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June  2021, 11(2): 329-351. doi: 10.3934/mcrf.2020039

On switching properties of time optimal controls for linear ODEs

Shulin Qin 1, , Gengsheng Wang 2, and  Huaiqiang Yu 3,,

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

* Corresponding author: Huaiqiang Yu

Received  November 2019 Revised  June 2020 Published  June 2021 Early access  October 2020

Fund Project: This work was partially supported by the NNSF of China under grants 11601377, 11901432, 11971022

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.

Keywords: Time optimal control, switching problem, maximum principle, ordinary differential equation, piecewise control.
Mathematics Subject Classification: Primary: 93C15; Secondary: 49J15.
Citation: Shulin Qin, Gengsheng Wang, Huaiqiang Yu. On switching properties of time optimal controls for linear ODEs. Mathematical Control & Related Fields, 2021, 11 (2) : 329-351. doi: 10.3934/mcrf.2020039
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[5]

C. Biolo, Switching in Time-Optimal Problem, Ph.D thesis, Scuola Internazionale Superiore di Studi Avanzati - Trieste, 2017. Google Scholar

[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.  Google Scholar

[7]

R. Conti, Teoia del Controllo e del Controllo Ottimo, UTET, Torino, Italy, 1974. Google Scholar

[8]

L. C. Evans, An Introduction to Mathematical Optimal Control Theory, Lecture Notes, Univerisity of California, Berkeley, 2005. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

G. Wang and Y. Zhang, Decompositions and bang-bang properties, Math. Control Relat. Fields, 7 (2017), 73-170.  doi: 10.3934/mcrf.2017005.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

C. Biolo, Switching in Time-Optimal Problem, Ph.D thesis, Scuola Internazionale Superiore di Studi Avanzati - Trieste, 2017. Google Scholar

[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.  Google Scholar

[7]

R. Conti, Teoia del Controllo e del Controllo Ottimo, UTET, Torino, Italy, 1974. Google Scholar

[8]

L. C. Evans, An Introduction to Mathematical Optimal Control Theory, Lecture Notes, Univerisity of California, Berkeley, 2005. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

G. Wang and Y. Zhang, Decompositions and bang-bang properties, Math. Control Relat. Fields, 7 (2017), 73-170.  doi: 10.3934/mcrf.2017005.  Google Scholar

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