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

Lijuan Wang; Yashan Xu

doi:10.3934/mcrf.2018043

Article Contents

2018, Volume 8, Issue 3&4: 1001-1019. Doi: 10.3934/mcrf.2018043

This issue Previous Article Switching between a pair of stocks: An optimal trading rule Next Article Forward backward SDEs in weak formulation

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
* Corresponding author: Yashan Xu

Received: August 2017

Revised: June 2018

Published: September 2018

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

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 49J15, 93C15; Secondary: 93B05.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

	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.
	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.
	V. Barbu, Optimal Control of Variational Inequalities, Research Notes in Mathematics, 100, Pitman, Boston, MA, 1984.
	R. Conti, Teoria del Controllo e del Controllo Ottimo, UTET, Torino, Italy, 1974.
	A. L. Dontchev , On the admissible controls of constrained linear systems, C. R. Acad. Bulgare Sci., 42 (1989) , 33-36.
	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.
	J. B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of Convex Analysis, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-56468-0.
	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.
	S. R. Musaev, A certain sufficient condition for the existence of admissible controls for a multimensional optimal control problem, (Russian) Akad. Nauk $Azerba\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} d\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over z} an$, SSR Dokl., 32 (1976), 3-7.
	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.
	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.
	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.
	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.
	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.
	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.
	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.