-
Previous Article
Embedded geodesic problems and optimal control for matrix Lie groups
- JGM Home
- This Issue
- Next Article
Lyapunov constraints and global asymptotic stabilization
1. | Centro Atmico Bariloche and Instituto Balseiro, 8400 S.C. de Bariloche, and CONICET, Argentina |
2. | Control and Dynamical Systems 107-81, California Institute of Technology, Pasadena, CA 91125, United States |
3. | United Technologies Research Center, East Hartford, CT 06118, United States |
References:
[1] |
R. Abraham and J. E. Marsden, "Foundation of Mechanics,", New York, (1985). Google Scholar |
[2] |
V. I. Arnold, "Mathematical Models in Classical Mechanics,", Graduate Texts in Mathematics, 60 (1978).
|
[3] |
A. M. Bloch, "Nonholonomic Mechanics and Control,", volume 24 of Interdisciplinary Applied Mathematics, (2003).
|
[4] |
A. M. Bloch, D. E. Chang, N. E. Leonard and J. E. Marsden, Controlled Lagrangian and the stabilization of mechanical systems II: Potential shaping,, IEEE Trans. Automat. Control, 46 (2001), 1556.
doi: 10.1109/9.956051. |
[5] |
A. M. Bloch, N. E. Leonard and J. E. Marsden, Controlled Lagrangian and the stabilization of mechanical systems I: The first matching theorem,, IEEE Trans. Automat. Control, 45 (2000), 2253.
doi: 10.1109/9.895562. |
[6] |
W. M. Boothby, "An Introduction to Differentiable Manifolds and Riemannian Geometry," 2nd edition, Pure and Applied Mathematics, 120,, Academic Press, (1986).
|
[7] |
F. Bullo and A. Lewis, "Geometric Control of Mechanical Systems," Texts in Applied Mathematics, 49,, Springer-Verlag, (2005).
|
[8] |
H. Cendra and S. Grillo, Generalized nonholonomic mechanics, servomechanisms and related brackets,, J. Math. Phys., 47 (2006), 2209.
doi: 10.1063/1.2165797. |
[9] |
H. Cendra and S. Grillo, Lagrangian systems with higher order constraints,, J. Math. Phys., 48 (2007).
|
[10] |
H. Cendra, A. Ibort, M. de León and D. Martin de Diego, A generalization of Chetaev's principle for a class of higher order non-holonomic constraints,, J. Math. Phys., 45 (2004), 2785.
doi: 10.1063/1.1763245. |
[11] |
D. Chang, A. M. Bloch, N. E. Leonard, J. E. Marsden and C. Woolsey, "The Equivalence of Controlled Lagrangian and Controlled Hamiltonian Systems,", \emph{ESIAM: Control, (2001). Google Scholar |
[12] |
B. Gharesifard, A. D. Lewis and A.-R. Mansouri, A geometric framework for stabilization by energy shaping: Sufficient conditions for existence of solutions,, Communications for Information and Systems, 8 (2008), 353.
|
[13] |
S. Grillo, "Sistemas Noholónomos Generalizados,", Ph.D thesis, (2007). Google Scholar |
[14] |
S. Grillo, Higher order constrained Hamiltonian systems,, J. Math. Phys., 50 (2009).
|
[15] |
S. Grillo, F. Maciel and D. Pérez, Closed-loop and constrained mechanical systems,, International Journal of Geometric Methods in Modern Physics, (2010).
doi: 10.1142/S0219887810004580. |
[16] |
H. Khalil, "Nonlinear Systems,", Upper Saddle River NJ, (1996). Google Scholar |
[17] |
S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry,", New York, (1963). Google Scholar |
[18] |
C.-M. Marle, Kinematic and geometric constraints, servomechanism and control of mechanical systems, Geometrical Structures for Physical Theories, II (Vietri, 1996),, Rend. Sem. Mat. Univ. Pol. Torino \textbf{54} (1996), 54 (1996), 353.
|
[19] |
J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," Texts in Applied Mathematics, 17,, Springer-Verlag, (1994).
|
[20] |
J. E. Marsden and T. S. Ratiu, "Manifolds, Tensor Analysis and Applications,", New York, (2001). Google Scholar |
[21] |
R. Ortega, M. W. Spong, F. Gómez-Estern and G. Blankenstein, Stabilization of underactuated mechanical systems via interconnection and damping assignment,, IEEE Trans. Aut. Control, 47 (2002), 1281.
doi: 10.1109/TAC.2002.800770. |
[22] |
D. Pérez, Sistemas noholónomos generalizados y su aplicación a la teoría de control automático mediante vínculos cinemáticos,, Proyecto Integrador, (2006). Google Scholar |
[23] |
D. Pérez, "Sistemas con vínculos de orden superior y su aplicación a la teoría de control automático,", Master thesis, (2007). Google Scholar |
[24] |
J. Rayleigh, "The Theory of Sound," 2nd edition,, Dover Publications, (1945).
|
[25] |
A. Shiriaev, J. W. Perram and C. Canudas-de-Wit, Constructive tool for orbital stabilization of underactuated nonlinear systems: Virtual constraints approach,, IEEE Transactions on Automatic Control, 50 (2005), 1164.
doi: 10.1109/TAC.2005.852568. |
[26] |
E. Sontag, "Mathematical Control Theory," Texts in Applied Mathematics, 6,, Springer-Verlag, (1998).
|
[27] |
M. W. Spong, P. Corke and R. Lozano, Nonlinear control of the inertia wheel pendulum,, Automatica, 37 (2001), 1845.
doi: 10.1016/S0005-1098(01)00145-5. |
[28] |
E. T. Whittaker, "A Treatise on The Analytical Dynamics of Particles and Rigid Bodies,", Cambridge University Press, (1937).
|
[29] |
C. Woolsey, C. Reddy, A. Bloch, D. Chang, N. Leonard and J. Marsden, Controlled Lagrangian systems with gyroscopic forcing and dissipation,, European Journal of Control, 10 (2004), 478.
doi: 10.3166/ejc.10.478-496. |
show all references
References:
[1] |
R. Abraham and J. E. Marsden, "Foundation of Mechanics,", New York, (1985). Google Scholar |
[2] |
V. I. Arnold, "Mathematical Models in Classical Mechanics,", Graduate Texts in Mathematics, 60 (1978).
|
[3] |
A. M. Bloch, "Nonholonomic Mechanics and Control,", volume 24 of Interdisciplinary Applied Mathematics, (2003).
|
[4] |
A. M. Bloch, D. E. Chang, N. E. Leonard and J. E. Marsden, Controlled Lagrangian and the stabilization of mechanical systems II: Potential shaping,, IEEE Trans. Automat. Control, 46 (2001), 1556.
doi: 10.1109/9.956051. |
[5] |
A. M. Bloch, N. E. Leonard and J. E. Marsden, Controlled Lagrangian and the stabilization of mechanical systems I: The first matching theorem,, IEEE Trans. Automat. Control, 45 (2000), 2253.
doi: 10.1109/9.895562. |
[6] |
W. M. Boothby, "An Introduction to Differentiable Manifolds and Riemannian Geometry," 2nd edition, Pure and Applied Mathematics, 120,, Academic Press, (1986).
|
[7] |
F. Bullo and A. Lewis, "Geometric Control of Mechanical Systems," Texts in Applied Mathematics, 49,, Springer-Verlag, (2005).
|
[8] |
H. Cendra and S. Grillo, Generalized nonholonomic mechanics, servomechanisms and related brackets,, J. Math. Phys., 47 (2006), 2209.
doi: 10.1063/1.2165797. |
[9] |
H. Cendra and S. Grillo, Lagrangian systems with higher order constraints,, J. Math. Phys., 48 (2007).
|
[10] |
H. Cendra, A. Ibort, M. de León and D. Martin de Diego, A generalization of Chetaev's principle for a class of higher order non-holonomic constraints,, J. Math. Phys., 45 (2004), 2785.
doi: 10.1063/1.1763245. |
[11] |
D. Chang, A. M. Bloch, N. E. Leonard, J. E. Marsden and C. Woolsey, "The Equivalence of Controlled Lagrangian and Controlled Hamiltonian Systems,", \emph{ESIAM: Control, (2001). Google Scholar |
[12] |
B. Gharesifard, A. D. Lewis and A.-R. Mansouri, A geometric framework for stabilization by energy shaping: Sufficient conditions for existence of solutions,, Communications for Information and Systems, 8 (2008), 353.
|
[13] |
S. Grillo, "Sistemas Noholónomos Generalizados,", Ph.D thesis, (2007). Google Scholar |
[14] |
S. Grillo, Higher order constrained Hamiltonian systems,, J. Math. Phys., 50 (2009).
|
[15] |
S. Grillo, F. Maciel and D. Pérez, Closed-loop and constrained mechanical systems,, International Journal of Geometric Methods in Modern Physics, (2010).
doi: 10.1142/S0219887810004580. |
[16] |
H. Khalil, "Nonlinear Systems,", Upper Saddle River NJ, (1996). Google Scholar |
[17] |
S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry,", New York, (1963). Google Scholar |
[18] |
C.-M. Marle, Kinematic and geometric constraints, servomechanism and control of mechanical systems, Geometrical Structures for Physical Theories, II (Vietri, 1996),, Rend. Sem. Mat. Univ. Pol. Torino \textbf{54} (1996), 54 (1996), 353.
|
[19] |
J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," Texts in Applied Mathematics, 17,, Springer-Verlag, (1994).
|
[20] |
J. E. Marsden and T. S. Ratiu, "Manifolds, Tensor Analysis and Applications,", New York, (2001). Google Scholar |
[21] |
R. Ortega, M. W. Spong, F. Gómez-Estern and G. Blankenstein, Stabilization of underactuated mechanical systems via interconnection and damping assignment,, IEEE Trans. Aut. Control, 47 (2002), 1281.
doi: 10.1109/TAC.2002.800770. |
[22] |
D. Pérez, Sistemas noholónomos generalizados y su aplicación a la teoría de control automático mediante vínculos cinemáticos,, Proyecto Integrador, (2006). Google Scholar |
[23] |
D. Pérez, "Sistemas con vínculos de orden superior y su aplicación a la teoría de control automático,", Master thesis, (2007). Google Scholar |
[24] |
J. Rayleigh, "The Theory of Sound," 2nd edition,, Dover Publications, (1945).
|
[25] |
A. Shiriaev, J. W. Perram and C. Canudas-de-Wit, Constructive tool for orbital stabilization of underactuated nonlinear systems: Virtual constraints approach,, IEEE Transactions on Automatic Control, 50 (2005), 1164.
doi: 10.1109/TAC.2005.852568. |
[26] |
E. Sontag, "Mathematical Control Theory," Texts in Applied Mathematics, 6,, Springer-Verlag, (1998).
|
[27] |
M. W. Spong, P. Corke and R. Lozano, Nonlinear control of the inertia wheel pendulum,, Automatica, 37 (2001), 1845.
doi: 10.1016/S0005-1098(01)00145-5. |
[28] |
E. T. Whittaker, "A Treatise on The Analytical Dynamics of Particles and Rigid Bodies,", Cambridge University Press, (1937).
|
[29] |
C. Woolsey, C. Reddy, A. Bloch, D. Chang, N. Leonard and J. Marsden, Controlled Lagrangian systems with gyroscopic forcing and dissipation,, European Journal of Control, 10 (2004), 478.
doi: 10.3166/ejc.10.478-496. |
[1] |
Francisco Braun, Jaume Llibre, Ana Cristina Mereu. Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5245-5255. doi: 10.3934/dcds.2016029 |
[2] |
Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329 |
[3] |
Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166 |
[4] |
Yuri Chekanov, Felix Schlenk. Notes on monotone Lagrangian twist tori. Electronic Research Announcements, 2010, 17: 104-121. doi: 10.3934/era.2010.17.104 |
[5] |
Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1 |
[6] |
Christopher Bose, Rua Murray. Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 597-615. doi: 10.3934/dcds.2006.14.597 |
[7] |
Yves Dumont, Frederic Chiroleu. Vector control for the Chikungunya disease. Mathematical Biosciences & Engineering, 2010, 7 (2) : 313-345. doi: 10.3934/mbe.2010.7.313 |
[8] |
Mohsen Abdolhosseinzadeh, Mir Mohammad Alipour. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 321-332. doi: 10.3934/naco.2020028 |
[9] |
Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4329-4346. doi: 10.3934/dcds.2017185 |
[10] |
Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243 |
[11] |
Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044 |
[12] |
Zhi-Min Chen, Philip A. Wilson. Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2329-2341. doi: 10.3934/dcdsb.2012.17.2329 |
[13] |
J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008 |
[14] |
Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 |
[15] |
Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267 |
[16] |
Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017 |
[17] |
Olena Naboka. On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1933-1956. doi: 10.3934/cpaa.2009.8.1933 |
[18] |
Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185 |
[19] |
Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206 |
[20] |
Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448 |
2019 Impact Factor: 0.649
Tools
Metrics
Other articles
by authors
[Back to Top]