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A mathematical model for hepatitis B with infection-age structure
Planar tilting maneuver of a spacecraft: Singular arcs in the minimum time problem and chattering
1. | Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France, France |
2. | Sorbonne Universités, UPMC Univ. Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Institut Universitaire de France, F-75005, Paris |
References:
[1] |
A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint,, Springer, (2004).
doi: 10.1007/978-3-662-06404-7. |
[2] |
E. Bakolas and Tsiotras, Optimal synthesis of the Zermelo-Markov-Dubins problem in a constant drift field,, Journal of Optimization Theory and Applications, 156 (2013), 469.
doi: 10.1007/s10957-012-0128-0. |
[3] |
J. T. Betts, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming,, Second edition, (2010).
doi: 10.1137/1.9780898718577. |
[4] |
K. D. Bilimoria and B. B. Wie, Time-optimal three-axis reorientation of rigid spacecraft,, Journal of Guidance, 16 (1993), 446.
doi: 10.2514/3.21030. |
[5] |
J. H. Blakelock, Automatic control of aircraft and missiles,, John Wiley and Sons, (1991), 251. Google Scholar |
[6] |
J. F. Bonnans and A. Hermant, Well-posedness of the shooting algorithm for state constrained optimal control problems with a single constraint and control,, SIAM Journal on Control and Optimization, 46 (2007), 1398.
doi: 10.1137/06065756X. |
[7] |
B. Bonnard and M. Chyba, The Role of Singular Trajectories in Control Theory,, Springer Verlag, (2003).
|
[8] |
B. Bonnard, J. B. Caillau and E. Trélat, Geometric optimal control of elliptic Keplerian orbits,, Discrete and Continuous Dynamical Systems, 5 (2005), 929.
doi: 10.3934/dcdsb.2005.5.929. |
[9] |
B. Bonnard, J. B. Caillau and E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control,, ESAIM: Control, 13 (2007), 207.
doi: 10.1051/cocv:2007012. |
[10] |
B. Bonnard, L. Faubourg and E. Trélat, Mécanique Céleste et Contrôle de Systèmes Spatiaux,, Mathématiques and Applications, (2006).
doi: 10.1007/3-540-37640-2. |
[11] |
B. Bonnard and I. Kupka, Generic properties of singular trajectories,, Annales de l'Institut Henri Poincaré, 14 (1997), 167.
doi: 10.1016/S0294-1449(97)80143-6. |
[12] |
L. Cesari, Optimization - Theory and Applications. Problems with Ordinary Differential Equations,, Applications of Mathematics 17, (1983).
doi: 10.1007/978-1-4613-8165-5. |
[13] |
Y. Chitour, F. Jean and E. Trélat, Singular trajectories of control-affine systems,, SIAM Journal on Control and Optimization, 47 (2008), 1078.
doi: 10.1137/060663003. |
[14] |
L. E. Dubins, On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents,, American Journal of mathematics, 79 (1957), 497.
doi: 10.2307/2372560. |
[15] |
A. Fleming and I. M. Ross, Optimal control of spinning axisymmetric spacecraft: A pseudospectral approach,, AIAA Guidance, (2008), 7164.
doi: 10.2514/6.2008-7164. |
[16] |
R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Mathematical Programming Language,, Murray Hill, (1987).
doi: 10.1007/978-3-642-83724-1_12. |
[17] |
A. T. Fuller, Relay control systems optimized for various performance criteria,, In Proceedings of the 1st World Congress IFAC, (1960), 510. Google Scholar |
[18] |
A. T. Fuller, Study of an optimum non-linear control system,, International Journal of Electronics, 15 (1963), 63.
doi: 10.1080/00207216308937555. |
[19] |
S. Gong, H. Baoyin and J. Li, Coupled attitude-orbit dynamics and control for displaced solar orbits,, Acta Astronautica, 65 (2009), 730.
doi: 10.1016/j.actaastro.2009.03.006. |
[20] |
T. Haberkorn and E. Trélat, Convergence results for smooth regularizations of hybrid nonlinear optimal control problems,, SIAM Journal on Control and Optimization, 49 (2011), 1498.
doi: 10.1137/100809209. |
[21] |
H. J. Kelley, R. E. Kopp, H. G. Moyer and H. Gardner, Singular extremals,, in Topics in Optimization (G. Leitmann, (1967), 63.
|
[22] |
D. Kim and J. D. Turner, Near-minimum-time control of asymmetric rigid spacecraft using two controls,, Automatica, 50 (2014), 2084.
doi: 10.1016/j.automatica.2014.05.038. |
[23] |
A. J. Knutson and K. C. Howell, Coupled orbit and attitude dynamics for spacecraft comprised of multiple bodies in Earth-Moon Halo orbits,, In Proceedings of 63rd International Astronautical Congress, (2012), 5951. Google Scholar |
[24] |
A. J. Krener, The high order maximal principle and its application to singular extremals,, SIAM Journal on Control and Optimization, 15 (1977), 256.
doi: 10.1137/0315019. |
[25] |
I. A. K. Kupka, The ubiquity of Fuller's phenomenon,, Nonlinear controllability and optimal control, 133 (1990), 313.
|
[26] |
J. P. Laumond, Robot Motion Planning and Control,, Lecture Notes in Control and Information Sciences, (1998).
doi: 10.1007/BFb0036069. |
[27] |
C. Marchal, Chattering arcs and chattering controls,, Journal of Optimization Theory and Applications, 11 (1973), 441.
doi: 10.1007/BF00935659. |
[28] |
A. A. Markov, Some examples of the solution of a special kind of problem in greatest and least quantities,, (in Russian) Soobshch. Karkovsk. Mat. Obshch. 1, (1887), 250. Google Scholar |
[29] |
J. P. Mcdanell and W. F. Powers, Necessary conditions joining optimal singular and nonsingular subarcs,, SIAM Journal on Control, 9 (1971), 161.
doi: 10.1137/0309014. |
[30] |
T. G. McGee and J. K. Hedrick, Optimal path planning with a kinematic airplane model,, Journal of Guidance, 30 (2007), 629.
doi: 10.2514/1.25042. |
[31] |
V. Y. Glizer, Optimal planar interception with fixed end conditions: Approximate solutions,, Journal of Optimization Theory and Applications, 93 (1997), 1.
doi: 10.1023/A:1022675631937. |
[32] |
L. S. Pontryagin, Mathematical Theory of Optimal Processes,, CRC Press, (1987). Google Scholar |
[33] |
R. Proulx and I. M. Ross, Time-optimal reorientation of asymmetric rigid bodies,, Advances in the Astronautical Sciences, 109 (2001), 1207. Google Scholar |
[34] |
J. A. Reeds and L. A. Shepp, Optimal paths for a car that goes both forwards and backwards,, Pacific journal of mathematics, 145 (1990), 367.
doi: 10.2140/pjm.1990.145.367. |
[35] |
H. Schättler and U. Ledzewicz, Synthesis of optimal controlled trajectories with chattering arcs,, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 19 (2012), 161.
|
[36] |
H. Shen and Tsiotras, Time-optimal control of axisymmetric rigid spacecraft using two controls,, Journal of Guidance, 22 (1999), 682.
doi: 10.2514/2.4436. |
[37] |
C. J. Silva and E. Trélat, Smooth regularization of bang-bang optimal control problems,, IEEE Trans. Automatic Control, 55 (2010), 2488.
doi: 10.1109/TAC.2010.2047742. |
[38] |
J. Stoer and R. Bulirsch, Introduction to Numerical Analysis,, Translated from the German by R. Bartels, (1993).
doi: 10.1007/978-1-4757-2272-7. |
[39] |
H. J. Sussmann and G. Tang, Shortest paths for the reeds-shepp car: a worked out example of the use of geometric techniques in nonlinear optimal control,, Rutgers Center for Systems and Control Technical Report, 10 (1991), 1. Google Scholar |
[40] |
H. J. Sussmann, The Markov-Dubins problem with angular acceleration control,, In Proccedings of the 36th IEEE Conference on Decision and Control, 3 (1997), 2639.
doi: 10.1109/CDC.1997.657778. |
[41] |
L. Techy and C. A. Woolsey, Minimum-time path-planning for unmanned aerial vehicles in steady uniform winds,, Journal of Guidance, 32 (2009), 1736.
doi: 10.2514/1.44580. |
[42] |
J. D. Thorne and C. D. Hall, Minimum-time continuous-thrust orbit transfers using the Kustaanheimo-Stiefel transformation,, Journal of Guidance, 20 (1997), 836. Google Scholar |
[43] |
E. Trélat, Optimal control and applications to aerospace: Some results and challenges,, Journal of Optimization Theory and Applications, 154 (2012), 713.
doi: 10.1007/s10957-012-0050-5. |
[44] |
E. Trélat, Contrôle Optimal: Théorie & Applications,, Vuibert, (2005).
|
[45] |
A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming,, Mathematical Programming, 106 (2006), 25.
doi: 10.1007/s10107-004-0559-y. |
[46] |
P. K. C. Wang and F. Hadaegh, Coordination and Control of Multiple Microspacecraft Moving in Formation,, Journal of the Astronautical Sciences, 44 (1996), 315. Google Scholar |
[47] |
W. M. Wonham, Note on a problem in optimal non-linear control,, Journal of Electronics and Control, 15 (1963), 59.
doi: 10.1080/00207216308937554. |
[48] |
X. Yue, Y. Yang and Z. Geng, Indirect optimization for finite-thrust time-optimal orbital maneuver,, Journal of Guidance, 33 (2010), 628.
doi: 10.2514/1.44885. |
[49] |
M. I. Zelikin and V. F. Borisov, Theory of Chattering Control, with Applications to Astronautics, Robotics, Economics and Engineering,, Systems & Control: Foundations & Applications, (1994).
doi: 10.1007/978-1-4612-2702-1. |
[50] |
M. I. Zelikin and V. F. Borisov, Optimal chattering feedback control,, Journal of Mathematical Sciences, 114 (2003), 1227.
doi: 10.1023/A:1022082011808. |
[51] |
J. Zhu, E. Trélat and M. Cerf, Minimum time control of the rocket attitude reorientation associated with orbit dynamics,, SIAM J. Control Optim., 54 (2016), 391.
doi: 10.1137/15M1028716. |
show all references
References:
[1] |
A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint,, Springer, (2004).
doi: 10.1007/978-3-662-06404-7. |
[2] |
E. Bakolas and Tsiotras, Optimal synthesis of the Zermelo-Markov-Dubins problem in a constant drift field,, Journal of Optimization Theory and Applications, 156 (2013), 469.
doi: 10.1007/s10957-012-0128-0. |
[3] |
J. T. Betts, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming,, Second edition, (2010).
doi: 10.1137/1.9780898718577. |
[4] |
K. D. Bilimoria and B. B. Wie, Time-optimal three-axis reorientation of rigid spacecraft,, Journal of Guidance, 16 (1993), 446.
doi: 10.2514/3.21030. |
[5] |
J. H. Blakelock, Automatic control of aircraft and missiles,, John Wiley and Sons, (1991), 251. Google Scholar |
[6] |
J. F. Bonnans and A. Hermant, Well-posedness of the shooting algorithm for state constrained optimal control problems with a single constraint and control,, SIAM Journal on Control and Optimization, 46 (2007), 1398.
doi: 10.1137/06065756X. |
[7] |
B. Bonnard and M. Chyba, The Role of Singular Trajectories in Control Theory,, Springer Verlag, (2003).
|
[8] |
B. Bonnard, J. B. Caillau and E. Trélat, Geometric optimal control of elliptic Keplerian orbits,, Discrete and Continuous Dynamical Systems, 5 (2005), 929.
doi: 10.3934/dcdsb.2005.5.929. |
[9] |
B. Bonnard, J. B. Caillau and E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control,, ESAIM: Control, 13 (2007), 207.
doi: 10.1051/cocv:2007012. |
[10] |
B. Bonnard, L. Faubourg and E. Trélat, Mécanique Céleste et Contrôle de Systèmes Spatiaux,, Mathématiques and Applications, (2006).
doi: 10.1007/3-540-37640-2. |
[11] |
B. Bonnard and I. Kupka, Generic properties of singular trajectories,, Annales de l'Institut Henri Poincaré, 14 (1997), 167.
doi: 10.1016/S0294-1449(97)80143-6. |
[12] |
L. Cesari, Optimization - Theory and Applications. Problems with Ordinary Differential Equations,, Applications of Mathematics 17, (1983).
doi: 10.1007/978-1-4613-8165-5. |
[13] |
Y. Chitour, F. Jean and E. Trélat, Singular trajectories of control-affine systems,, SIAM Journal on Control and Optimization, 47 (2008), 1078.
doi: 10.1137/060663003. |
[14] |
L. E. Dubins, On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents,, American Journal of mathematics, 79 (1957), 497.
doi: 10.2307/2372560. |
[15] |
A. Fleming and I. M. Ross, Optimal control of spinning axisymmetric spacecraft: A pseudospectral approach,, AIAA Guidance, (2008), 7164.
doi: 10.2514/6.2008-7164. |
[16] |
R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Mathematical Programming Language,, Murray Hill, (1987).
doi: 10.1007/978-3-642-83724-1_12. |
[17] |
A. T. Fuller, Relay control systems optimized for various performance criteria,, In Proceedings of the 1st World Congress IFAC, (1960), 510. Google Scholar |
[18] |
A. T. Fuller, Study of an optimum non-linear control system,, International Journal of Electronics, 15 (1963), 63.
doi: 10.1080/00207216308937555. |
[19] |
S. Gong, H. Baoyin and J. Li, Coupled attitude-orbit dynamics and control for displaced solar orbits,, Acta Astronautica, 65 (2009), 730.
doi: 10.1016/j.actaastro.2009.03.006. |
[20] |
T. Haberkorn and E. Trélat, Convergence results for smooth regularizations of hybrid nonlinear optimal control problems,, SIAM Journal on Control and Optimization, 49 (2011), 1498.
doi: 10.1137/100809209. |
[21] |
H. J. Kelley, R. E. Kopp, H. G. Moyer and H. Gardner, Singular extremals,, in Topics in Optimization (G. Leitmann, (1967), 63.
|
[22] |
D. Kim and J. D. Turner, Near-minimum-time control of asymmetric rigid spacecraft using two controls,, Automatica, 50 (2014), 2084.
doi: 10.1016/j.automatica.2014.05.038. |
[23] |
A. J. Knutson and K. C. Howell, Coupled orbit and attitude dynamics for spacecraft comprised of multiple bodies in Earth-Moon Halo orbits,, In Proceedings of 63rd International Astronautical Congress, (2012), 5951. Google Scholar |
[24] |
A. J. Krener, The high order maximal principle and its application to singular extremals,, SIAM Journal on Control and Optimization, 15 (1977), 256.
doi: 10.1137/0315019. |
[25] |
I. A. K. Kupka, The ubiquity of Fuller's phenomenon,, Nonlinear controllability and optimal control, 133 (1990), 313.
|
[26] |
J. P. Laumond, Robot Motion Planning and Control,, Lecture Notes in Control and Information Sciences, (1998).
doi: 10.1007/BFb0036069. |
[27] |
C. Marchal, Chattering arcs and chattering controls,, Journal of Optimization Theory and Applications, 11 (1973), 441.
doi: 10.1007/BF00935659. |
[28] |
A. A. Markov, Some examples of the solution of a special kind of problem in greatest and least quantities,, (in Russian) Soobshch. Karkovsk. Mat. Obshch. 1, (1887), 250. Google Scholar |
[29] |
J. P. Mcdanell and W. F. Powers, Necessary conditions joining optimal singular and nonsingular subarcs,, SIAM Journal on Control, 9 (1971), 161.
doi: 10.1137/0309014. |
[30] |
T. G. McGee and J. K. Hedrick, Optimal path planning with a kinematic airplane model,, Journal of Guidance, 30 (2007), 629.
doi: 10.2514/1.25042. |
[31] |
V. Y. Glizer, Optimal planar interception with fixed end conditions: Approximate solutions,, Journal of Optimization Theory and Applications, 93 (1997), 1.
doi: 10.1023/A:1022675631937. |
[32] |
L. S. Pontryagin, Mathematical Theory of Optimal Processes,, CRC Press, (1987). Google Scholar |
[33] |
R. Proulx and I. M. Ross, Time-optimal reorientation of asymmetric rigid bodies,, Advances in the Astronautical Sciences, 109 (2001), 1207. Google Scholar |
[34] |
J. A. Reeds and L. A. Shepp, Optimal paths for a car that goes both forwards and backwards,, Pacific journal of mathematics, 145 (1990), 367.
doi: 10.2140/pjm.1990.145.367. |
[35] |
H. Schättler and U. Ledzewicz, Synthesis of optimal controlled trajectories with chattering arcs,, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 19 (2012), 161.
|
[36] |
H. Shen and Tsiotras, Time-optimal control of axisymmetric rigid spacecraft using two controls,, Journal of Guidance, 22 (1999), 682.
doi: 10.2514/2.4436. |
[37] |
C. J. Silva and E. Trélat, Smooth regularization of bang-bang optimal control problems,, IEEE Trans. Automatic Control, 55 (2010), 2488.
doi: 10.1109/TAC.2010.2047742. |
[38] |
J. Stoer and R. Bulirsch, Introduction to Numerical Analysis,, Translated from the German by R. Bartels, (1993).
doi: 10.1007/978-1-4757-2272-7. |
[39] |
H. J. Sussmann and G. Tang, Shortest paths for the reeds-shepp car: a worked out example of the use of geometric techniques in nonlinear optimal control,, Rutgers Center for Systems and Control Technical Report, 10 (1991), 1. Google Scholar |
[40] |
H. J. Sussmann, The Markov-Dubins problem with angular acceleration control,, In Proccedings of the 36th IEEE Conference on Decision and Control, 3 (1997), 2639.
doi: 10.1109/CDC.1997.657778. |
[41] |
L. Techy and C. A. Woolsey, Minimum-time path-planning for unmanned aerial vehicles in steady uniform winds,, Journal of Guidance, 32 (2009), 1736.
doi: 10.2514/1.44580. |
[42] |
J. D. Thorne and C. D. Hall, Minimum-time continuous-thrust orbit transfers using the Kustaanheimo-Stiefel transformation,, Journal of Guidance, 20 (1997), 836. Google Scholar |
[43] |
E. Trélat, Optimal control and applications to aerospace: Some results and challenges,, Journal of Optimization Theory and Applications, 154 (2012), 713.
doi: 10.1007/s10957-012-0050-5. |
[44] |
E. Trélat, Contrôle Optimal: Théorie & Applications,, Vuibert, (2005).
|
[45] |
A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming,, Mathematical Programming, 106 (2006), 25.
doi: 10.1007/s10107-004-0559-y. |
[46] |
P. K. C. Wang and F. Hadaegh, Coordination and Control of Multiple Microspacecraft Moving in Formation,, Journal of the Astronautical Sciences, 44 (1996), 315. Google Scholar |
[47] |
W. M. Wonham, Note on a problem in optimal non-linear control,, Journal of Electronics and Control, 15 (1963), 59.
doi: 10.1080/00207216308937554. |
[48] |
X. Yue, Y. Yang and Z. Geng, Indirect optimization for finite-thrust time-optimal orbital maneuver,, Journal of Guidance, 33 (2010), 628.
doi: 10.2514/1.44885. |
[49] |
M. I. Zelikin and V. F. Borisov, Theory of Chattering Control, with Applications to Astronautics, Robotics, Economics and Engineering,, Systems & Control: Foundations & Applications, (1994).
doi: 10.1007/978-1-4612-2702-1. |
[50] |
M. I. Zelikin and V. F. Borisov, Optimal chattering feedback control,, Journal of Mathematical Sciences, 114 (2003), 1227.
doi: 10.1023/A:1022082011808. |
[51] |
J. Zhu, E. Trélat and M. Cerf, Minimum time control of the rocket attitude reorientation associated with orbit dynamics,, SIAM J. Control Optim., 54 (2016), 391.
doi: 10.1137/15M1028716. |
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