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Admissible controls and controllable sets for a linear time-varying ordinary differential equation
1. | School of Mathematics and Statistics, Wuhan University, Wuhan, MO 430072, China |
2. | School of Mathematical Sciences, Fudan University, KLMNS, Shanghai, MO 200433, China |
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.
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. |
[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. |
[3] |
V. Barbu, Optimal Control of Variational Inequalities, Research Notes in Mathematics, 100, Pitman, Boston, MA, 1984. |
[4] |
R. Conti, Teoria del Controllo e del Controllo Ottimo, UTET, Torino, Italy, 1974. |
[5] |
A. L. Dontchev,
On the admissible controls of constrained linear systems, C. R. Acad. Bulgare Sci., 42 (1989), 33-36.
|
[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. |
[7] |
J. B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of Convex Analysis, Springer-Verlag, Berlin, 2001.
doi: 10.1007/978-3-642-56468-0. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[3] |
V. Barbu, Optimal Control of Variational Inequalities, Research Notes in Mathematics, 100, Pitman, Boston, MA, 1984. |
[4] |
R. Conti, Teoria del Controllo e del Controllo Ottimo, UTET, Torino, Italy, 1974. |
[5] |
A. L. Dontchev,
On the admissible controls of constrained linear systems, C. R. Acad. Bulgare Sci., 42 (1989), 33-36.
|
[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. |
[7] |
J. B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of Convex Analysis, Springer-Verlag, Berlin, 2001.
doi: 10.1007/978-3-642-56468-0. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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