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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, Applied Mathematical Sciences, 75, Springer-Verlag, 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), 467-532. |
[3] |
A. A. Agrachev and Y. Sachkov, Control Theory from the Geometric Viewpoint, Encyclopedia of Mathematical Sciences, 87, Springer-Verlag, New York-Heidelberg-Berlin, 2004.
doi: 10.1007/978-3-662-06404-7. |
[4] |
C. O. Aguilar, Local Controllability of Affine Distributions, PhD thesis, Queen's University, Kingston, Kingston, ON, Canada, 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), 1250073, 33pp.
doi: 10.1142/S0219887812500739. |
[6] |
R. Beckmann and A. Deitmar, Strong vector valued integrals, 2011,, , ().
|
[7] |
N. Bourbaki, Algebra I, Elements of Mathematics, Springer-Verlag, New York-Heidelberg-Berlin, 1989. |
[8] |
G. E. Bredon, Sheaf Theory, 2nd edition, Graduate Texts in Mathematics, 170, Springer-Verlag, New York-Heidelberg-Berlin, 1997.
doi: 10.1007/978-1-4612-0647-7. |
[9] |
R. W. Brockett, Finite Dimensional Linear Systems, John Wiley and Sons, New York, 1970. |
[10] |
D. L. Cohn, Measure Theory, Birkhäuser, Boston-Basel-Stuttgart, 1980. |
[11] |
C. T. J. Dodson and T. Poston, Tensor Geometry, Graduate Texts in Mathematics, 130, Springer-Verlag, New York-Heidelberg-Berlin, 1991.
doi: 10.1007/978-3-642-10514-2. |
[12] |
H. Federer, Geometric Measure Theory, Reprint of 1969 edition, Classics in Mathematics, Springer-Verlag, New York-Heidelberg-Berlin, 1996. |
[13] |
R. Godement, Topologie Algébrique et Théorie Des Faisceaux, Publications de l'Institut de mathématique de l'Université de Strasbourg, 13, Hermann, Paris, 1958. |
[14] |
M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics, 33, Springer-Verlag, New York-Heidelberg-Berlin, 1976. |
[15] |
A. Isidori, Nonlinear Control Systems, 3rd edition, Communications and Control Engineering Series, Springer-Verlag, New York-Heidelberg-Berlin, 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, Signals and Systems, 2015. |
[18] |
H. Jarchow, Locally Convex Spaces, Mathematical Textbooks, Teubner, Leipzig, 1981. |
[19] |
M. Kashiwara and P. Schapira, Sheaves on Manifolds, Grundlehren der Mathematischen Wissenschaften, 292, Springer-Verlag, New York-Heidelberg-Berlin, 1990.
doi: 10.1007/978-3-662-02661-8. |
[20] |
H. K. Khalil, Nonlinear Systems, 3rd edition, Prentice-Hall, Englewood Cliffs, NJ, 2001. |
[21] |
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Volume I, Interscience Tracts in Pure and Applied Mathematics, 15, Interscience Publishers, New York, 1963. |
[22] |
I. Kolář, P. W. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, New York-Heidelberg-Berlin, 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, IEEE, Maui, HI, 2012, 7511-7516.
doi: 10.1109/CDC.2012.6427046. |
[24] |
A. D. Lewis, Tautological Control Systems, Springer Briefs in Electrical and Computer Engineering-Control, Automation and Robotics, Springer-Verlag, New York-Heidelberg-Berlin, 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), 397-440.
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, 213, Cambridge University Press, New York-Port Chester-Melbourne-Sydney, 2005.
doi: 10.1017/CBO9781107325883. |
[27] |
H. Nijmeijer and A. J. van der Schaft, Nonlinear Dynamical Control Systems, Springer-Verlag, New York-Heidelberg-Berlin, 1990.
doi: 10.1007/978-1-4757-2101-0. |
[28] |
S. Ramanan, Global Calculus, Graduate Studies in Mathematics, 65, American Mathematical Society, Providence, RI, 2005. |
[29] |
W. Rudin, Functional Analysis, 2nd edition, International Series in Pure and Applied Mathematics, McGraw-Hill, New York, 1991. |
[30] |
S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tôhoku Math. J. (2), 10 (1958), 338-354.
doi: 10.2748/tmj/1178244668. |
[31] |
S. Sastry, Nonlinear Systems: Analysis, Stability, and Control, Interdisciplinary Applied Mathematics, 10, Springer-Verlag, New York-Heidelberg-Berlin, 1999.
doi: 10.1007/978-1-4757-3108-8. |
[32] |
D. J. Saunders, The Geometry of Jet Bundles, London Mathematical Society Lecture Note Series, 142, Cambridge University Press, New York-Port Chester-Melbourne-Sydney, 1989.
doi: 10.1017/CBO9780511526411. |
[33] |
H. H. Schaefer and M. P. Wolff, Topological Vector Spaces, 2nd edition, Graduate Texts in Mathematics, 3, Springer-Verlag, New York-Heidelberg-Berlin, 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, Universität Bonn, SFB 256, 2000. |
[35] |
E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd edition, Texts in Applied Mathematics, 6, Springer-Verlag, New York-Heidelberg-Berlin, 1998.
doi: 10.1007/978-1-4612-0577-7. |
[36] |
Stacks Project Authors, Stacks project, http://stacks.math.columbia.edu, 2014. |
[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), Dekker Marcel Dekker, New York, 1997, 463-557. |
[38] |
W. M. Wonham, Linear Multivariable Control, A Geometric Approach, 3rd edition, Applications of Mathematics, 10, Springer-Verlag, New York-Heidelberg-Berlin, 1985.
doi: 10.1007/978-1-4612-1082-5. |
[39] |
K. Yano and S. Ishihara, Tangent and Cotangent Bundles, Pure and Applied Mathematics, 16, Dekker Marcel Dekker, New York, 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), 194-210.
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, Applied Mathematical Sciences, 75, Springer-Verlag, 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), 467-532. |
[3] |
A. A. Agrachev and Y. Sachkov, Control Theory from the Geometric Viewpoint, Encyclopedia of Mathematical Sciences, 87, Springer-Verlag, New York-Heidelberg-Berlin, 2004.
doi: 10.1007/978-3-662-06404-7. |
[4] |
C. O. Aguilar, Local Controllability of Affine Distributions, PhD thesis, Queen's University, Kingston, Kingston, ON, Canada, 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), 1250073, 33pp.
doi: 10.1142/S0219887812500739. |
[6] |
R. Beckmann and A. Deitmar, Strong vector valued integrals, 2011,, , ().
|
[7] |
N. Bourbaki, Algebra I, Elements of Mathematics, Springer-Verlag, New York-Heidelberg-Berlin, 1989. |
[8] |
G. E. Bredon, Sheaf Theory, 2nd edition, Graduate Texts in Mathematics, 170, Springer-Verlag, New York-Heidelberg-Berlin, 1997.
doi: 10.1007/978-1-4612-0647-7. |
[9] |
R. W. Brockett, Finite Dimensional Linear Systems, John Wiley and Sons, New York, 1970. |
[10] |
D. L. Cohn, Measure Theory, Birkhäuser, Boston-Basel-Stuttgart, 1980. |
[11] |
C. T. J. Dodson and T. Poston, Tensor Geometry, Graduate Texts in Mathematics, 130, Springer-Verlag, New York-Heidelberg-Berlin, 1991.
doi: 10.1007/978-3-642-10514-2. |
[12] |
H. Federer, Geometric Measure Theory, Reprint of 1969 edition, Classics in Mathematics, Springer-Verlag, New York-Heidelberg-Berlin, 1996. |
[13] |
R. Godement, Topologie Algébrique et Théorie Des Faisceaux, Publications de l'Institut de mathématique de l'Université de Strasbourg, 13, Hermann, Paris, 1958. |
[14] |
M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics, 33, Springer-Verlag, New York-Heidelberg-Berlin, 1976. |
[15] |
A. Isidori, Nonlinear Control Systems, 3rd edition, Communications and Control Engineering Series, Springer-Verlag, New York-Heidelberg-Berlin, 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, Signals and Systems, 2015. |
[18] |
H. Jarchow, Locally Convex Spaces, Mathematical Textbooks, Teubner, Leipzig, 1981. |
[19] |
M. Kashiwara and P. Schapira, Sheaves on Manifolds, Grundlehren der Mathematischen Wissenschaften, 292, Springer-Verlag, New York-Heidelberg-Berlin, 1990.
doi: 10.1007/978-3-662-02661-8. |
[20] |
H. K. Khalil, Nonlinear Systems, 3rd edition, Prentice-Hall, Englewood Cliffs, NJ, 2001. |
[21] |
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Volume I, Interscience Tracts in Pure and Applied Mathematics, 15, Interscience Publishers, New York, 1963. |
[22] |
I. Kolář, P. W. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, New York-Heidelberg-Berlin, 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, IEEE, Maui, HI, 2012, 7511-7516.
doi: 10.1109/CDC.2012.6427046. |
[24] |
A. D. Lewis, Tautological Control Systems, Springer Briefs in Electrical and Computer Engineering-Control, Automation and Robotics, Springer-Verlag, New York-Heidelberg-Berlin, 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), 397-440.
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, 213, Cambridge University Press, New York-Port Chester-Melbourne-Sydney, 2005.
doi: 10.1017/CBO9781107325883. |
[27] |
H. Nijmeijer and A. J. van der Schaft, Nonlinear Dynamical Control Systems, Springer-Verlag, New York-Heidelberg-Berlin, 1990.
doi: 10.1007/978-1-4757-2101-0. |
[28] |
S. Ramanan, Global Calculus, Graduate Studies in Mathematics, 65, American Mathematical Society, Providence, RI, 2005. |
[29] |
W. Rudin, Functional Analysis, 2nd edition, International Series in Pure and Applied Mathematics, McGraw-Hill, New York, 1991. |
[30] |
S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tôhoku Math. J. (2), 10 (1958), 338-354.
doi: 10.2748/tmj/1178244668. |
[31] |
S. Sastry, Nonlinear Systems: Analysis, Stability, and Control, Interdisciplinary Applied Mathematics, 10, Springer-Verlag, New York-Heidelberg-Berlin, 1999.
doi: 10.1007/978-1-4757-3108-8. |
[32] |
D. J. Saunders, The Geometry of Jet Bundles, London Mathematical Society Lecture Note Series, 142, Cambridge University Press, New York-Port Chester-Melbourne-Sydney, 1989.
doi: 10.1017/CBO9780511526411. |
[33] |
H. H. Schaefer and M. P. Wolff, Topological Vector Spaces, 2nd edition, Graduate Texts in Mathematics, 3, Springer-Verlag, New York-Heidelberg-Berlin, 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, Universität Bonn, SFB 256, 2000. |
[35] |
E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd edition, Texts in Applied Mathematics, 6, Springer-Verlag, New York-Heidelberg-Berlin, 1998.
doi: 10.1007/978-1-4612-0577-7. |
[36] |
Stacks Project Authors, Stacks project, http://stacks.math.columbia.edu, 2014. |
[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), Dekker Marcel Dekker, New York, 1997, 463-557. |
[38] |
W. M. Wonham, Linear Multivariable Control, A Geometric Approach, 3rd edition, Applications of Mathematics, 10, Springer-Verlag, New York-Heidelberg-Berlin, 1985.
doi: 10.1007/978-1-4612-1082-5. |
[39] |
K. Yano and S. Ishihara, Tangent and Cotangent Bundles, Pure and Applied Mathematics, 16, Dekker Marcel Dekker, New York, 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), 194-210.
doi: 10.2969/jmsj/01820194. |
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