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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

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 & 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.  doi: 10.1007/s00020-003-1356-3.  Google Scholar

[2]

Madhu Belur, Control in a Behavioral Context,, Ph.D Thesis, (2003).   Google Scholar

[3]

R. W. Carroll, Abstract Methods in Partial Differential Equations,, Harper's Series in Modern Mathematics, (1969).   Google Scholar

[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.  doi: 10.1016/j.automatica.2009.03.005.  Google Scholar

[5]

R. F. Curtain and A. J. Sasane, On Riccati equations in Banach algebras,, SIAM J. Control Optim., 49 (2011), 464.  doi: 10.1137/100806011.  Google Scholar

[6]

W. F. Donoghue, Jr., Distributions and Fourier Transforms,, Pure and Applied Mathematics, (1969).   Google Scholar

[7]

L. Hörmander, Null solutions of partial differential equations,, Arch. Rational Mech. Anal., 4 (1960), 255.   Google Scholar

[8]

L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis,, 2nd edition, (1990).   Google Scholar

[9]

U. Oberst and M. Scheicher, Time-autonomy and time-controllability of discrete multidimensional behaviors,, Internat. J. Control, 85 (2012), 990.  doi: 10.1080/00207179.2012.673135.  Google Scholar

[10]

J. W. Polderman and J. C. Willems, Introduction to Mathematical Systems Theory. A Behavioral Approach,, Texts in Applied Mathematics, (1998).   Google Scholar

[11]

H. K. Pillai and S. Shankar, A behavioral approach to control of distributed systems,, SIAM J. Control Optim., 37 (1999), 388.  doi: 10.1137/S0363012997321784.  Google Scholar

[12]

A. J. Sasane, E. G. F. Thomas and J. C. Willems, Time-autonomy versus time-controllability,, Systems Control Lett., 45 (2002), 145.  doi: 10.1016/S0167-6911(01)00174-8.  Google Scholar

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.  doi: 10.1007/s00020-003-1356-3.  Google Scholar

[2]

Madhu Belur, Control in a Behavioral Context,, Ph.D Thesis, (2003).   Google Scholar

[3]

R. W. Carroll, Abstract Methods in Partial Differential Equations,, Harper's Series in Modern Mathematics, (1969).   Google Scholar

[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.  doi: 10.1016/j.automatica.2009.03.005.  Google Scholar

[5]

R. F. Curtain and A. J. Sasane, On Riccati equations in Banach algebras,, SIAM J. Control Optim., 49 (2011), 464.  doi: 10.1137/100806011.  Google Scholar

[6]

W. F. Donoghue, Jr., Distributions and Fourier Transforms,, Pure and Applied Mathematics, (1969).   Google Scholar

[7]

L. Hörmander, Null solutions of partial differential equations,, Arch. Rational Mech. Anal., 4 (1960), 255.   Google Scholar

[8]

L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis,, 2nd edition, (1990).   Google Scholar

[9]

U. Oberst and M. Scheicher, Time-autonomy and time-controllability of discrete multidimensional behaviors,, Internat. J. Control, 85 (2012), 990.  doi: 10.1080/00207179.2012.673135.  Google Scholar

[10]

J. W. Polderman and J. C. Willems, Introduction to Mathematical Systems Theory. A Behavioral Approach,, Texts in Applied Mathematics, (1998).   Google Scholar

[11]

H. K. Pillai and S. Shankar, A behavioral approach to control of distributed systems,, SIAM J. Control Optim., 37 (1999), 388.  doi: 10.1137/S0363012997321784.  Google Scholar

[12]

A. J. Sasane, E. G. F. Thomas and J. C. Willems, Time-autonomy versus time-controllability,, Systems Control Lett., 45 (2002), 145.  doi: 10.1016/S0167-6911(01)00174-8.  Google Scholar

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