# American Institute of Mathematical Sciences

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

 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.

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. [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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