Advertisement

Overdetermined Multidimensional Systems: State Space and Frequency Domain Methods

  • Joseph A. Ball
  • Victor Vinnikov
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 134)

Abstract

We consider 2D input-state-output linear systems where the evolution of the whole state is specified in two independent directions. The requirement that the value of the state at a given point be independent of the path from the origin chosen to arrive at the given point leads to nontrivial consistency conditions: the transient evolutions (i.e., state evolution with zero inputs) should commute, and the input signal (and then also the output signal) should solve a compatibility partial differential equation. We show that many of the standard structural properties (e.g., controllability, observability, minimality, pairing with adjoint system, feedback coupling, equivalence between conservative systems and Lax-Phillips scattering theory) and standard problems (e.g., pole placement, linear-quadratic-regulator problem, H -control problems) for ID linear systems carry over for this setting. There is also a frequency-domain theory for this class of systems: the transfer function is a bundle map between flat vector bundles on a compact Riemann surface, or equivalently, between kernel bundles for determinantal representations of an algebraic curve C embedded in ℂ2 (or rather in the projective plane ℙ 2 ). The transform from the time domain to the frequency domain is implemented by a “Laplace transform along the curve C”. Just as optimal control for continuous-time systems leads to control-theoretic interpretations for Hardy-space function theory on the right half plane, control theory for this class of overdetermined systems leads to control-theoretic interpretations for function theory on a finite bordered Riemann surface. We expect such a mathematically rich theory to have use in control applications yet to be discovered as well as applications beyond the scope of traditional system theory; we mention two such possibilities of the latter type: wave-particle duality in quantum mechanics and a mathematical model for DNA chains.

Keywords

Vector Bundle Riemann Surface Compact Riemann Surface Frequency Domain Analysis Adjoint System 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M.B. Abrahamse, The Pick interpolation theorem for finitely connected domains, Michigan Math. J., 26: 195–203, 1979.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    D. Alpay and V. Vinnikov, Indefinite Hardy spaces on finite bordered Riemann surfaces, J. Functional Analysis, 172: 221–248, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  3. E. Arbarello, M. Arbarello, M. Griffiths, and J. Harris, Geometry of Algebraic Curves: Volume I, Springer-Verlag, New York, 1985.zbMATHGoogle Scholar
  4. [4]
    J.A. Ball and V. Vinnikov, Zero-pole interpolation for meromorphic matrix functions on an algebraic curve and transfer functions of 2D systems. Acta Applicandae Mathematicae, 45: 239–316, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    J.A. Ball and V. Vinnikov, Zero-pole interpolation for matrix meromorphic functions on a compact Riemann surface and a matrix Fay trisecant identity, American J. Math., 121: 841–888, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    J.A. Ball and V. Vinnikov, Multidimensional discrete-time systems, algebraic curves and commuting nonunitary operators, in preparation.Google Scholar
  7. [7]
    J.A. Ball and V. Vinnikov, Hardy spaces on a finite bordered Riemann surface, multivariable operator model theory and Fourier analysis along a unimodular curve, in Systems, Approximation, Singular Integral Operators, and Related Topics, International Workshop on Operator Theory and Applications, IWOTA 2000 (eds. Alexander A. Borichev and Nikolai K. Nikolski), OT129, Birkhäuser, pages 37–56, 2001.CrossRefGoogle Scholar
  8. [8]
    H. Bart, I. Gohberg, and M.A. Kaashoek, Minimal Factorization of Matrix and Operator Functions, OTl, Birkhäuser-Verlag, Basel-Boston, 1979.zbMATHGoogle Scholar
  9. [9]
    N.K. Bose, Applied Multidimensional Systems Theory, Van Nostrand Reinhold, 1982.zbMATHGoogle Scholar
  10. [10]
    F.M. Gallier and C.A. Desoer, Linear System Theory, Springer-Verlag, Berlin-New York, 1991.Google Scholar
  11. [11]
    B.A. Francis, A Course in Hoo Control, LNCIS 88, Springer-Verlag, Berlin-New York, 1987.CrossRefGoogle Scholar
  12. [12]
    W. Fulton, Algebraic Curves: An Introduction to Algebraic Geometry, Benjamin, New York, 1969.Google Scholar
  13. [13]
    M. Green and D.J.N. Limebeer, Linear Robust Control, Prentice Hall, Englewood Cliffs, 1995.zbMATHGoogle Scholar
  14. [14]
    P.A. Griffiths, Introduction to Algebraic Curves, Transl. Math. Monographs 76, Amer. Math. Soc., Providence, 1989.zbMATHGoogle Scholar
  15. [15]
    H. Gauchman, Connection colligations on Hilbert bundles, Integral Equations Operator Theory, 6: 31–58, 1983.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    H. Gauchman, Connection colligations of the second order. Integral Equations Operator Theory, 6: 184–205, 1983.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    F. John, Partial Differential Equations (Fourth Edition), Applied Mathematical Sciences 1, Springer-Verlag, New York-Heidelberg-Berlin, 1982.Google Scholar
  18. [18]
    T. Kaczorek, Two-dimensional Linear Systems, LNCIS #68, Springer-Verlag, Berlin-New York, 1985.zbMATHGoogle Scholar
  19. [19]
    T. Kailath, Linear Systems, Prentice-Hall, Englewood Cliffs, 1980.zbMATHGoogle Scholar
  20. [20]
    N. Kravitsky, Regular colligations for several commuting operators in Banach space. Integral Equations and Operator Theory, 6: 224–249, 1983.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    N. Kravitsky, Discriminant varieties and discriminant ideals for operator vessels in Banach space. Integral Equations Operator Theory, 23: 441–458, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    N. Kravitsky, The joint characteristic function of a commutative operator vessel in Banach space. Integral Equations Operator Theory, 25: 199–215, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    M.S. Livšic, Operator waves in Hilbert space and related partial differential equations. Integral Equations Operator Theory, 2(1): 25–47, 1979.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    M.S. Livšic, A method for constructing triangular canonical models of commuting operators based on connections with algebraic curves. Integral Equations Operator Theory, 3(4): 489–507, 1980.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    M.S. Livšic, Cayley-Hamilton theorem, vector bundles and divisors of commuting operators. Integral Equations Operator Theory, 6: 250–273, 1983.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    M.S. Livšic, Collective motions of spatio-temporcd systems, J. Math. Anal. Appl., 116: 22–41, 1986.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    M.S. Livšic, Commuting nonselfadjoint operators and mappings of vector bundles on algebraic curves, in H. Bart, L Gohberg, and M.A. Kaashoek, editors. Operator Theory and Systems, Vol. 19 of Operator Theory: Adv. Appl., pages 255–279. Birkhäuser Verlag, Basel, 1986.Google Scholar
  28. [28]
    M.S. Livšic, What is a particle from the standpoint of systems theory? Integral Equations Operator Theory, 14: 552–563, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  29. [29]
    M.S. Livšic, Commuting nonselfadjoint operators and a unified theory of waves and corpuscles, in L Gohberg and Yu. Lyubich, editors. New Results in Operator Theory and its Applications, Vol. 98 of Operator Theory: Adv. Appl., pages 163–185. Birkhäuser Verlag, Basel, 1997.Google Scholar
  30. [30]
    M.S. Livšic, Vortices of 2D systems, in D. Alpay and V. Vinnikov, editors. Operator Theory, System Theory and Related Topics (The Moshe Livšic Anniversary Volume), Vol. 123 of Operator Theory: Adv. Appl., pages 7–42. Birkhäuser Verlag, Basel, 2001.Google Scholar
  31. [31]
    M.S. Livšic, Chains of space-time open systems and DNA, in D. Alpay, L Gohberg, and V. Vinnikov, editors. Interpolation Theory, System Theory and Related Topics (The Harry Dym Anniversary Volume), Vol. 134 of Operator Theory: Adv. Appl., Birkhäuser Verlag, Basel, pages 319–336, 2002.Google Scholar
  32. [32]
    M.S. Livšic and Y. Avishai, A study of solitonic combinations based on the theory of commuting nonselfadjoint operators. Linear Algebra Appl., 122/123/124: 357–414, 1989.CrossRefGoogle Scholar
  33. [33]
    M.S. Livšic, N. Kravitsky, A.S. Markus, and V. Vinnikov, Theory of Commuting Nonselfadjoint Operators, Kluwer, 1995.zbMATHGoogle Scholar
  34. [34]
    C.S. Seshadri, Fihrés Vectoriels sur les Courbes Algébriques, Astérisque 96, 1982.Google Scholar
  35. [35]
    F. Severi, Funzioni Quasi Abeliane, Rome, 1947.zbMATHGoogle Scholar
  36. [36]
    E.D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems (Second Edition), Springer, 1998.zbMATHGoogle Scholar
  37. [37]
    O.J. Staffans, Passive and conservative infinite-dimensional impedance and scattering systems (from a personal point of view). Proceedings of the International Symposium on the Mathematiceil Theory of Networks and Systems (University of Notre Dame, August, 2002). Pages 375–413 of this volume.Google Scholar
  38. [38]
    O.J. Staffans and G. Weiss, Transfer functions of regular linear sytems. Part II: the system operator and the Lax-Phillips semigroup, Trans. Amer. Math. Soc., to appear.Google Scholar
  39. [39]
    V. Vinnikov, Commuting nonselfadjoint operators and algebraic curves, in T. Ando and I. Gohberg, editors. Operator Theory and Complex Analysis, Vol. 59 of Operator Theory: Adv. Appl, pages 348–371. Birkhäuser Verlag, Basel, 1992.CrossRefGoogle Scholar
  40. [40]
    V. Vinnikov, 2D systems and realization of bundle mappings on compact Riemann surfaces, in U. Helmke, R. Mennicken, and J. Saurer, editors. Systems and Networks: Mathematical Theory and Applications (Vol II), Vol. 79 of Math. Res., pages 909–912. Akademie Verlag, Berlin, 1994.Google Scholar
  41. [41]
    V. Vinnikov, Commuting operators and function theory on a Riemann surface, in Holomorphic Spaces (Ed. S. Axler, J.E. McCarthy, and D. Sarason), MSRI Publications, Cambridge University Press, 1998.Google Scholar
  42. [42]
    J.C. Willems, Dissipative dynamical systems. Part I: General theory. Arch. Rational Mech. Anal., 45: 321–351, 1972.MathSciNetzbMATHCrossRefGoogle Scholar
  43. [43]
    J.C. Willems, Dissipative dynamical systems. Part II: Linear systems with quadratic supply rates. Arch. Rational Mech. Anal., 45: 352–393, 1972.MathSciNetzbMATHCrossRefGoogle Scholar
  44. [44]
    E. Zerz, Topics in Multidimensional Linear Systems Theory, LNCIS #256, Springer-Verlag, Berlin-New York, 2000.zbMATHGoogle Scholar
  45. [45]
    K. Zhou (with J.C. Doyle), Essentials of Robust Control, Prentice-Hall, Upper Saddle River, NJ, 1998.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 2003

Authors and Affiliations

  • Joseph A. Ball
    • 1
  • Victor Vinnikov
    • 2
  1. 1.Department of MathematicsVirginia TechBlacksburgUSA
  2. 2.Department of MathematicsBen Gurion University of the NegevBeer-ShevaIsrael

Personalised recommendations