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

In this paper, we study the restrictions of solutions of a scalar system of PDEs to a proper subspace of the domain $$\mathbb {R}^n$$Rn. 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.

[1]  Jeffrey Wood,et al.  A formal theory of matrix primeness , 1998, Math. Control. Signals Syst..

[2]  Michiel Hazewinkel,et al.  Constructive algebra and systems theory , 2006 .

[3]  Harish K. Pillai,et al.  Lossless and Dissipative Distributed Systems , 2001, SIAM J. Control. Optim..

[4]  Charles Riquier,et al.  Les Systèmes D'équations aux Dérivées Partielles , 2009 .

[5]  Maria Paula Macedo Rocha.  Structure and representation of 2-D systems , 1990 .

[6]  U. Oberst.  Multidimensional constant linear systems , 1990, EUROCAST.

[7]  Harish K. Pillai,et al.  On restrictions of n-d systems to 1-d subspaces , 2014, Multidimens. Syst. Signal Process..

[8]  David A. Cox,et al.  Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics) , 2007 .

[9]  J. Willems.  Paradigms and puzzles in the theory of dynamical systems , 1991 .

[10]  U. Oberst,et al.  The canonical Cauchy problem for linear systems of partial difference equations with constant coefficients over the completer-dimensional integral lattice ℕ2r , 1993 .

[11]  Shiva Shankar.  Can One Control the Vibrations of a Drum? , 2000, Multidimens. Syst. Signal Process..

[12]  Diego Napp Avelli,et al.  Autonomous multidimensional systems and their implementation by behavioral control , 2010, Syst. Control. Lett..

[13]  S. Zampieri,et al.  State space realization of 2-D finite-dimensional behaviours , 1993 .

[14]  R. Rogers,et al.  An introduction to partial differential equations , 1993 .

[15]  Maria Elena Valcher,et al.  Characteristic cones and stability properties of two-dimensional autonomous behaviors , 2000 .

[16]  Diego Napp Avelli,et al.  Lyapunov stability of 2D finite-dimensional behaviours , 2011, Int. J. Control.

[17]  David A. Cox,et al.  Ideals, Varieties, and Algorithms , 1997 .

[18]  Martin Kreuzer,et al.  Computational Commutative Algebra 1 , 2000 .

[19]  Jean-François Pommaret.  Partial Differential Equations and Group Theory: New Perspectives for Applications , 2014 .

[20]  Wilhelm Plesken,et al.  Janet’s approach to presentations and resolutions for polynomials and linear pdes , 2005 .

[21]  D. Eisenbud.  Commutative Algebra: with a View Toward Algebraic Geometry , 1995 .

[22]  Ulrich Oberst,et al.  The Constructive Solution of Linear Systems of Partial Difference and Differential Equations with Constant Coefficients , 2001, Multidimens. Syst. Signal Process..

[23]  Kiyotsugu Takaba,et al.  Lyapunov stability analysis of higher-order 2-D systems , 2011, Multidimens. Syst. Signal Process..