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| 1 | Solution to the Generalized Champagne Problem on simultaneous stabilization of linear systems显示文摘The well-known Generalized Champagne Problem on simultaneous stabilization of linear systems is solved by using complex analysis and Blondel’s technique. We give a complete answer to the open problem proposed by Patel et al., which auto-matically includes the solution to the original Champagne Problem. Based on the recent development in automated inequality-type theorem proving, a new stabiliz-ing controller design method is established. Our numerical examples significantly improve the relevant results in the literature. | GUAN Qiang WANG Long XIA BiCan YANG Lu YU WenSheng ZENG ZhenBing | 2007 | Science in China(Series F)2007，50，5： | 4 |
| 2 | Generic Regular Decompositions for Parametric Polynomial Systems显示文摘This paper presents a generalization of the authors' earlier work. In this paper, the two concepts, generic regular decomposition(GRD) and regular-decomposition-unstable(RDU) variety introduced in the authors' previous work for generic zero-dimensional systems, are extended to the case where the parametric systems are not necessarily zero-dimensional. An algorithm is provided to compute GRDs and the associated RDU varieties of parametric systems simultaneously on the basis of the algorithm for generic zero-dimensional systems proposed in the authors' previous work. Then the solutions of any parametric system can be represented by the solutions of finitely many regular systems and the decomposition is stable at any parameter value in the complement of the associated RDU variety of the parameter space. The related definitions and the results presented in the authors' previous work are also generalized and a further discussion on RDU varieties is given from an experimental point of view. The new algorithm has been implemented on the basis of DISCOVERER with Maple 16 and experimented with a number of benchmarks from the literature. | CHEN Zhenghong TANG Xiaoxian XIA Bican | 2015 | Journal of Systems Engineering and Electronics2015，26，5： | 2 |
| 3 | Generating Semi-Algebraic Invariants for Non-Autonomous Polynomial Hybrid Systems显示文摘Hybrid systems are dynamical systems with interacting discrete computation and continuous physical processes, which have become more common, more indispensable, and more complicated in our modern life. Particularly, many of them are safety-critical, and therefore are required to meet a critical safety standard. Invariant generation plays a central role in the verification and synthesis of hybrid systems. In the previous work, the fourth author and his coauthors gave a necessary and sufficient condition for a semi-algebraic set being an invariant of a polynomial autonomous dynamical system, which gave a confirmative answer to the open problem. In addition, based on which a complete algorithm for generating all semi-algebraic invariants of a given polynomial autonomous hybrid system with the given shape was proposed. This paper considers how to extend their work to non-autonomous dynamical and hybrid systems. Non-autonomous dynamical and hybrid systems are with inputs, which are very common in practice; in contrast, autonomous ones are without inputs. Furthermore, the authors present a sound and complete algorithm to verify semi-algebraic invariants for non-autonomous polynomial hybrid systems. Based on which, the authors propose a sound and complete algorithm to generate all invariants with a pre-defined template. | WANG Qiuye LI Yangjia XIA Bican ZHAN Naijun | 2017 | Journal of Systems Science & Complexity2017，30，1： | 1 |
| 4 | A Hybrid Procedure for Finding Real Points on a Real Algebraic Set显示文摘Motivated by the idea of Shen, et al.'s work, which proposed a hybrid procedure for real root isolation of polynomial equations based on homotopy continuation methods and interval analysis,this paper presents a hybrid procedure to compute sample points on each connected component of a real algebraic set by combining a special homotopy method and interval analysis with a better estimate on initial intervals. For a real algebraic set given by a polynomial system, the new method ?rst constructs a square polynomial system which represents the sample points, and then solve this system by a special homotopy continuation method introduced recently by Wang, et al.(2017). For each root returned by the homotopy continuation method, which is a complex approximation of some(complex/real) root of the polynomial system, interval analysis is used to verify whether it is an approximation of a real root and ?nally get real points on the given real algebraic set. A new estimate on initial intervals is presented which helps compute smaller initial intervals before performing interval iteration and thus saves computation. Experiments show that the new method works pretty well on tested examples. | WANG Yu XIA Bican | 2019 | Journal of Systems Science & Complexity2019，32，1： | 1 |
| 5 | Symbolic decision procedure for termination of linear programs显示文摘 | Bican Xia Lu Yang Naijun Zhan | 2011 | Formal Aspects of Computing2011，23，2： | 1 |
| 6 | Logcf: An Efficient Tool for Real Root Isolation显示文摘Computing upper bounds of the positive real roots of some polynomials is a key step of those real root isolation algorithms based on continued fraction expansion and Vincent's theorem.The authors give a new algorithm for computing an upper bound of positive roots in this paper.The complexity of the algorithm is O(n log(uH-l))additions and multiplications where u is the optimal upper bound satisfying Theorem 3.1 of this paper and n is the degree of the polynomial.The method together w辻h some tricks have been implemented as a software package logcf using C language.Experiments on many benchmarks show that logcf is competitive with Root Intervals of Mathematica and the function realroot of Maple averagely and it is much faster than existing open source real root solvers in many test cases. | DAI Liyun FAN Zhe XIA Bican ZHANG Hanwen | 2019 | Journal of Systems Science & Complexity2019，32，6： | 1 |
| 7 | An algorithm for isolating the real so- lutions of semi-algebraic systems 显示文摘 | Xia Bican Yang Lu | 2002 | J Symb Comput2002，34，5： | 1 |
| 8 | Recent advances in program verification through computer algebra显示文摘 | Lu Yang Chaochen Zhou Naijun Zhan Bican Xia | 2010 | Frontiers of Computer Science in China2010，，1： | 1 |
| 9 | Real solution isolation using inter- val arithmetic 显示文摘 | Xia Bican Zhang Ting | 2006 | Computers and Mathematics with Appli- cations2006，52，67： | 1 |
| 10 | Real solution isolation with multiplicity of zero-dimensional triangular systems显示文摘Existing algorithms for isolating real solutions of zero-dimensional polynomial systems do not compute the multiplicities of the solutions.In this paper,we define in a natural way the multiplicity of solutions of zero-dimensional triangular polynomial systems and prove that our definition is equivalent to the classical definition of local(intersection)multiplicity.Then we present an effective and complete algorithm for isolating real solutions with multiplicities of zero-dimensional triangular polynomial systems using our definition. The algorithm is based on interval arithmetic and square-free factorization of polynomials with real algebraic coeffcients.The computational results on some examples from the literature are presented. | ZHANG ZhiHai FANG Tian XIA BiCan | 2011 | Science China(Information Sciences)2011，54，1： | 1 |
| 11 | Generic regular decompositions for generic zero-dimensional systems显示文摘Two new concepts,generic regular decomposition and regular-decomposition-unstable(RDU)variety for generic zero-dimensional systems,are introduced in this paper and an algorithm is proposed for computing a generic regular decomposition and the associated RDU variety of a given generic zero-dimensional system simultaneously.The solutions of the given system can be expressed by finitely many zero-dimensional regular chains if the parameter value is not on the RDU variety.The so called weakly relatively simplicial decomposition plays a crucial role in the algorithm,which is based on the theories of subresultants.Furthermore,the algorithm can be naturally adopted to compute a non-redundant Wu’s decomposition and the decomposition is stable at any parameter value that is not on the RDU variety.The algorithm has been implemented with Maple 16 and experimented with a number of benchmarks from the literature.Empirical results are also presented to show the good performance of the algorithm. | TANG XiaoXian CHEN ZhengHong XIA BiCan | 2014 | Science China(Information Sciences)2014，57，9： | 1 |
| 12 | Termination of linear programs with nonlinear constraints 显示文摘 | Xia Bican Zhang Zhihai | 2010 | Journal of Symbolic Computation2010，45，： | 1 |
| 13 | Termination of linear programs with nonlinear constraints显示文摘 | Bican Xia Zhihai Zhang | 2010 | Journal of Symbolic Computation2010，，11： | 1 |
| 14 | Symbolic decision procedure for termination of linear programs显示文摘 | Xia Bican Yang Lu Zhan Naijun | 2011 | Formal Aspects of Computing2011，23，2： | 1 |
| 15 | Square-Free Pure Triangular Decomposition of Zero-Dimensional Polynomial Systems显示文摘Triangular decomposition with different properties has been used for various types of problem solving.In this paper,the concepts of pure chains and square-free pure triangular decomposition(SFPTD)of zero-dimensional polynomial systems are defined.Because of its good properties,SFPTD may be a key way to many problems related to zero-dimensional polynomial systems.Inspired by the work of Wang(2016)and of Dong and Mou(2019),the authors propose an algorithm for computing SFPTD based on Gr¨obner bases computation.The novelty of the algorithm is that the authors make use of saturated ideals and separant to ensure that the zero sets of any two pure chains are disjoint and every pure chain is square-free,respectively.On one hand,the authors prove the arithmetic complexity of the new algorithm can be single exponential in the square of the number of variables,which seems to be among the rare complexity analysis results for triangular-decomposition methods.On the other hand,the authors show experimentally that,on a large number of examples in the literature,the new algorithm is far more efficient than a popular triangular-decomposition method based on pseudodivision,and the methods based on SFPTD for real solution isolation and for computing radicals of zero-dimensional ideals are very efficient. | LI Haokun XIA Bican ZHAO Tianqi | 2023 | Journal of Systems Science & Complexity2023，36，6： | 0 |
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