Abstract
In these notes we motive the study of Liouville equations having control terms using examples from problem areas as diverse as atomic physics (NMR), biological motion control and minimum attention control. On one hand, the Liouville model is interpreted as applying to multiple trials involving a single system and on the other, as applying to the control of many identical copies of a single system; e.g., control of a flock. We illustrate the important role the Liouville formulation has in distinguishing between open loop and feedback control. Mathematical results involving controllability and optimization are discussed along with a theorem establishing the controllability of multiple moments associated with linear models. The methods used succeed by relating the behavior of the solutions of the Liouville equation to the behavior of the underlying ordinary differential equation, the related stochastic differential equation, and the consideration of the related moment equations.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
R.W. Brockett, Minimum Attention Control, in Proceedings of the 36th IEEE Conference on Decision and Control (1997), pp. 2628–2632
R. Brockett, Control of Stochastic Ensembles, ed. by B. Wittenmark, A. Rantzer. Åström Symposium on Control (Studentlitteratur, Lund, 1999), pp. 199–216
R. Brockett, N. Khaneja, in System Theory Modeling, Analysis, and Control ed. by T. Djaferis, I. Schick. On the Stochastic Control of Quantum Ensembles (Kluwer Academic, Norwell, 1999), pp. 75–96
R. Brockett, On the control of a Flock by a leader. Proc. Skeklov Inst. Math. 268(1), 49–57 (2010)
R.W. Brockett, Reduced Complexity Control Systems, in Proceedings of the 17th World Congress, The International Federation of Automatic Control, Seoul, 2008
J. Moser, On the volume elements of a manifold. Trans. Am. Math. Soc. 120, 286–294 (1965)
B. Dacorogna, J. Moser, On a partial differential equation involving the Jacobian determinant. Ann. Inst. Henri Poincare C 7(1), 1–26 (1990)
R. Brockett, in Proceedings of the International Conference on Complex Geometry and Related Fields, ed. by Z. Chen et al. Optimal Control of the Liouville Equation (American Mathematical Society, Providence, 2007), pp. 23–35
A.A. Agrachev, M. Caponigro, Controllability on the group of diffeomorphisms. Ann. Inst. Henri Poincare C Non Lin. Anal. 26(6), 2503–2509 (2009)
V. Jurdjevic, H.J. Sussmann, Control systems on Lie groups, J. Differ. Equat. 12(2), 313–329 (1972)
R. Brockett, Nonlinear Systems and Differential Geometry, in Proceedings of the IEEE, vol 64 (1976), pp. 61–72
R.W. Brockett, Linear feedback systems and the groups of Lie and Galois. Lin. Algebra Appl. 50, 45–60 (1983)
Yu.L. Sachkov, in Positive orthant scalar controllability of bilinear systems. Springer Mathematical Notes, Mat. Zametki, 58(3), pp. 666–669 (1995)
Y.L. Sachkov, On positive orthant controllability of bilinear systems in small codimensions. SIAM J. Contr. Optim. 35(1), 29–35, (1997)
W.M. Boothby, Some comments on positive orthant controllability of bilinear systems. SIAM J. Contr. Optim. 20(5), 634–644 (1982)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Brockett, R. (2012). Notes on the Control of the Liouville Equation. In: Control of Partial Differential Equations. Lecture Notes in Mathematics(), vol 2048. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27893-8_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-27893-8_2
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-27892-1
Online ISBN: 978-3-642-27893-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)