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

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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

Amol Sasane ^1,

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

Received August 2012 Revised December 2012 Published December 2013

Abstract
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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 $M\in ({\mathbb{C}}[\xi_1,\dots, \xi_d, \tau])^{m\times n}$.

Keywords: controllability, distributions that are periodic in the spatial directions, Partial differential equations, autonomy, Fourier transformation, spatially invariant systems., behaviours.

Mathematics Subject Classification: Primary: 35A24; Secondary: 93B05, 93C2.

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.

show all references

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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