Planar tilting maneuver of a spacecraft: Singular arcs in the minimum time problem and chattering

Previous Article
DCDS-B Home
This Issue
Next Article
A mathematical model for hepatitis B with infection-age structure

June 2016, 21(4): 1347-1388. doi: 10.3934/dcdsb.2016.21.1347

Planar tilting maneuver of a spacecraft: Singular arcs in the minimum time problem and chattering

Jiamin Zhu ^1, , Emmanuel Trélat ^2, and Max Cerf ^1,

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

Received April 2015 Published March 2016

Abstract
Related Papers

In this paper, we study the minimum time planar tilting maneuver of a spacecraft, from the theoretical as well as from the numerical point of view, with a particular focus on the chattering phenomenon. We prove that there exist optimal chattering arcs when a singular junction occurs. Our study is based on the Pontryagin Maximum Principle and on results by M.I. Zelikin and V.F. Borisov. We give sufficient conditions on the terminal values under which the optimal solutions do not contain any singular arc, and are bang-bang with a finite number of switchings. Moreover, we implement sub-optimal strategies by replacing the chattering control with a fixed number of piecewise constant controls. Numerical simulations illustrate our results.

Keywords: singular control, Pontryagin Maximum Principle, Spacecraft planar tilting maneuver, chattering arcs, sub-optimal strategy., minimum time control.

Mathematics Subject Classification: Primary: 49K15, 49M37, 65H2.

Citation: Jiamin Zhu, Emmanuel Trélat, Max Cerf. Planar tilting maneuver of a spacecraft: Singular arcs in the minimum time problem and chattering. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1347-1388. doi: 10.3934/dcdsb.2016.21.1347

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), 469-492. doi: 10.1007/s10957-012-0128-0.
[3]	J. T. Betts, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Second edition, Advances in Design and Control, SIAM, Philadelphia, PA, 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, Control, and Dynamics, 16 (1993), 446-452. doi: 10.2514/3.21030.
[5]	J. H. Blakelock, Automatic control of aircraft and missiles, John Wiley and Sons, 1991, 251-252.
[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), 1398-1430. 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, Series B, 5 (2005), 929-956. 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, Optimisation and Calculus of Variations, 13 (2007), 207-236. doi: 10.1051/cocv:2007012.
[10]	B. Bonnard, L. Faubourg and E. Trélat, Mécanique Céleste et Contrôle de Systèmes Spatiaux, Mathématiques and Applications, Springer Verlag, 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, Analyse Non Linéaire, 14 (1997), 167-186. doi: 10.1016/S0294-1449(97)80143-6.
[12]	L. Cesari, Optimization - Theory and Applications. Problems with Ordinary Differential Equations, Applications of Mathematics 17, Springer Verlag, 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), 1078-1095. 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), 497-516. doi: 10.2307/2372560.
[15]	A. Fleming and I. M. Ross, Optimal control of spinning axisymmetric spacecraft: A pseudospectral approach, AIAA Guidance, Navigation and Control Conference and Exhibit, 2008, 7164-7173. 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, Moscow, 1960, 510-519.
[18]	A. T. Fuller, Study of an optimum non-linear control system, International Journal of Electronics, 15 (1963), 63-71. 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), 730-737. 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), 1498-1522. 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, ed.), 63-101, Academic Press, New York, 1967.
[22]	D. Kim and J. D. Turner, Near-minimum-time control of asymmetric rigid spacecraft using two controls, Automatica, 50 (2014), 2084-2089. 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), 5951-5966.
[24]	A. J. Krener, The high order maximal principle and its application to singular extremals, SIAM Journal on Control and Optimization, 15 (1977), 256-293. doi: 10.1137/0315019.
[25]	I. A. K. Kupka, The ubiquity of Fuller's phenomenon, Nonlinear controllability and optimal control, 133 (1990), 313-350.
[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), 441-468. 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, 250-276.
[29]	J. P. Mcdanell and W. F. Powers, Necessary conditions joining optimal singular and nonsingular subarcs, SIAM Journal on Control, 9 (1971), 161-173. doi: 10.1137/0309014.
[30]	T. G. McGee and J. K. Hedrick, Optimal path planning with a kinematic airplane model, Journal of Guidance, Control, and Dynamics, 30 (2007), 629-633. 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), 1-25. doi: 10.1023/A:1022675631937.
[32]	L. S. Pontryagin, Mathematical Theory of Optimal Processes, CRC Press, 1987.
[33]	R. Proulx and I. M. Ross, Time-optimal reorientation of asymmetric rigid bodies, Advances in the Astronautical Sciences, 109 (2001), 1207-1227.
[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), 367-393. 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), 161-186.
[36]	H. Shen and Tsiotras, Time-optimal control of axisymmetric rigid spacecraft using two controls, Journal of Guidance, Control, and Dynamics, 22 (1999), 682-694. 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), 2488-2499. doi: 10.1109/TAC.2010.2047742.
[38]	J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Translated from the German by R. Bartels, W. Gautschi and C. Witzgall. Second edition. Texts in Applied Mathematics, 12. Springer Verlag, New York, 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), 1-71.
[40]	H. J. Sussmann, The Markov-Dubins problem with angular acceleration control, In Proccedings of the 36th IEEE Conference on Decision and Control, INSTITUTE OF ELECTRICAL ENGINEERS INC (IEE), 3, (1997), 2639-2643. 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, Control, and Dynamics, 32 (2009), 1736-1746. 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, Control, and Dynamics, 20 (1997), 836-838.
[43]	E. Trélat, Optimal control and applications to aerospace: Some results and challenges, Journal of Optimization Theory and Applications, 154 (2012), 713-758. doi: 10.1007/s10957-012-0050-5.
[44]	E. Trélat, Contrôle Optimal: Théorie & Applications, Vuibert, Paris, 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), 25-57. 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), 315-355.
[47]	W. M. Wonham, Note on a problem in optimal non-linear control, Journal of Electronics and Control, 15 (1963), 59-62. doi: 10.1080/00207216308937554.
[48]	X. Yue, Y. Yang and Z. Geng, Indirect optimization for finite-thrust time-optimal orbital maneuver, Journal of Guidance, Control, and Dynamics, 33 (2010), 628-634. 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, Birkhäuser Boston, Inc., Boston, MA, 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), 1227-1344. 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), 391-422, arXiv:1507.00172. 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), 469-492. doi: 10.1007/s10957-012-0128-0.
[3]	J. T. Betts, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Second edition, Advances in Design and Control, SIAM, Philadelphia, PA, 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, Control, and Dynamics, 16 (1993), 446-452. doi: 10.2514/3.21030.
[5]	J. H. Blakelock, Automatic control of aircraft and missiles, John Wiley and Sons, 1991, 251-252.
[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), 1398-1430. 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, Series B, 5 (2005), 929-956. 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, Optimisation and Calculus of Variations, 13 (2007), 207-236. doi: 10.1051/cocv:2007012.
[10]	B. Bonnard, L. Faubourg and E. Trélat, Mécanique Céleste et Contrôle de Systèmes Spatiaux, Mathématiques and Applications, Springer Verlag, 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, Analyse Non Linéaire, 14 (1997), 167-186. doi: 10.1016/S0294-1449(97)80143-6.
[12]	L. Cesari, Optimization - Theory and Applications. Problems with Ordinary Differential Equations, Applications of Mathematics 17, Springer Verlag, 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), 1078-1095. 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), 497-516. doi: 10.2307/2372560.
[15]	A. Fleming and I. M. Ross, Optimal control of spinning axisymmetric spacecraft: A pseudospectral approach, AIAA Guidance, Navigation and Control Conference and Exhibit, 2008, 7164-7173. 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, Moscow, 1960, 510-519.
[18]	A. T. Fuller, Study of an optimum non-linear control system, International Journal of Electronics, 15 (1963), 63-71. 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), 730-737. 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), 1498-1522. 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, ed.), 63-101, Academic Press, New York, 1967.
[22]	D. Kim and J. D. Turner, Near-minimum-time control of asymmetric rigid spacecraft using two controls, Automatica, 50 (2014), 2084-2089. 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), 5951-5966.
[24]	A. J. Krener, The high order maximal principle and its application to singular extremals, SIAM Journal on Control and Optimization, 15 (1977), 256-293. doi: 10.1137/0315019.
[25]	I. A. K. Kupka, The ubiquity of Fuller's phenomenon, Nonlinear controllability and optimal control, 133 (1990), 313-350.
[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), 441-468. 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, 250-276.
[29]	J. P. Mcdanell and W. F. Powers, Necessary conditions joining optimal singular and nonsingular subarcs, SIAM Journal on Control, 9 (1971), 161-173. doi: 10.1137/0309014.
[30]	T. G. McGee and J. K. Hedrick, Optimal path planning with a kinematic airplane model, Journal of Guidance, Control, and Dynamics, 30 (2007), 629-633. 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), 1-25. doi: 10.1023/A:1022675631937.
[32]	L. S. Pontryagin, Mathematical Theory of Optimal Processes, CRC Press, 1987.
[33]	R. Proulx and I. M. Ross, Time-optimal reorientation of asymmetric rigid bodies, Advances in the Astronautical Sciences, 109 (2001), 1207-1227.
[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), 367-393. 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), 161-186.
[36]	H. Shen and Tsiotras, Time-optimal control of axisymmetric rigid spacecraft using two controls, Journal of Guidance, Control, and Dynamics, 22 (1999), 682-694. 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), 2488-2499. doi: 10.1109/TAC.2010.2047742.
[38]	J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Translated from the German by R. Bartels, W. Gautschi and C. Witzgall. Second edition. Texts in Applied Mathematics, 12. Springer Verlag, New York, 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), 1-71.
[40]	H. J. Sussmann, The Markov-Dubins problem with angular acceleration control, In Proccedings of the 36th IEEE Conference on Decision and Control, INSTITUTE OF ELECTRICAL ENGINEERS INC (IEE), 3, (1997), 2639-2643. 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, Control, and Dynamics, 32 (2009), 1736-1746. 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, Control, and Dynamics, 20 (1997), 836-838.
[43]	E. Trélat, Optimal control and applications to aerospace: Some results and challenges, Journal of Optimization Theory and Applications, 154 (2012), 713-758. doi: 10.1007/s10957-012-0050-5.
[44]	E. Trélat, Contrôle Optimal: Théorie & Applications, Vuibert, Paris, 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), 25-57. 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), 315-355.
[47]	W. M. Wonham, Note on a problem in optimal non-linear control, Journal of Electronics and Control, 15 (1963), 59-62. doi: 10.1080/00207216308937554.
[48]	X. Yue, Y. Yang and Z. Geng, Indirect optimization for finite-thrust time-optimal orbital maneuver, Journal of Guidance, Control, and Dynamics, 33 (2010), 628-634. 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, Birkhäuser Boston, Inc., Boston, MA, 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), 1227-1344. 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), 391-422, arXiv:1507.00172. doi: 10.1137/15M1028716.

[1]	Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations and Control Theory, 2022, 11 (2) : 347-371. doi: 10.3934/eect.2020110
[2]	Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control and Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195
[3]	Guy Barles, Ariela Briani, Emmanuel Trélat. Value function for regional control problems via dynamic programming and Pontryagin maximum principle. Mathematical Control and Related Fields, 2018, 8 (3&4) : 509-533. doi: 10.3934/mcrf.2018021
[4]	Xiao-Li Ding, Iván Area, Juan J. Nieto. Controlled singular evolution equations and Pontryagin type maximum principle with applications. Evolution Equations and Control Theory, 2021 doi: 10.3934/eect.2021059
[5]	Zhen Wu, Feng Zhang. Maximum principle for discrete-time stochastic optimal control problem and stochastic game. Mathematical Control and Related Fields, 2022, 12 (2) : 475-493. doi: 10.3934/mcrf.2021031
[6]	Shi'an Wang, N. U. Ahmed. Optimal control and stabilization of building maintenance units based on minimum principle. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1713-1727. doi: 10.3934/jimo.2020041
[7]	Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control and Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61
[8]	Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control and Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018
[9]	Md. Haider Ali Biswas, Maria do Rosário de Pinho. A nonsmooth maximum principle for optimal control problems with state and mixed constraints - convex case. Conference Publications, 2011, 2011 (Special) : 174-183. doi: 10.3934/proc.2011.2011.174
[10]	Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161
[11]	Omid S. Fard, Javad Soolaki, Delfim F. M. Torres. A necessary condition of Pontryagin type for fuzzy fractional optimal control problems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (1) : 59-76. doi: 10.3934/dcdss.2018004
[12]	Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control and Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019
[13]	Mehdi Badra. Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1169-1208. doi: 10.3934/dcds.2012.32.1169
[14]	Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581
[15]	Ellina Grigorieva, Evgenii Khailov. Chattering and its approximation in control of psoriasis treatment. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2251-2280. doi: 10.3934/dcdsb.2019094
[16]	H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete and Continuous Dynamical Systems, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77
[17]	Shaolin Ji, Xiaole Xue. A stochastic maximum principle for linear quadratic problem with nonconvex control domain. Mathematical Control and Related Fields, 2019, 9 (3) : 495-507. doi: 10.3934/mcrf.2019022
[18]	Andrei V. Dmitruk, Nikolai P. Osmolovskii. Necessary conditions for a weak minimum in optimal control problems with integral equations on a variable time interval. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4323-4343. doi: 10.3934/dcds.2015.35.4323
[19]	Andrei V. Dmitruk, Nikolai P. Osmolovski. Necessary conditions for a weak minimum in a general optimal control problem with integral equations on a variable time interval. Mathematical Control and Related Fields, 2017, 7 (4) : 507-535. doi: 10.3934/mcrf.2017019
[20]	Yujing Wang, Changjun Yu, Kok Lay Teo. A new computational strategy for optimal control problem with a cost on changing control. Numerical Algebra, Control and Optimization, 2016, 6 (3) : 339-364. doi: 10.3934/naco.2016016

2021 Impact Factor: 1.497

Tools

Metrics

Other articles
by authors

on AIMS
Jiamin Zhu Emmanuel Trélat Max Cerf
on Google Scholar
Jiamin Zhu Emmanuel Trélat Max Cerf

[Back to Top]