANGER ZONE APPROACH

This paper proposes an obstacle avoidance algorithm for unmanned vehicles in unknown environment. The vehicle uses an ultrasonic sensor and a servo motor which rotates from 0 to 180 degrees to obtain the distance data, and the profile of the obstacle. In this avoidance algorithm we will use the danger zone concept to judge whether the obstacle will cause a possible collision. The danger zone concept surrounds the vehicle through the intersection of semi-algebraic sets. These semi-algebraic sets use the relative velocity of the obstacle to calculate the area in which obstacles will collide with the vehicle within a pre-specified time period. Combining the profile of the boundary of the obstacle with the danger zone concept, a method for determining the safe maneuvers to avoid collisions is also provided.

On orbit types of semialgebraically proper actions

In this paper we consider semialgebraically proper actions of semialgebraic groups on semialgebraic sets. Let G be a semialgebraic group. We prove that every semialgebraically proper G-set has only finitely many orbit types.

Discovering polynomial Lyapunov functions for continuous dynamical systems

In this paper we analyze locally asymptotic stability of polynomial dynamical systems by discovering local Lyapunov functions beyond quadratic forms. We first derive an algebraizable sufficient condition for the existence of a polynomial Lyapunov function. Then we apply a real root classification based method step by step to under-approximate this derived condition as a semi-algebraic system such that the semi-algebraic system only involves the coefficients of the pre-assumed polynomial. Afterward, we compute a sample point in the corresponding semi-algebraic set for the coefficients resulting in a local Lyapunov function. Moreover, we test our approach on some examples using a prototype implementation and compare it with the generic quantifier elimination based method and the sum of squares based method. These computation and comparison results show the applicability and efficiency of our approach.

THE EQUIVALENCE CONDITIONS FOR SEMIALGEBRAICALLY PROPER MAPS

In this paper we compare the notion of proper map in the category of topological spaces with that in the category of semialgebraic sets. To do this, we find some equivalence conditions for semialgebraically proper maps. In particular, we prove that a continuous semialgebraic map is semialgebraically proper if and only if it is proper. Moreover, we compare the semialgebraically proper map with the proper map in the sense of Delfs and Knebush [4].

Guaranteed cost [formula omitted] controller synthesis for switched systems defined on semi-algebraic sets

A methodology to design guaranteed cost H∞ controllers for a class of switched systems with polynomial vector fields is proposed. To this end, we use sum of squares programming techniques. In addition, instead of the customary Carathéodory solutions, the analysis is performed in the framework of Filippov solutions which subsumes solutions with infinite switching in finite time and sliding modes. Firstly, conditions assuring asymptotic stability of Filippov solutions pertained to a switched system defined on semi-algebraic sets are formulated. Accordingly, we derive a set of sum of squares feasibility tests leading to a stabilizing switching controller. Finally, we propose a scheme to synthesize stabilizing switching controllers with a guaranteed cost H∞ disturbance attenuation performance. The applicability of the proposed methods is elucidated thorough simulation analysis.

Algebraic Boundaries of $$\mathrm{SO}(2)$$ SO ( 2 ) -Orbitopes

Let $$X\subset \mathbb{A }^{2r}$$XźA2r be a real curve embedded into an even-dimensional affine space. We characterise when the $$r$$rth secant variety to $$X$$X is an irreducible component of the algebraic boundary of the convex hull of the real points $$X(\mathbb{R })$$X(R) of $$X$$X. This fact is then applied to $$4$$4-dimensional $$\mathrm{SO}(2)$$SO(2)-orbitopes and to the so called Barvinok---Novik orbitopes to study when they are basic closed semi-algebraic sets. In the case of $$4$$4-dimensional $$\mathrm{SO}(2)$$SO(2)-orbitopes, we find all irreducible components of their algebraic boundary.

The Łojasiewicz–Siciak Condition of the Pluricomplex Green Function

The aim of this paper is to address a problem raised originally by L. Gendre, later by W. Pleśniak and recently by L. Bialas–Ciez and M. Kosek. This problem concerns the pluricomplex Green function and consists in finding new examples of sets with so–called Łojasiewicz–Siciak ((ŁS) for short) property. So far, the known examples of such sets are rather of particular nature. We prove that each compact subset of ℝ N , treated as a subset of ℂ N , satisfies the Łojasiewicz–Siciak condition. We also give a sufficient geometric criterion for a semialgebraic set in ℝ2, but treated as a subset of ℂ, to satisfy this condition. This criterion applies more generally to a set in ℂ definable in a polynomially bounded o–minimal structure.

Real ideal and the duality of semidefinite programming for polynomial optimization

We study the ideal generated by polynomials vanishing on a semialgebraic set and give elementary proofs for some equivalent conditions for reality of ideals or S-radical ideals. These results can be applied for modifying polynomial optimization problems so that the associated semidefinite programming relaxation problems have no duality gap.

Curves testing boundedness of polynomials on subsets of the real plane

Let S⊂R2 be a semialgebraic set. We exhibit a family of semialgebraic plane curves Γc, c⩾0, such that a polynomial f∈R[X,Y] is bounded on S if and only if it is bounded on a finite number of curves from this family. This number depends on S and degf. More precisely, each Γc is a sum of at most l continuous semialgebraic curves Γic, each parametrized by a Puiseux polynomial, where the number l and the family of curves Γic depend on the set S only. To this aim we describe the algebras of bounded polynomials on tentacles of the set S which determine the algebra of polynomials bounded on S.

Rational Lyapunov functions for estimating and controlling the robust domain of attraction

This paper addresses the estimation and control of the robust domain of attraction (RDA) of equilibrium points through rational Lyapunov functions (LFs) and sum of squares (SOS) techniques. Specifically, continuous-time uncertain polynomial systems are considered, where the uncertainty is represented by a vector that affects polynomially the system and is constrained into a semialgebraic set. The estimation problem consists of computing the largest estimate of the RDA (LERDA) provided by a given rational LF. The control problem consists of computing a polynomial static output controller of given degree for maximizing such a LERDA. In particular, the paper shows that the computation of the best lower bound of the LERDA for chosen degrees of the SOS polynomials, which requires the solution of a nonconvex optimization problem with bilinear matrix inequalities (BMIs), can be reformulated as a quasi-convex optimization problem under some conditions. Moreover, the paper provides a necessary and sufficient condition for establishing tightness of this lower bound. Lastly, the paper discusses the search for optimal rational LFs using the proposed strategy.

Robust Consensus for a Class of Uncertain Multi-Agent Dynamical Systems

This paper investigates robust consensus for a class of uncertain multi-agent dynamical systems. Specifically, it is supposed that the system is described by a weighted adjacency matrix whose entries are polynomial functions of an uncertain vector constrained in a semi-algebraic set. For this uncertain topology, we provide necessary and sufficient conditions for ensuring robust first-order consensus and robust second-order consensus, in both cases of positive and non-positive weighted adjacency matrices. Moreover, we show how these conditions can be investigated through convex programming by using standard software. Some numerical examples illustrate the proposed results.

Dimensions, lengths, and separability in finite-dimensional quantum systems

Many important sets of normalized states in a multipartite quantum system of finite dimension d, such as the set \documentclass[12pt]{minimal}\begin{document}${\cal S}$\end{document}S of all separable states, are real semialgebraic sets. We compute dimensions of many such sets in several low-dimensional systems. By using dimension arguments, we show that there exist separable states which are not convex combinations of d or less pure product states. For instance, such states exist in bipartite M⊗N systems when (M − 2)(N − 2) > 1. This solves an open problem proposed by DiVincenzo, Terhal and Thapliyal about 12 years ago. We prove that there exist a separable state ρ and a pure product state, whose mixture has smaller length than that of ρ. We show that any real \documentclass[12pt]{minimal}\begin{document}$\rho \in {\cal S}$\end{document}ρ∈S, which is invariant under all partial transpose operations, is a convex sum of real pure product states. In the case of the 2⊗N system, the number r of product states can be taken to be \documentclass[12pt]{minimal}\begin{document}$r=\mathop {\rm rank}\rho$\end{document}r= rank ρ. We also show that the general multipartite separability problem can be reduced to the case of real states. Regarding the separability problem, we propose two conjectures describing \documentclass[12pt]{minimal}\begin{document}${\cal S}$\end{document}S as a semialgebraic set, which may eventually lead to an analytic solution in some low-dimensional systems such as 2⊗4, 3⊗3, and 2⊗2⊗2.

Robust consensus for uncertain multi-agent systems with discrete-time dynamics

This paper investigates robust consensus for multi-agent systems with discrete-time dynamics affected by uncertainty. In particular, the paper considers multi-agent systems with single and double integrators, where the weighted adjacency matrix is a polynomial function of uncertain parameters constrained into a semialgebraic set. Firstly, necessary and sufficient conditions are provided for robust consensus based on the existence of a Lyapunov function polynomially dependent on the uncertainty. In particular, an upper bound on the degree required for achieving necessity is provided. Secondly, a necessary and sufficient condition is provided for robust consensus with single integrator and nonnegative weighted adjacency matrices based on the zeros of a polynomial. Lastly, it is shown how these conditions can be investigated through convex programming by exploiting linear matrix inequalities and sums of squares of polynomials. Some numerical examples illustrate the proposed results. Copyright © 2013 John Wiley & Sons, Ltd.

Integration of semialgebraic functions and integrated Nash functions

When integrating semialgebraic functions one has to leave the semialgebraic setting. For example, one gets the global logarithm and iterated antiderivatives of algebraic power series such as the arctangent. We show that it is enough to enlarge the semialgebraic functions by these functions to completely describe parameterized integrals of semialgebraic functions. To realize this we close the rings of algebraic power series in arbitrary dimension under taking antiderivatives. We analyze these rings profoundly. In particular we show that the Weierstrass division theorem and the Weierstrass preparation theorem hold. This allows us to apply model theoretic results to obtain an explicit description of parameterized integrals of semialgebraic functions. Finally, we investigate the structure generated by the integrated algebraic power series.

On Range Searching with Semialgebraic Sets. II

Let $P$ be a set of $n$ points in $\mathbb{R}^d$. We present a linear-size data structure for answering range queries on $P$ with constant-complexity semialgebraic sets as ranges, in time close to $O(n^{1-1/d})$. It essentially matches the performance of similar structures for simplex range searching, and, for $d\ge 5$, significantly improves earlier solutions by the first two authors obtained in 1994. This almost settles a long-standing open problem in range searching. The data structure is based on a partitioning technique of Guth and Katz [On the Erdös distinct distances problem in the plane, arXiv:1011.4105, 2010], which shows that for a parameter $r$, $1 < r \le n$, there exists a $d$-variate polynomial $f$ of degree $O(r^{1/d})$ such that each connected component of $\mathbb{R}^d\setminus Z(f)$ contains at most $n/r$ points of $P$, where $Z(f)$ is the zero set of $f$. We present an efficient randomized algorithm for computing such a polynomial partition, which is of independent interest and is likely to have additional applications.

Realization Theory of Nash Systems

This paper deals with realization theory of so-called Nash systems, i.e., nonlinear systems the right-hand sides of which are defined by Nash functions. A Nash function is a semialgebraic analytic function. The class of Nash systems is an extension of the class of polynomial and rational systems and it is a subclass of analytic nonlinear systems. Nash systems occur in many applications, including systems biology. Formulation of the realization problem for Nash systems and a partial solution to it are presented. More precisely, necessary and sufficient conditions for realizability of a response map by a Nash system are provided. The concepts of semialgebraic observability and semialgebraic reachability are formulated and their relationship with minimality is explained. In addition to their importance for systems theory, the obtained results are expected to contribute to system identification and model reduction of Nash systems.

A Lagrangian Relaxation View of Linear and Semidefinite Hierarchies

We consider the general polynomial optimization problem $\mathbf{P}: f^*=\min\{f(\mathbf{x}): \mathbf{x}\in\mathbf{K}\}$, where $\mathbf{K}$ is a compact basic semialgebraic set. We first show that the standard Lagrangian relaxation yields a lower bound as close as desired to the global optimum $f^*$, provided that it is applied to a problem $\tilde{\mathbf{P}}$ equivalent to $\mathbf{P}$, in which sufficiently many redundant constraints (products of the initial ones) are added to the initial description of $\mathbf{P}$. Next we show that the standard hierarchy of linear program (LP-)relaxations of $\mathbf{P}$ (in the spirit of the Sherali--Adams reformulation-linearization technique) can be interpreted as a brute force simplification of the above Lagrangian relaxation in which a nonnegative polynomial (with coefficients to be determined) is replaced with a constant polynomial equal to zero. Inspired by this interpretation, we provide a systematic improvement of the LP-hierarchy by doing a much less brutal simplification which results into a parametrized hierarchy of semidefinite programs (and not LPs any more). For each semidefinite program in the parametrized hierarchy, the semidefinite constraint has a fixed size $O(n^k)$, independent of the rank in the hierarchy. When applied to a nontrivial class of convex problems, the first relaxation of the parametrized hierarchy is exact, in contrast with the LP-hierarchy, where convergence cannot be finite.

Optimization on linear matrix inequalities for polynomial systems control

Many problems of systems control theory boil down to solving polynomial equations, polynomial inequalities or polyomial differential equations. Recent advances in convex optimization and real algebraic geometry can be combined to generate approximate solutions in floating point arithmetic.