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On switching properties of time optimal controls for linear ODEs
| 1. | School of Science, Tianjin University of Commerce, Tianjin, 300134, China |
| 2. | Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China |
| 3. | School of Mathematics, Tianjin University, Tianjin, 300354, China |
In this paper, we present some properties of time optimal controls for linear ODEs with the ball-type control constraint. More precisely, given an optimal control, we build up an upper bound for the number of its switching points; show that it jumps from one direction to the reverse direction at each switching point; give its dynamic behaviour between two consecutive switching points; and study its switching directions.
References:
| [1] |
A. A. Agrachev and C. Biolo,
Switching in time-optimal problem with control in a ball, SIAM J. Control Optim., 56 (2018), 183-120.
doi: 10.1137/16M110304X. |
| [2] |
A. A. Agrachev and C. Biolo,
Switching in time-optimal problem: The 3D Case with 2D control, J. Dyn. Control Syst., 23 (2017), 577-595.
doi: 10.1007/s10883-016-9342-7. |
| [3] |
A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, Encyclopaedia of Mathematical Sciences, 87, Springer-Verlag, Berlin, 2004.
doi: 10.1007/978-3-662-06404-7. |
| [4] |
R. Bellman, I. Glicksberg and O. Gross,
On the "bang-bang" control problem, Quart. Appl. Math., 14 (1956), 11-18.
doi: 10.1090/qam/78516. |
| [5] |
C. Biolo, Switching in Time-Optimal Problem, Ph.D thesis, Scuola Internazionale Superiore di Studi Avanzati - Trieste, 2017. |
| [6] |
C. K. Chui and G. Chen, Linear Systems and Optimal Control, Springer Series in Information Sciences, 18, Springer-Verlag, Berlin, 1989.
doi: 10.1007/978-3-642-61312-8. |
| [7] |
R. Conti, Teoia del Controllo e del Controllo Ottimo, UTET, Torino, Italy, 1974. |
| [8] |
L. C. Evans, An Introduction to Mathematical Optimal Control Theory, Lecture Notes, Univerisity of California, Berkeley, 2005. |
| [9] |
H. O. Fattorini, Infinite Dimensional Linear Control Systems. The Time Optimal and Norm Optimal Control Problems, North-Holland Mathematics Studies, 201, Elsevier Science B.V., Amsterdam, 2005. |
| [10] |
H. O. Fattorini,
Time and norm optimal controls: A survey of recent results and open problems, Acta Math. Sci. Ser. B (Engl. Ed.), 31 (2011), 2203-2218.
doi: 10.1016/S0252-9602(11)60394-9. |
| [11] |
J. P. LaSalle,
Time optimal control systems, Proc. Nat. Acad. Sci. U.S.A., 45 (1959), 573-577.
doi: 10.1073/pnas.45.4.573. |
| [12] |
P. Lin and G. Wang,
Blowup time optimal control for ordinary differential equations, SIAM J. Control Optim., 49 (2011), 73-105.
doi: 10.1137/090764232. |
| [13] |
L. Poggiolini,
Structural stability of bang-bang trajectories with a double switching time in the minimum time problem, SIAM J. Control Optim., 55 (2017), 3779-3798.
doi: 10.1137/16M1083761. |
| [14] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers John Wiley & Sons, Inc., New York-London 1962. |
| [15] |
S. Qin and G. Wang,
Controllability of impulse controlled systems of heat equations coupled by constant matrices, J. Differential Equations, 263 (2017), 6456-6493.
doi: 10.1016/j.jde.2017.07.018. |
| [16] |
E. D. Sontag, Mathematical Control Theory. Deterministic Finite-Dimensional Systems, Texts in Applied Mathematics, 6, Springer-Verlag, New York, 1998.
doi: 10.1007/978-1-4612-0577-7. |
| [17] |
H. J. Sussmann,
A bang-bang theorem with bounds on the number of switchings, SIAM J. Control Optim., 17 (1979), 629-651.
doi: 10.1137/0317045. |
| [18] |
G. Wang, L. Wang, Y. Xu and Y. Zhang, Time Optimal Control of Evolution Equations, Progress in Nonlinear Differential Equations and Their Applications, Subseries in Control, 92, Birkhäuser/Springer, Cham, 2018.
doi: 10.1007/978-3-319-95363-2. |
| [19] |
G. Wang and Y. Zhang,
Decompositions and bang-bang properties, Math. Control Relat. Fields, 7 (2017), 73-170.
doi: 10.3934/mcrf.2017005. |
show all references
References:
| [1] |
A. A. Agrachev and C. Biolo,
Switching in time-optimal problem with control in a ball, SIAM J. Control Optim., 56 (2018), 183-120.
doi: 10.1137/16M110304X. |
| [2] |
A. A. Agrachev and C. Biolo,
Switching in time-optimal problem: The 3D Case with 2D control, J. Dyn. Control Syst., 23 (2017), 577-595.
doi: 10.1007/s10883-016-9342-7. |
| [3] |
A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, Encyclopaedia of Mathematical Sciences, 87, Springer-Verlag, Berlin, 2004.
doi: 10.1007/978-3-662-06404-7. |
| [4] |
R. Bellman, I. Glicksberg and O. Gross,
On the "bang-bang" control problem, Quart. Appl. Math., 14 (1956), 11-18.
doi: 10.1090/qam/78516. |
| [5] |
C. Biolo, Switching in Time-Optimal Problem, Ph.D thesis, Scuola Internazionale Superiore di Studi Avanzati - Trieste, 2017. |
| [6] |
C. K. Chui and G. Chen, Linear Systems and Optimal Control, Springer Series in Information Sciences, 18, Springer-Verlag, Berlin, 1989.
doi: 10.1007/978-3-642-61312-8. |
| [7] |
R. Conti, Teoia del Controllo e del Controllo Ottimo, UTET, Torino, Italy, 1974. |
| [8] |
L. C. Evans, An Introduction to Mathematical Optimal Control Theory, Lecture Notes, Univerisity of California, Berkeley, 2005. |
| [9] |
H. O. Fattorini, Infinite Dimensional Linear Control Systems. The Time Optimal and Norm Optimal Control Problems, North-Holland Mathematics Studies, 201, Elsevier Science B.V., Amsterdam, 2005. |
| [10] |
H. O. Fattorini,
Time and norm optimal controls: A survey of recent results and open problems, Acta Math. Sci. Ser. B (Engl. Ed.), 31 (2011), 2203-2218.
doi: 10.1016/S0252-9602(11)60394-9. |
| [11] |
J. P. LaSalle,
Time optimal control systems, Proc. Nat. Acad. Sci. U.S.A., 45 (1959), 573-577.
doi: 10.1073/pnas.45.4.573. |
| [12] |
P. Lin and G. Wang,
Blowup time optimal control for ordinary differential equations, SIAM J. Control Optim., 49 (2011), 73-105.
doi: 10.1137/090764232. |
| [13] |
L. Poggiolini,
Structural stability of bang-bang trajectories with a double switching time in the minimum time problem, SIAM J. Control Optim., 55 (2017), 3779-3798.
doi: 10.1137/16M1083761. |
| [14] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers John Wiley & Sons, Inc., New York-London 1962. |
| [15] |
S. Qin and G. Wang,
Controllability of impulse controlled systems of heat equations coupled by constant matrices, J. Differential Equations, 263 (2017), 6456-6493.
doi: 10.1016/j.jde.2017.07.018. |
| [16] |
E. D. Sontag, Mathematical Control Theory. Deterministic Finite-Dimensional Systems, Texts in Applied Mathematics, 6, Springer-Verlag, New York, 1998.
doi: 10.1007/978-1-4612-0577-7. |
| [17] |
H. J. Sussmann,
A bang-bang theorem with bounds on the number of switchings, SIAM J. Control Optim., 17 (1979), 629-651.
doi: 10.1137/0317045. |
| [18] |
G. Wang, L. Wang, Y. Xu and Y. Zhang, Time Optimal Control of Evolution Equations, Progress in Nonlinear Differential Equations and Their Applications, Subseries in Control, 92, Birkhäuser/Springer, Cham, 2018.
doi: 10.1007/978-3-319-95363-2. |
| [19] |
G. Wang and Y. Zhang,
Decompositions and bang-bang properties, Math. Control Relat. Fields, 7 (2017), 73-170.
doi: 10.3934/mcrf.2017005. |
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