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Engineering Systems and Free Semi-Algebraic Geometry

  • Mauricio C. De OliveiraEmail author
  • J. William Helton
  • Scott A. Mccullough
  • Mihai Putinar
Chapter
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 149)

Abstract

This article sketches a few of the developments in the recently emerging area of real algebraic geometry (in short RAG) in a free* algebra, in particular on “noncommutative inequalities”. Also we sketch the engineering problems which both motivated them and are expected to provide directions for future developments. The free* algebra is forced on us when we want to manipulate expressions where the unknowns enter naturally as matrices. Conditions requiring positive definite matrices force one to noncommutative inequalities. The theory developed to treat such situations has two main parts, one parallels classical semialgebraic geometry with sums of squares representations (Positivstellensatze) and the other has a new flavor focusing on how noncommutative convexity (similarly, a variety with positive curvature) is very constrained, so few actually exist.

Keywords

Linear Matrix Inequality Positive Semidefinite Free Algebra Symmetric Polynomial Weyl Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Ami57.
    S.A. AMITSUR. A generalization of Hilbert’s Nullstellensatz. Proceedings of the American Mathematical Societu, 8:649~56, 1957.Google Scholar
  2. BDG96.
    C.L. BECK, J.e. DOYLE, AND K. GLOVER. Model reduction of multidimensional and uncertain systems. IEEE Transactions on Automatic Control, 41(10):1406-1477, 1996.Google Scholar
  3. Bec01.
    C.L. BECK. On formal power series representations for uncertain systems. IEEE Transactions on Automatic Control, 46(2):314-319, 200l.Google Scholar
  4. BEGFB94.
    S.P. BOYD, L.EL GHAOUI, E. FERON, AND V. BALAKR1SHNAN. Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia, PA, 1994.Google Scholar
  5. BGM05.
    J.A. BALL, G. GROENEWALD, AND T. MALAKORN. Structured noncommutative multidimensional linear systems. SIAM Journal On Control And Optimization, 44(4):1474-1528, 2005.Google Scholar
  6. BGM06a.
    J.A. BALL, G. GROENEWALD, AND T. MALAKORN. Bounded real lemma for structured noncornmutative multidimensional linear systems and robust control. Multidimensional Systems And Signal Processing, 17(2-3):119-150, July 2006.Google Scholar
  7. BGM06b.
    J.A. BALL, G. GROENEWALD, AND T. MALAKORN. Conservative structured noncommutative multidirnensionallinear systems. In The state space method generalizations and applications, volume 161 of Operator Theory Advances and Applications, pages 179-223. BlrkhauserVerlag, Basel-Boston-Berlin, 2006.Google Scholar
  8. BHN99.
    R.H. BYRD, M.E. HRIBAR, AND J. NOCEDAL. An interior point algorithm for large scale nonlinear programming. SIAM Journal on Optimization, 9(4):877-900, 1999.Google Scholar
  9. BR84.
    J. BERSTEL AND C. REUTENAUER. Rational series and their languages. EATCS Monographs on Theoretical Computer Science. SpringerVerlag, Berlin-New York, 1984.Google Scholar
  10. CHS06.
    J.F. CAMINO, J.W. HELTON, AND R.E. SKELTON. Solving matrix inequalities whose unknowns are matrices. SIAM Journal on Optimization, 17(1):1-36,2006.Google Scholar
  11. CHSY03.
    J.F. CAMINO, J.W. HELTON, R.E. SKELTON, AND J. YEo Matrix inequalities: a symbolic procedure to determine convexity automatically. Integral Equation and Operator Theory, 46(4):399-454, 2003.Google Scholar
  12. Ci.
    J. CIMPRIC Maximal quadratic modules on *-rings. Algebr. Represent. Theory. to appear.Google Scholar
  13. DGKF89.
    J.C. DOYLE, KEITH GLOVER, P.P. KHARGONEKAR, AND B.A. FRANCIS. State-space solutions to standard H2 and Hoo control problems. IEEE Transactions on Automatic Control, 34(8):831-847, 1989.Google Scholar
  14. dOH06a.
    M.C. DE OLIVEIRA AND J.W. HELTON. Computer algebra tailored to matrix inequalities in control. International Journal of Control, 79(11):1382-1400, November 2006.Google Scholar
  15. dOH06b.
    M.G. DE OLIVEIRA AND J.W. HELTON. A symbolic procedure for computing semidefinite program duals. In Proceedings of the 45th IEEE Conference on Decision and Control, pages 5192-5197, San Diego, CA,2006.Google Scholar
  16. GM82.
    P.E. GILL AND W. MURRAY. Practical Optimization. Academic Press, 1982.Google Scholar
  17. Han97.
    F. HANSEN. Operator convex functions of several variables. Publ. RIMS, Kyoto Univ., 33:443-464, 1997.Google Scholar
  18. HdOSM05.
    J.W. HELTON, M.C. DE OLIVEIRA, M. STANKUS, AND R.L. MILLER. NCAlgebra and NCGB, 2005 release edition, 2005. Available at http://math.ucsd.edu/rvncalg.
  19. Hel02.
    J.W. HELTON. ’Positive’ noncommutative polynomials are sums of squares. Annals of Mathematics, 156(2):675--694, September 2002.Google Scholar
  20. HHLM.
    J.W. HELTON, P.C. LIM, AND S.A. MCCULLOUGH. in preparation.Google Scholar
  21. HM03.
    J.W. HELTON AND S. MCCULLOUGH. Convex noncommutative polynomials have degree two or less. SIAM Journal on Matrix Analysis and Applications, 25(4):1124-1139, 2003.Google Scholar
  22. HM04.
    J.W. HELTON AND S.A. MCCULLOUGH. A posi tivstcllensatz for noncommutative polynomials. Transactions of the American Mathematical Society, 356(9):3721-3737, 2004.Google Scholar
  23. HMP04.
    J.W. HELTON, S.A. MCCULLOUGH, AND M. PUTINAR. A noncommutative positivstellensatz on isometries. Journal fur die reine und angewandte Mathematik, 568:71-80, March 2004.Google Scholar
  24. HMP05.
    J.W. HELTON, S.A. MCCULLOUGH, AND M. PUTINAR. Non-negative hereditary polynomials in a free *-algebra. Mathematische Zeitechrift; 250(3):515-522, July 2005.Google Scholar
  25. HMP07.
    J.W. HELTON, S. MCCULLOUGH, AND M. PUTINAR. Strong majorization in a free *-algebra. Mathematische Zeitschrift, 255(3):579-596, March 2007.Google Scholar
  26. HMV06.
    J.W. HELTON, S.A. MCCULLOUGH, AND V. VINNIKOV. Noncommutative convexity arises from linear matrix inequalities. Journal Of Functional Analysis, 240(1):105-191, November 2006.Google Scholar
  27. HP07.
    J.W. HELTON AND M. PUTINAR. Positive polynomials in scalar and mar trix variables, the spectral theorem and optimization; in vol. Operator Theory, Structured Matrices and Dilations. A volume dedicated to the memory ofT. Constantinescu (M. Bakony! et al., eds.), Theta, Bucharest, pp. 229-306, 2007.Google Scholar
  28. HPMV.
    J.W. HELTON, M. PUTINAR, S. MCCULLOUGH, AND V. VINNIKOV. Convex matrix inequalities versus linear matrix inequalities. Preprint.Google Scholar
  29. HT07.
    F. HANSEN AND J. TOMIYAMA. Differential analysis of matrix convex functions. Linear Algebra And Its Applications, 420(1):102-116, January 2007.Google Scholar
  30. Kra36.
    F. KRAUS. Uber konvexe matrixfunctionen. Math. Zeit., 41:18-42, 1936.Google Scholar
  31. KS07.
    I. KLEP AND M. SCHWEIGHOFER. A nichtnegtaivstellensatz for polynomials in nancommuting variables. Israel Journal of Mathematics, 161(1):17-27, 2007.Google Scholar
  32. KS08.
    I. KLEP AND M. SCHWEIGHOFER. Cannes’ embedding conjecture and sums of Hermitian squares. Advances in Mathematics, 217:18161837, 2008. to appear.Google Scholar
  33. KVV07.
    D.S. KALIUZHNYI-VERBOVETSKYI ANDV. VINNIKOV. Singularities of noncommutative rational functions and minimal factorizations, 2007. Preprint.Google Scholar
  34. Las01.
    J.B. LASSERRE. Global optimization with polynomials and the problem of moments. SIAM Journal on Optimization, 11(3):796-817, 200l.Google Scholar
  35. LZD96.
    W.-M Lu, K. ZHOU, AND J.C. DOYLE. Stabilization of uncertain linear systems: an 1ft approach. IEEE Transactions on Automatic Control, 41(1):50-65, 1996.Google Scholar
  36. OST07.
    H. OSAKA, S. SILVESTROV, AND J. TOMIYAMA. Monotone operator functions, gaps and power moment problem. Mathematica Scandinavica, 100(1):161-183, 2007.Google Scholar
  37. ParOO.
    P.A. PARRILO. Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. PhD thesis, California Institute of Technology, Passadena, CA, May 2000.Google Scholar
  38. PDOl.
    A. PRESTEL AND C.N. DELZELL. Positive polynomials. From Hilbert’s 17th problem to real algebra. Springer Monographs in Mathematics. Springer, Berlin, 2001.Google Scholar
  39. PS76.
    C. PROCESI AND M. SCHACHER. A non-commutative real Nullstellensatz and Hilbert’s 17th problem. Ann. of Math., 104(2):395-406, 1976.Google Scholar
  40. SIG98.
    R.E. SKELTON, T. IWASAKI, AND K. GRIGORIADIS. A Unified Algebraic Approach to Control Design. Taylor & Francis, London, UK, 1998.Google Scholar
  41. Uch05.
    M. UCHIYAMA. Operator monotone functions and operator inequalities. Sugaku Expositions, 18(1):39-52, 2005.Google Scholar
  42. VS99.
    R.J. VANDERBEI AND D.F. SHANNO. An interior-point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications, 13;231-252, 1999.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Mauricio C. De Oliveira
    • 1
    Email author
  • J. William Helton
    • 2
  • Scott A. Mccullough
    • 3
  • Mihai Putinar
    • 4
  1. 1.Mechanics and Aerospace Engineering DepartmentUC at San DiegoLa JollaUSA
  2. 2.Department of MathematicsUniversity of California at San DiegoLa JollaUSA
  3. 3.Department of MathematicsUniversity of FloridaGainesvilleUSA
  4. 4.Department of MathematicsUniversity of CaliforniaSanta BarbaraUSA

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