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Algorithms for the theory of restrictions of scalar \(n\)-D systems to proper subspaces of \(\mathbb {R}^n\)

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Abstract

In this paper, we study the restrictions of solutions of a scalar system of PDEs to a proper subspace of the domain \(\mathbb {R}^n\). The object of study is associated with certain intersection ideals. In the paper, we provide explicit algorithms to calculate these intersection ideals. We next deal with when a given subspace is “free” with respect to the solution set of a system of PDEs—this notion of freeness is related to restrictions and intersection ideals. We again provide algorithms and checkable algebraic criterion to answer the question of freeness of a subspace. Finally, we provide an upper bound to the dimension of free subspaces that can be associated with the solution set of a system of PDEs.

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Notes

  1. We came to know from an anonymous reviewer that Janet had obtained an explicit way of returning this (usually infinite) sequence.

  2. It was brought to our notice by an anonymous reviewer that this result, too, was contained in Janet’s work on more general classes of systems of PDEs.

References

  • Cox, D., Little, J., & O’Shea, D. (2007). Ideals, varieties, and algorithms: Undergraduate texts in mathematics (3rd ed.). Berlin: Springer.

    Book  Google Scholar 

  • Eisenbud, D. (1995). Commutative algebra with a view toward algebraic geometry. Berlin: Springer.

    MATH  Google Scholar 

  • Fornasini, E., Rocha, P., & Zampieri, S. (1993). State space realization of 2-D finite-dimensional behaviours. SIAM Journal on Control and Optimization, 31(6), 1502–1517.

    Article  MATH  MathSciNet  Google Scholar 

  • Kojima, C., Rapisarda, P., & Takaba, K. (2010). Lyapunov stability analysis of higher order 2-D systems. Multidimensional Systems and Signal Processing, 22, 287–302.

    Article  MathSciNet  Google Scholar 

  • Kreuzer, M., & Robbiano, L. (2000). Computational commutative algebra 2. Berlin: Springer.

    Book  Google Scholar 

  • Napp Avelli, D., & Rocha, P. (2010). Autonomous multidimensional systems and their implementation by behavioral control. Systems and Control Letters, 59(34), 203–208.

    Article  MATH  MathSciNet  Google Scholar 

  • Napp Avelli, D., Rapisarda, P., & Rocha, P. (2011). Lyapunov stability of 2D finite-dimensional behaviours. International Journal of Control, 84(4), 737–745.

    Article  MATH  MathSciNet  Google Scholar 

  • Oberst, U. (1990). Multidimensional constant linear systems. Acta Applicandae Mathematicae, 20, 1–175.

    Article  MATH  MathSciNet  Google Scholar 

  • Oberst, U., & Pauer, F. (2001). The constructive solution of linear systems of partial difference and differential equations with constant coefficients. Multidimensional Systems and Signal Processing, 12, 253–308.

    Article  MATH  MathSciNet  Google Scholar 

  • Oberst, U. (2006). The constructive solution of linear systems of partial difference and differential equations with constant coefficients. In B. Hanzon & M. Hazewinkel (Eds.), Constructive algebra and systems theory (pp. 205–233). Amsterdam: Royal Netherlands Academy of Arts and Sciences.

    Google Scholar 

  • Pal, D., & Pillai, H. K. (2014). On restrictions of \(n\)-D systems to 1-D subspaces. Multidimensional Systems and Signal Processing, 25, 115–144.

    Article  MATH  MathSciNet  Google Scholar 

  • Pillai, H. K., & Willems, J. C. (2002). Lossless and dissipative distributed systems. SIAM Journal on Control and Optimization, 40(5), 1406–1430.

    Article  MATH  MathSciNet  Google Scholar 

  • Plesken, W., & Robertz, D. (2005). Janets approach to presentations and resolutions for polynomials and linear pdes. Archiv der Mathematik, 84(1), 22–37.

    Article  MATH  MathSciNet  Google Scholar 

  • Pommaret, J.-F. (1994). Partial differential equations and group theory: New perspectives for applications. Dordrecht: Kluwer.

    Book  MATH  Google Scholar 

  • Renardy, M., & Rogers, R. C. (2004). An introduction to partial differential equations. New York: Springer.

    MATH  Google Scholar 

  • Riquier, C. (1910). Les Systèmes d’Équations aux Dérivées Partielles. Paris: Gauthiers-Villars.

    Google Scholar 

  • Rocha, P. (1990) Structure and representation of 2-D systems. PhD thesis, University of Groningen, The Netherlands

  • Shankar, S. (2000). Can one control the vibrations of a drum? Multidimensional Systems and Signal Processing, 11, 67–81.

    Article  MATH  MathSciNet  Google Scholar 

  • Valcher, M. E. (2001). Characteristic cones and stability properties of two-dimensional autonomous behaviors. IEEE Transactions on Circuits and Systems-Part I: Fundamental Theory and Applications, 47(3), 290–302.

    Article  Google Scholar 

  • Willems, J. C. (1991). Paradigms and puzzles in the theory of dynamical systems. IEEE Transactions on Automatic Control, 36(3), 259–294.

    Article  MATH  MathSciNet  Google Scholar 

  • Wood, J., Rogers, E., & Owens, D. H. (1998). A formal theory of matrix primeness. Mathematics of Control, Signals, and Systems, 11, 40–78.

    Article  MATH  MathSciNet  Google Scholar 

  • Zerz, E., & Oberst, U. (1993). The canonical cauchy problem for linear systems of partial difference equations with constant coefficients over the complete \(r\)-dimensional integral lattice \(\mathbb{Z}^r\). Acta Applicandae Mathematicae., 31, 249–273.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Electronics and Electrical Engineering, Indian Institute of Technology Guwahati, North Guwahati, 781 039, India

    Debasattam Pal

  2. Department of Electrical Engineering, Indian Institute of Technology Bombay, Powai, 400 076, Mumbai, India

    Harish K. Pillai

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  1. Debasattam Pal
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  2. Harish K. Pillai
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Correspondence to Harish K. Pillai.

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Pal, D., Pillai, H.K. Algorithms for the theory of restrictions of scalar \(n\)-D systems to proper subspaces of \(\mathbb {R}^n\) . Multidim Syst Sign Process 26, 439–457 (2015). https://doi.org/10.1007/s11045-014-0285-4

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  • DOI: https://doi.org/10.1007/s11045-014-0285-4

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