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September  2018, 8(3&4): 1001-1019. doi: 10.3934/mcrf.2018043

Admissible controls and controllable sets for a linear time-varying ordinary differential equation

Lijuan Wang 1, and  Yashan Xu 2,,

1. 

School of Mathematics and Statistics, Wuhan University, Wuhan, MO 430072, China
2. 

School of Mathematical Sciences, Fudan University, KLMNS, Shanghai, MO 200433, China

* Corresponding author: Yashan Xu

Received  August 2017 Revised  June 2018 Published  September 2018

Fund Project: The first author is supported by the National Natural Science Foundation under grants 11771344 and 11371285; the second author is supported by the National Natural Science Foundation under grants 11471080 and 11631004

In this paper, for a time optimal control problem governed by a linear time-varying ordinary differential equation, we give a description to check whether the set of admissible controls is nonempty or not by finite times.

Keywords: Ordinary differential equation, time optimal control, admissible control.
Mathematics Subject Classification: Primary: 49J15, 93C15; Secondary: 93B05.
Citation: Lijuan Wang, Yashan Xu. Admissible controls and controllable sets for a linear time-varying ordinary differential equation. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1001-1019. doi: 10.3934/mcrf.2018043
References:
[1]

M. E. Achhab, F. M. Callier and V. Wertz, Admissible controls and attainable states for a class of nonlinear systems with general constraints, Internat. J. Robust Nonlinear Control, 4 (1994), 267-288.  doi: 10.1002/rnc.4590040204.  Google Scholar

[2]

S. A. A$\breve{{\rm{i}}}$sagaliev and M. K. Ospanova, Existence of admissible controls for ordinary differential equations with fixed end-points of trajectories in the presence of phase and integral constraints, (Russian) Vestn. Minist. Obraz. Nauki Nats. Akad. Nauk Resp. Kaz., (2003), 16-26.  Google Scholar

[3]

V. Barbu, Optimal Control of Variational Inequalities, Research Notes in Mathematics, 100, Pitman, Boston, MA, 1984.  Google Scholar

[4]

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

[5]

A. L. Dontchev, On the admissible controls of constrained linear systems, C. R. Acad. Bulgare Sci., 42 (1989), 33-36.   Google Scholar

[6]

H. Hermes, On the closure and convexity of attainable sets in finite and infinite dimensions, SIAM J. Control, 5 (1967), 409-417.  doi: 10.1137/0305025.  Google Scholar

[7]

J. B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of Convex Analysis, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-56468-0.  Google Scholar

[8]

V. A. Komarov, Estimates for the accessibility set and the construction of admissible controls for linear systems, (Russian) Dokl. Akad. Nauk SSSR, 268 (1983), 537-541.  Google Scholar

[9]

S. R. Musaev, A certain sufficient condition for the existence of admissible controls for a multimensional optimal control problem, (Russian) Akad. Nauk , SSR Dokl., 32 (1976), 3-7.  Google Scholar

[10]

S. R. Musaev and T. M. Èfendiev, Construction of scalar admissible controls by the Picard-Rakovshchik method, (Russian) Questions of Mathematical Cybernetics and Applied Mathematics, "Èlm", Baku, 1980,134-145.  Google Scholar

[11]

L. D. Pustyl'nikov, On a method for finding admissible controls in a linear system with phase constraints, (Russian) Differentsial'nye Uravneniya, 17 (1981), 2176-2184, 2300.  Google Scholar

[12]

E. O. Roxin, The attainable set in control systems, in Mathematical Theory Of Control (Bombay, 1990), 307-319, Lecture Notes in Pure and Appl. Math., 142, Dekker, New York, 1993.  Google Scholar

[13]

W. E. Schmitendorf and B. R. Barmish, Null controllability of linear systems with constrained controls, SIAM J. Control and Optim., 18 (1980), 327-345.  doi: 10.1137/0318025.  Google Scholar

[14]

G. Wang, The existence of time optimal control of semilinear parabolic equations, Systems Control Lett., 53 (2004), 171-175.  doi: 10.1016/j.sysconle.2004.04.002.  Google Scholar

[15]

G. Wang, Y. Xu and Y. Zhang, Attainable subspaces and the bang-bang property of time optimal controls for heat equations, SIAM J. Control Optim., 53 (2015), 592-621.  doi: 10.1137/140966022.  Google Scholar

[16]

L. Wang and Q. Yan, Bang-bang property of time optimal null controls for some semilinear heat equation, SIAM J. Control Optim., 54 (2016), 2949-2964.  doi: 10.1137/140997452.  Google Scholar

show all references

References:
[1]

M. E. Achhab, F. M. Callier and V. Wertz, Admissible controls and attainable states for a class of nonlinear systems with general constraints, Internat. J. Robust Nonlinear Control, 4 (1994), 267-288.  doi: 10.1002/rnc.4590040204.  Google Scholar

[2]

S. A. A$\breve{{\rm{i}}}$sagaliev and M. K. Ospanova, Existence of admissible controls for ordinary differential equations with fixed end-points of trajectories in the presence of phase and integral constraints, (Russian) Vestn. Minist. Obraz. Nauki Nats. Akad. Nauk Resp. Kaz., (2003), 16-26.  Google Scholar

[3]

V. Barbu, Optimal Control of Variational Inequalities, Research Notes in Mathematics, 100, Pitman, Boston, MA, 1984.  Google Scholar

[4]

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

[5]

A. L. Dontchev, On the admissible controls of constrained linear systems, C. R. Acad. Bulgare Sci., 42 (1989), 33-36.   Google Scholar

[6]

H. Hermes, On the closure and convexity of attainable sets in finite and infinite dimensions, SIAM J. Control, 5 (1967), 409-417.  doi: 10.1137/0305025.  Google Scholar

[7]

J. B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of Convex Analysis, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-56468-0.  Google Scholar

[8]

V. A. Komarov, Estimates for the accessibility set and the construction of admissible controls for linear systems, (Russian) Dokl. Akad. Nauk SSSR, 268 (1983), 537-541.  Google Scholar

[9]

S. R. Musaev, A certain sufficient condition for the existence of admissible controls for a multimensional optimal control problem, (Russian) Akad. Nauk , SSR Dokl., 32 (1976), 3-7.  Google Scholar

[10]

S. R. Musaev and T. M. Èfendiev, Construction of scalar admissible controls by the Picard-Rakovshchik method, (Russian) Questions of Mathematical Cybernetics and Applied Mathematics, "Èlm", Baku, 1980,134-145.  Google Scholar

[11]

L. D. Pustyl'nikov, On a method for finding admissible controls in a linear system with phase constraints, (Russian) Differentsial'nye Uravneniya, 17 (1981), 2176-2184, 2300.  Google Scholar

[12]

E. O. Roxin, The attainable set in control systems, in Mathematical Theory Of Control (Bombay, 1990), 307-319, Lecture Notes in Pure and Appl. Math., 142, Dekker, New York, 1993.  Google Scholar

[13]

W. E. Schmitendorf and B. R. Barmish, Null controllability of linear systems with constrained controls, SIAM J. Control and Optim., 18 (1980), 327-345.  doi: 10.1137/0318025.  Google Scholar

[14]

G. Wang, The existence of time optimal control of semilinear parabolic equations, Systems Control Lett., 53 (2004), 171-175.  doi: 10.1016/j.sysconle.2004.04.002.  Google Scholar

[15]

G. Wang, Y. Xu and Y. Zhang, Attainable subspaces and the bang-bang property of time optimal controls for heat equations, SIAM J. Control Optim., 53 (2015), 592-621.  doi: 10.1137/140966022.  Google Scholar

[16]

L. Wang and Q. Yan, Bang-bang property of time optimal null controls for some semilinear heat equation, SIAM J. Control Optim., 54 (2016), 2949-2964.  doi: 10.1137/140997452.  Google Scholar

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