Causal and Stable Input/Output Structures on Multidimensional Behaviours

In this work we study multidimensional (nD) linear differential behaviours with a distinguished independent variable, called "time". We define in a natural way causality and stability of input/output structures with respect to this distinguished direction. We make an extension of some results in the theory of partial differential equations, demonstrating that causality is equivalent to a property of the transfer matrix. We also quote results which in effect characterize time-autonomy for the general systems case. Stability is likewise characterized by a property of the transfer matrix.

[1]  J. Wood.  Key problems in the extension of module-behaviour duality , 2002 .

[2]  Eric Rogers,et al.  Causal and Stable Input/Output Structures on Multidimensional Behaviours , 2005, CDC 2005.

[3]  Shiva Shankar.  The Nullstellensatz for Systems of PDE , 1999 .

[4]  J. WOOD,et al.  A Behavioral Approach to the Pole Structure of One-Dimensional and Multidimensional Linear Systems , 2000, SIAM J. Control. Optim..

[5]  L. Hörmander,et al.  The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients , 1983 .

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

[7]  Tsit Yuen Lam,et al.  Lectures on modules and rings , 1998 .

[8]  Eva Zerz.  Extension Modules in Behavioral Linear Systems Theory , 2001, Multidimens. Syst. Signal Process..

[9]  Ulrich Oberst,et al.  Variations on the fundamental principle for linear systems of partial differential and difference equations with constant coefficients , 1995, Applicable Algebra in Engineering, Communication and Computing.

[10]  B. Malgrange.  Division des distributions , 1958 .

[11]  Heide Gluesing-Luerssen,et al.  Linear Delay-Differential Systems with Commensurate Delays: An Algebraic Approach , 2001 .

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

[13]  E. Rogers,et al.  On Homomorphisms of nD Behaviors , 2001 .

[14]  L.C.G.J.M. Habets,et al.  Book review: Introduction to mathematical systems theory, a behavioral approach , 2000 .

[15]  Virendra Sule,et al.  Algebraic geometric aspects of feedback stabilization , 1992 .

[16]  Sandro Zampieri,et al.  Causal Input/Output Representation of 2D Systems in the Behavioral Approach , 1998 .

[17]  Jan C. Willems,et al.  Introduction to mathematical systems theory: a behavioral approach, Texts in Applied Mathematics 26 , 1999 .

[18]  Shiva Shankar,et al.  An obstruction to the simultaneous stabilization of two n-D plants , 1994 .

[19]  S. Gindikin,et al.  The Cauchy Problem , 1991 .

[20]  L. Hörmander,et al.  On the theory of general partial differential operators , 1955 .

[21]  V. Palamodov,et al.  Linear Differential Operators with Constant Coefficients , 1970 .

[22]  Sang Bum Lee.  h-Divisible Modules , 2003 .

[23]  Shiva Shankar.  Geometric Completeness of Distribution Spaces , 2003 .

[24]  Jan C. Willems,et al.  Time-autonomy versus time-controllability , 2002, Syst. Control. Lett..

[25]  Jeffrey Wood,et al.  Controllable and Uncontrollable Poles and Zeros of nD Systems , 2001, Math. Control. Signals Syst..

[26]  Amol Sasane.  On the Willems closure with respect to Ws , 2003, IMA J. Math. Control. Inf..

[27]  S. Matsuura.  On general systems of partial differential operators with constant coefficients , 1961 .

[28]  F. John.  Partial differential equations , 1967 .

[29]  M. Nacinovich.  Cauchy problem for overdetermined systems , 1990 .

[30]  Jeffrey Wood,et al.  Modules and Behaviours in nD Systems Theory , 2000, Multidimens. Syst. Signal Process..

[31]  L. Hörmander,et al.  The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis , 1983 .

[32]  L. Hörmander,et al.  An introduction to complex analysis in several variables , 1973 .

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

[34]  Harish K. Pillai,et al.  A Behavioral Approach to Control of Distributed Systems , 1999 .

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

[36]  Amol Sasane,et al.  Time-Autonomy and Time-Controllability of 2-D Behaviours that are Tempered in the Spatial Direction , 2004, Multidimens. Syst. Signal Process..

[37]  B. F. Caviness,et al.  Quantifier Elimination and Cylindrical Algebraic Decomposition , 2004, Texts and Monographs in Symbolic Computation.

[38]  B. Malgrange,et al.  Systèmes différentiels à coefficients constants , 1964 .