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Controllability and Observability of Linear Time-Invariant Control System on Superspace

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Abstract

There is an abundance of literature available in which control theory has a profound impact on understanding and controlling many physical systems. The role of superspace in understanding many dynamical systems where both fermions and bosons are treated in a unified manner is well known in physics. Due to the formalism of dynamical systems involving both bosonic and fermionic variables such as Schrödinger equations, quantum Kepler problems, quantum harmonic and (an-)harmonic oscillators, and sKdV equation, we stand at the edge of introducing the control problems for such dynamical systems in superspace. With this motivation, we consider the introductory control problem such as the linear time-invariant control system in superspace, and discuss its controllability. We mathematically derive controllability conditions for such systems, which we call the extended Kalman rank condition in superspace, and illustrate this with the help of some simple but interesting examples of mixed bosonic-fermionic control systems. We also derive the observability criterion for such systems.

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References

  1. Das A, Roy S. The zero curvature formulation of the sKdV equations. Journal of mathematical physics. 1990;31(9):2145–9.

    Article  ADS  MathSciNet  Google Scholar 

  2. Mirrahimi M, Rouchon P. Controllability of quantum harmonic oscillators. IEEE Trans Automat Contr. 2004;49(5):745–7. https://doi.org/10.1109/TAC.2004.825966.

    Article  MathSciNet  Google Scholar 

  3. Landi G, Marmo G, Vilasi G. Remarks on the complete integrability of dynamical systems with fermionic variables. J Phys A: Math Gen. 1992;25(16):4413. https://doi.org/10.1088/0305-4470/25/16/017.

    Article  ADS  MathSciNet  Google Scholar 

  4. Schwinger J. The theory of quantized fields I. Phys Rev. 1951;82(6):914. https://doi.org/10.1103/physrev.82.914.

    Article  ADS  MathSciNet  Google Scholar 

  5. Berezin FA. The method of second quantization. London: Academic Press; 1966.

    Google Scholar 

  6. Berezin FA, Kac GI. Lie groups with commuting and anticommuting parameters. Math USSR Sb. 1970;11(3):311.

    Article  Google Scholar 

  7. Ferrara S, Zumino B, Wess J. Supergauge multiplets and superfields. Phys Lett B. 1974;51(3):239–41. https://doi.org/10.1016/0370-2693(74)90283-4.

    Article  ADS  Google Scholar 

  8. Arnowitt R, Nath P, Zumino B. Superfield densities and action principle in curved superspace. Phys Lett B. 1975;56(1):81–4. https://doi.org/10.1016/0370-2693(75)90504-3.

    Article  ADS  MathSciNet  Google Scholar 

  9. Casalbuoni R. The classical mechanics for Bose-Fermi systems. Il Nuovo Cimento A. 1976;33(3):389–431. https://doi.org/10.1007/BF02729860.

    Article  ADS  MathSciNet  Google Scholar 

  10. Delbourgo R, Jones L, White M. Anharmonic Grassmann oscillator. Phys Rev D. 1989;40(8):2716. https://doi.org/10.1103/PhysRevD.40.2716.

    Article  ADS  MathSciNet  CAS  Google Scholar 

  11. De Bie H. Schrödinger equation with delta potential in superspace. Phys Lett A. 2008;372(24):4350–2. https://doi.org/10.1016/j.physleta.2008.04.005.

    Article  ADS  MathSciNet  CAS  Google Scholar 

  12. Zhang R-B. Orthosymplectic lie superalgebras in superspace analogues of quantum Kepler problems. Comm Math Phys. 2008;280(2):545–62. https://doi.org/10.1007/s00220-008-0450-4.

    Article  ADS  MathSciNet  Google Scholar 

  13. Macfarlane A, Majid S. Quantum group structure in a fermionic extension of the quantum harmonic oscillator. Phys Lett B. 1991;268(1):71–4. https://doi.org/10.1016/0370-2693(91)90924-F.

    Article  ADS  MathSciNet  Google Scholar 

  14. De Bie H, Sommen F. Hermite and Gegenbauer polynomials in superspace using Clifford analysis. J Phys A Math Theor. 2007;40(34):10441.

    Article  MathSciNet  Google Scholar 

  15. Dunne GV, Halliday I. Negative dimensional oscillators. Nucl Phys B. 1988;308(2–3):589–618.

    Article  ADS  MathSciNet  Google Scholar 

  16. Bandos I. On polarized scattering equations for superamplitudes of 11d supergravity and ambitwistor superstring. J High Energy Phys. 2019;2019(11):1–41. https://doi.org/10.1007/JHEP11(2019)087.

    Article  MathSciNet  Google Scholar 

  17. Volkov D, Pashnev A, Soroka V, Tkach V. Hamiltonian systems with even and odd Poisson brackets, duality of their conservation laws. J Exp Theor Phys Lett. 1986;44(70–72):35.

    Google Scholar 

  18. Soroka V. On Hamilton systems with even and odd Poisson brackets. Lett Math Phys. 1989;17(3):201–8. https://doi.org/10.1007/BF00401586.

    Article  ADS  MathSciNet  Google Scholar 

  19. Buchbinder IL, Kuzenko SM. Ideas and methods of supersymmetry and supergravity: a walk through superspace. Bristol: CRC Press; 1998.

    Google Scholar 

  20. Deligne P, Etingof P, Freed DS, Jeffrey LC, Kazhdan D, Morgan JW, Morrison DR, Witten E. Quantum fields and strings: a course for mathematicians, vol. 1. Providence, RI: American Mathematical Society; 1999.

    Google Scholar 

  21. Varadarajan VS. Supersymmetry for mathematicians: an introduction: an introduction, vol. 11. Providence, RI: American Mathematical Society; 2004.

    Google Scholar 

  22. Gates SJ, Grisaru MT, Rocek M, Siegeln W. Superspace, or one thousand and one lessons in supersymmetry. Switzerland: Frontiers in Physics; 1983.

    Google Scholar 

  23. Freedman DZ, Nieuwenhuizen PV. Properties of supergravity theory. Phys Rev D. 1976;14(4):912. https://doi.org/10.1103/PhysRevD.14.912.

    Article  ADS  MathSciNet  Google Scholar 

  24. Strathdee J. Super-gauge transformations. In: Ferrara S, editor. Supersymmetry. Singapore: World Scientific; 1987. p. 35–40.

    Google Scholar 

  25. Salam A, Ali A. Selected papers of Abdus Salam:(with Commentary) vol. 5. World Scientific, Singapore; 1994.

  26. DeWitt B. Supermanifolds. Cambridge: Cambridge University Press; 1992.

    Book  Google Scholar 

  27. Leites DA. Introduction to the theory of supermanifolds. Russ Math Surv. 1980;35(1):1. https://doi.org/10.1070/RM1980v035n01ABEH001545.

    Article  MathSciNet  Google Scholar 

  28. Kostant B. Graded manifolds, graded lie theory, and prequantization. In: Differential Geometrical Methods in Mathematical Physics, pp. 177–306. Springer, Berlin; 1977.

  29. Rogers A. Supermanifolds: theory and applications. Singapore: World Scientific; 2007.

    Book  Google Scholar 

  30. Tuynman GM. Supermanifolds and supergroups: basic theory, vol. 570. Berlin: Springer; 2004.

    Google Scholar 

  31. Kalman RE. On the general theory of control systems. In: Proceedings First International Conference on Automatic Control,Moscow, USSR, 1960; pp. 481–492.

  32. Kalman RE. Mathematical description of linear dynamical systems. JSIAM CONTROL Ser A. 1963;1(2):152–92. https://doi.org/10.1137/0301010.

    Article  MathSciNet  Google Scholar 

  33. Kirillov AA, editor. Introduction to superanalysis. Netherland: Springer; 2013.

    Google Scholar 

  34. Rogers A. A global theory of supermanifolds. J Math Phys. 1980;21(6):1352–65. https://doi.org/10.1063/1.524585.

    Article  ADS  MathSciNet  Google Scholar 

  35. Inoue A. Foundations of real analysis on the superspace \(\mathbb{R} ^{m|n}\) over \(\infty \)-dimensional Frechet-Grassmann algebra. J Fac Sci Univ Tokyo. 1992;39:419–74.

    Google Scholar 

  36. Batchelor M. Two approaches to supermanifolds. Trans Amer Math Soc. 1980;258(1):257–70. https://doi.org/10.1090/S0002-9947-1980-0554332-9.

    Article  MathSciNet  Google Scholar 

  37. Urrutia LF, Morales N. The Cayley-Hamilton theorem for supermatrices. J Phys A: Math Gen. 1994;27(6):1981–97. https://doi.org/10.1088/0305-4470/27/6/022.

    Article  ADS  MathSciNet  Google Scholar 

  38. Alpay D, Paiva IL, Struppa DC. Positivity, rational Schur functions, Blaschke factors, and other related results in the Grassmann algebra. Integral Equ Ope Theory. 2019;91(2):1–39. https://doi.org/10.1007/s00020-019-2506-6.

    Article  MathSciNet  Google Scholar 

  39. Vladimirov VS, Volovich IV. Superanalysis. I. Differential calculus. Theo Math Phys. 1984;59(1):317–35. https://doi.org/10.1007/BF01028510.

    Article  MathSciNet  Google Scholar 

  40. Kreyszig E. Introductory functional analysis with applications, vol. 17. New York: John Wiley & Sons; 1991.

    Google Scholar 

  41. Schepper HD, Adán AG, Sommen F. Spin actions in Euclidean and Hermitian Clifford analysis in superspace. J Math Anal Appl. 2018;457(1):23–50. https://doi.org/10.1016/j.jmaa.2017.08.009.

    Article  MathSciNet  Google Scholar 

  42. Carmeli C, Caston L, Fioresi R. Mathematical foundations of supersymmetry, vol. 15. Zürich: European Mathematical Society; 2011.

    Book  Google Scholar 

  43. Bonanos S, Kamimura K. On the Cayley-Hamilton theorem for supermatrices; 2010. Preprint at arxiv:1003.2667

  44. Kobayashi Y, Nagamachi S. Characteristic functions and invariants of supermatrices. J Math Phys. 1990;31(11):2726–30. https://doi.org/10.1063/1.528976.

    Article  ADS  MathSciNet  Google Scholar 

  45. Monterde J, Sánchez-Valenzuela O. Existence and uniqueness of solutions to superdifferential equations. Journal of Geometry and Physics. 1993;10(4):315–43.

    Article  ADS  MathSciNet  Google Scholar 

  46. Wegner F. Supermathematics and its applications in statistical physics. Heidelberg: Springer; 2016.

    Book  Google Scholar 

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Acknowledgements

The authors thank the editor and the anonymous referees for constructive and pertinent suggestions.

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Author notes

  1. Kishor Chandra Pati contributed equally to this work.

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  1. Department of Mathematics, National Institute of Technology Rourkela, Sector 1, Sundergarh, 769008, Odisha, India

    Aroonima Sahoo & Kishor Chandra Pati

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  1. Aroonima Sahoo
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Correspondence to Kishor Chandra Pati.

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Sahoo, A., Pati, K.C. Controllability and Observability of Linear Time-Invariant Control System on Superspace. J Dyn Control Syst (2024). https://doi.org/10.1007/s10883-024-09685-1

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