# American Institute of Mathematical Sciences

March  2014, 4(1): 115-124. doi: 10.3934/mcrf.2014.4.115

## Algebraic characterization of autonomy and controllability of behaviours of spatially invariant systems

 1 Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE

Received  August 2012 Revised  December 2012 Published  December 2013

We give algebraic characterizations of the properties of autonomy and of controllability of behaviours of spatially invariant dynamical systems, consisting of distributional solutions $w$, that are periodic in the spatial variables, to a system of partial differential equations $$M\left(\frac{\partial}{\partial x_1},\cdots, \frac{\partial}{\partial x_d} , \frac{\partial}{\partial t}\right) w=0,$$ corresponding to a polynomial matrix $M\in ({\mathbb{C}}[\xi_1,\dots, \xi_d, \tau])^{m\times n}$.
Citation: Amol Sasane. Algebraic characterization of autonomy and controllability of behaviours of spatially invariant systems. Mathematical Control and Related Fields, 2014, 4 (1) : 115-124. doi: 10.3934/mcrf.2014.4.115
##### References:
 [1] J. A. Ball and O. J. Staffans, Conservative state-space realizations of dissipative system behaviors, Integral Equations Operator Theory, 54 (2006), 151-213. doi: 10.1007/s00020-003-1356-3. [2] Madhu Belur, Control in a Behavioral Context, Ph.D Thesis, Rijksuniversiteit Groningen, The Netherlands, 2003. Available from: http://www.dissertations.ub.rug.nl/faculties/science/2003/m.n.belur/. [3] R. W. Carroll, Abstract Methods in Partial Differential Equations, Harper's Series in Modern Mathematics, Harper & Row, New York-London, 1969. [4] R. F. Curtain, O. V. Iftime and H. J. Zwart, System theoretic properties of a class of spatially invariant systems, Automatica J. IFAC, 45 (2009), 1619-1627. doi: 10.1016/j.automatica.2009.03.005. [5] R. F. Curtain and A. J. Sasane, On Riccati equations in Banach algebras, SIAM J. Control Optim., 49 (2011), 464-475. doi: 10.1137/100806011. [6] W. F. Donoghue, Jr., Distributions and Fourier Transforms, Pure and Applied Mathematics, 32, Academic Press, New York-London, 1969. [7] L. Hörmander, Null solutions of partial differential equations, Arch. Rational Mech. Anal., 4 (1960), 255-261. [8] L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis, 2nd edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 256, Springer-Verlag, Berlin, 1990. [9] U. Oberst and M. Scheicher, Time-autonomy and time-controllability of discrete multidimensional behaviors, Internat. J. Control, 85 (2012), 990-1009. doi: 10.1080/00207179.2012.673135. [10] J. W. Polderman and J. C. Willems, Introduction to Mathematical Systems Theory. A Behavioral Approach, Texts in Applied Mathematics, 26, Springer-Verlag, New York, 1998. [11] H. K. Pillai and S. Shankar, A behavioral approach to control of distributed systems, SIAM J. Control Optim., 37 (1999), 388-408. doi: 10.1137/S0363012997321784. [12] A. J. Sasane, E. G. F. Thomas and J. C. Willems, Time-autonomy versus time-controllability, Systems Control Lett., 45 (2002), 145-153. doi: 10.1016/S0167-6911(01)00174-8.

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##### References:
 [1] J. A. Ball and O. J. Staffans, Conservative state-space realizations of dissipative system behaviors, Integral Equations Operator Theory, 54 (2006), 151-213. doi: 10.1007/s00020-003-1356-3. [2] Madhu Belur, Control in a Behavioral Context, Ph.D Thesis, Rijksuniversiteit Groningen, The Netherlands, 2003. Available from: http://www.dissertations.ub.rug.nl/faculties/science/2003/m.n.belur/. [3] R. W. Carroll, Abstract Methods in Partial Differential Equations, Harper's Series in Modern Mathematics, Harper & Row, New York-London, 1969. [4] R. F. Curtain, O. V. Iftime and H. J. Zwart, System theoretic properties of a class of spatially invariant systems, Automatica J. IFAC, 45 (2009), 1619-1627. doi: 10.1016/j.automatica.2009.03.005. [5] R. F. Curtain and A. J. Sasane, On Riccati equations in Banach algebras, SIAM J. Control Optim., 49 (2011), 464-475. doi: 10.1137/100806011. [6] W. F. Donoghue, Jr., Distributions and Fourier Transforms, Pure and Applied Mathematics, 32, Academic Press, New York-London, 1969. [7] L. Hörmander, Null solutions of partial differential equations, Arch. Rational Mech. Anal., 4 (1960), 255-261. [8] L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis, 2nd edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 256, Springer-Verlag, Berlin, 1990. [9] U. Oberst and M. Scheicher, Time-autonomy and time-controllability of discrete multidimensional behaviors, Internat. J. Control, 85 (2012), 990-1009. doi: 10.1080/00207179.2012.673135. [10] J. W. Polderman and J. C. Willems, Introduction to Mathematical Systems Theory. A Behavioral Approach, Texts in Applied Mathematics, 26, Springer-Verlag, New York, 1998. [11] H. K. Pillai and S. Shankar, A behavioral approach to control of distributed systems, SIAM J. Control Optim., 37 (1999), 388-408. doi: 10.1137/S0363012997321784. [12] A. J. Sasane, E. G. F. Thomas and J. C. Willems, Time-autonomy versus time-controllability, Systems Control Lett., 45 (2002), 145-153. doi: 10.1016/S0167-6911(01)00174-8.
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