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Free Courant and derived Leibniz pseudoalgebras
Linearisation of tautological control systems
1. | Department of Mathematics and Statistics, Queen's University, Kingston, ON K7L 3N6, Canada |
References:
[1] |
R. Abraham, J. E. Marsden and T. S. Ratiu, Manifolds, Tensor Analysis, and Applications,, 2nd edition, (1988).
doi: 10.1007/978-1-4612-1029-0. |
[2] |
A. A. Agrachev and R. V. Gamkrelidze, The exponential representation of flows and the chronological calculus,, Math. USSR-Sb., 107 (1978), 467.
|
[3] |
A. A. Agrachev and Y. Sachkov, Control Theory from the Geometric Viewpoint,, Encyclopedia of Mathematical Sciences, (2004).
doi: 10.1007/978-3-662-06404-7. |
[4] |
C. O. Aguilar, Local Controllability of Affine Distributions,, PhD thesis, (2010).
|
[5] |
M. Barbero-Liñán and A. D. Lewis, Geometric interpretations of the symmetric product in affine differential geometry,, Int. J. Geom. Methods Mod. Phys., 9 (2012).
doi: 10.1142/S0219887812500739. |
[6] |
R. Beckmann and A. Deitmar, Strong vector valued integrals, 2011,, , (). Google Scholar |
[7] |
N. Bourbaki, Algebra I,, Elements of Mathematics, (1989). Google Scholar |
[8] |
G. E. Bredon, Sheaf Theory,, 2nd edition, (1997).
doi: 10.1007/978-1-4612-0647-7. |
[9] |
R. W. Brockett, Finite Dimensional Linear Systems,, John Wiley and Sons, (1970). Google Scholar |
[10] |
D. L. Cohn, Measure Theory,, Birkhäuser, (1980).
|
[11] |
C. T. J. Dodson and T. Poston, Tensor Geometry,, Graduate Texts in Mathematics, (1991).
doi: 10.1007/978-3-642-10514-2. |
[12] |
H. Federer, Geometric Measure Theory,, Reprint of 1969 edition, (1969).
|
[13] |
R. Godement, Topologie Algébrique et Théorie Des Faisceaux,, Publications de l'Institut de mathématique de l'Université de Strasbourg, (1958).
|
[14] |
M. W. Hirsch, Differential Topology,, Graduate Texts in Mathematics, (1976).
|
[15] |
A. Isidori, Nonlinear Control Systems,, 3rd edition, (1995).
doi: 10.1007/978-1-84628-615-5. |
[16] |
S. Jafarpour and A. D. Lewis, Time-varying Vector Fields and Their Flows,, To appear in Springer Briefs in Mathematics, (2014).
doi: 10.1007/978-3-319-10139-2. |
[17] |
S. Jafarpour and A. D. Lewis, Locally Convex Topologies and Control Theory,, Submitted to Mathematics of Control, (2015). Google Scholar |
[18] |
H. Jarchow, Locally Convex Spaces,, Mathematical Textbooks, (1981).
|
[19] |
M. Kashiwara and P. Schapira, Sheaves on Manifolds,, Grundlehren der Mathematischen Wissenschaften, (1990).
doi: 10.1007/978-3-662-02661-8. |
[20] |
H. K. Khalil, Nonlinear Systems,, 3rd edition, (2001). Google Scholar |
[21] |
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Volume I,, Interscience Tracts in Pure and Applied Mathematics, (1963).
|
[22] |
I. Kolář, P. W. Michor and J. Slovák, Natural Operations in Differential Geometry,, Springer-Verlag, (1993).
doi: 10.1007/978-3-662-02950-3. |
[23] |
A. D. Lewis, Fundamental problems of geometric control theory,, in Proceedings of the 51st IEEE Conference on Decision and Control, (2012), 7511.
doi: 10.1109/CDC.2012.6427046. |
[24] |
A. D. Lewis, Tautological Control Systems,, Springer Briefs in Electrical and Computer Engineering-Control, (2014).
doi: 10.1007/978-3-319-08638-5. |
[25] |
A. D. Lewis and D. R. Tyner, Geometric Jacobian linearization and LQR theory,, J. Geom. Mech., 2 (2010), 397.
doi: 10.3934/jgm.2010.2.397. |
[26] |
K. C. H. Mackenzie, The General Theory of Lie Groupoids and Lie Algebroids,, London Mathematical Society Lecture Note Series, (2005).
doi: 10.1017/CBO9781107325883. |
[27] |
H. Nijmeijer and A. J. van der Schaft, Nonlinear Dynamical Control Systems,, Springer-Verlag, (1990).
doi: 10.1007/978-1-4757-2101-0. |
[28] |
S. Ramanan, Global Calculus,, Graduate Studies in Mathematics, (2005).
|
[29] |
W. Rudin, Functional Analysis,, 2nd edition, (1991).
|
[30] |
S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds,, Tôhoku Math. J. (2), 10 (1958), 338.
doi: 10.2748/tmj/1178244668. |
[31] |
S. Sastry, Nonlinear Systems: Analysis, Stability, and Control,, Interdisciplinary Applied Mathematics, (1999).
doi: 10.1007/978-1-4757-3108-8. |
[32] |
D. J. Saunders, The Geometry of Jet Bundles,, London Mathematical Society Lecture Note Series, (1989).
doi: 10.1017/CBO9780511526411. |
[33] |
H. H. Schaefer and M. P. Wolff, Topological Vector Spaces,, 2nd edition, (1999).
doi: 10.1007/978-1-4612-1468-7. |
[34] |
F. Schuricht and H. von der Mosel, Ordinary Differential Equations with Measurable Right-Hand Side and Parameter Dependence,, Technical Report Preprint 676, (2000). Google Scholar |
[35] |
E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems,, 2nd edition, (1998).
doi: 10.1007/978-1-4612-0577-7. |
[36] |
Stacks Project Authors, Stacks project,, , (2014). Google Scholar |
[37] |
H. J. Sussmann, An introduction to the coordinate-free maximum principle,, in Geometry of Feedback and Optimal Control (eds. B. Jakubczyk and W. Respondek), (1997), 463.
|
[38] |
W. M. Wonham, Linear Multivariable Control, A Geometric Approach,, 3rd edition, (1985).
doi: 10.1007/978-1-4612-1082-5. |
[39] |
K. Yano and S. Ishihara, Tangent and Cotangent Bundles,, Pure and Applied Mathematics, (1973).
|
[40] |
K. Yano and S. Kobayashi, Prolongations of tensor fields and connections to tangent bundles I. General theory,, J. Math. Soc. Japan, 18 (1966), 194.
doi: 10.2969/jmsj/01820194. |
show all references
References:
[1] |
R. Abraham, J. E. Marsden and T. S. Ratiu, Manifolds, Tensor Analysis, and Applications,, 2nd edition, (1988).
doi: 10.1007/978-1-4612-1029-0. |
[2] |
A. A. Agrachev and R. V. Gamkrelidze, The exponential representation of flows and the chronological calculus,, Math. USSR-Sb., 107 (1978), 467.
|
[3] |
A. A. Agrachev and Y. Sachkov, Control Theory from the Geometric Viewpoint,, Encyclopedia of Mathematical Sciences, (2004).
doi: 10.1007/978-3-662-06404-7. |
[4] |
C. O. Aguilar, Local Controllability of Affine Distributions,, PhD thesis, (2010).
|
[5] |
M. Barbero-Liñán and A. D. Lewis, Geometric interpretations of the symmetric product in affine differential geometry,, Int. J. Geom. Methods Mod. Phys., 9 (2012).
doi: 10.1142/S0219887812500739. |
[6] |
R. Beckmann and A. Deitmar, Strong vector valued integrals, 2011,, , (). Google Scholar |
[7] |
N. Bourbaki, Algebra I,, Elements of Mathematics, (1989). Google Scholar |
[8] |
G. E. Bredon, Sheaf Theory,, 2nd edition, (1997).
doi: 10.1007/978-1-4612-0647-7. |
[9] |
R. W. Brockett, Finite Dimensional Linear Systems,, John Wiley and Sons, (1970). Google Scholar |
[10] |
D. L. Cohn, Measure Theory,, Birkhäuser, (1980).
|
[11] |
C. T. J. Dodson and T. Poston, Tensor Geometry,, Graduate Texts in Mathematics, (1991).
doi: 10.1007/978-3-642-10514-2. |
[12] |
H. Federer, Geometric Measure Theory,, Reprint of 1969 edition, (1969).
|
[13] |
R. Godement, Topologie Algébrique et Théorie Des Faisceaux,, Publications de l'Institut de mathématique de l'Université de Strasbourg, (1958).
|
[14] |
M. W. Hirsch, Differential Topology,, Graduate Texts in Mathematics, (1976).
|
[15] |
A. Isidori, Nonlinear Control Systems,, 3rd edition, (1995).
doi: 10.1007/978-1-84628-615-5. |
[16] |
S. Jafarpour and A. D. Lewis, Time-varying Vector Fields and Their Flows,, To appear in Springer Briefs in Mathematics, (2014).
doi: 10.1007/978-3-319-10139-2. |
[17] |
S. Jafarpour and A. D. Lewis, Locally Convex Topologies and Control Theory,, Submitted to Mathematics of Control, (2015). Google Scholar |
[18] |
H. Jarchow, Locally Convex Spaces,, Mathematical Textbooks, (1981).
|
[19] |
M. Kashiwara and P. Schapira, Sheaves on Manifolds,, Grundlehren der Mathematischen Wissenschaften, (1990).
doi: 10.1007/978-3-662-02661-8. |
[20] |
H. K. Khalil, Nonlinear Systems,, 3rd edition, (2001). Google Scholar |
[21] |
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Volume I,, Interscience Tracts in Pure and Applied Mathematics, (1963).
|
[22] |
I. Kolář, P. W. Michor and J. Slovák, Natural Operations in Differential Geometry,, Springer-Verlag, (1993).
doi: 10.1007/978-3-662-02950-3. |
[23] |
A. D. Lewis, Fundamental problems of geometric control theory,, in Proceedings of the 51st IEEE Conference on Decision and Control, (2012), 7511.
doi: 10.1109/CDC.2012.6427046. |
[24] |
A. D. Lewis, Tautological Control Systems,, Springer Briefs in Electrical and Computer Engineering-Control, (2014).
doi: 10.1007/978-3-319-08638-5. |
[25] |
A. D. Lewis and D. R. Tyner, Geometric Jacobian linearization and LQR theory,, J. Geom. Mech., 2 (2010), 397.
doi: 10.3934/jgm.2010.2.397. |
[26] |
K. C. H. Mackenzie, The General Theory of Lie Groupoids and Lie Algebroids,, London Mathematical Society Lecture Note Series, (2005).
doi: 10.1017/CBO9781107325883. |
[27] |
H. Nijmeijer and A. J. van der Schaft, Nonlinear Dynamical Control Systems,, Springer-Verlag, (1990).
doi: 10.1007/978-1-4757-2101-0. |
[28] |
S. Ramanan, Global Calculus,, Graduate Studies in Mathematics, (2005).
|
[29] |
W. Rudin, Functional Analysis,, 2nd edition, (1991).
|
[30] |
S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds,, Tôhoku Math. J. (2), 10 (1958), 338.
doi: 10.2748/tmj/1178244668. |
[31] |
S. Sastry, Nonlinear Systems: Analysis, Stability, and Control,, Interdisciplinary Applied Mathematics, (1999).
doi: 10.1007/978-1-4757-3108-8. |
[32] |
D. J. Saunders, The Geometry of Jet Bundles,, London Mathematical Society Lecture Note Series, (1989).
doi: 10.1017/CBO9780511526411. |
[33] |
H. H. Schaefer and M. P. Wolff, Topological Vector Spaces,, 2nd edition, (1999).
doi: 10.1007/978-1-4612-1468-7. |
[34] |
F. Schuricht and H. von der Mosel, Ordinary Differential Equations with Measurable Right-Hand Side and Parameter Dependence,, Technical Report Preprint 676, (2000). Google Scholar |
[35] |
E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems,, 2nd edition, (1998).
doi: 10.1007/978-1-4612-0577-7. |
[36] |
Stacks Project Authors, Stacks project,, , (2014). Google Scholar |
[37] |
H. J. Sussmann, An introduction to the coordinate-free maximum principle,, in Geometry of Feedback and Optimal Control (eds. B. Jakubczyk and W. Respondek), (1997), 463.
|
[38] |
W. M. Wonham, Linear Multivariable Control, A Geometric Approach,, 3rd edition, (1985).
doi: 10.1007/978-1-4612-1082-5. |
[39] |
K. Yano and S. Ishihara, Tangent and Cotangent Bundles,, Pure and Applied Mathematics, (1973).
|
[40] |
K. Yano and S. Kobayashi, Prolongations of tensor fields and connections to tangent bundles I. General theory,, J. Math. Soc. Japan, 18 (1966), 194.
doi: 10.2969/jmsj/01820194. |
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