Approximate controllability of second-order impulsive stochastic differential equations with state-dependent delay. (English) Zbl 1458.93030
Summary: In this paper we study a kind of second-order impulsive stochastic differential equations with state-dependent delay in a real separable Hilbert space. Some sufficient conditions for the approximate controllability of this system are formulated and proved under the assumption that the corresponding deterministic linear system is approximately controllable. The results concerning the existence and approximate controllability of mild solutions have been addressed by using strongly continuous cosine families of operators and the contraction mapping principle. At last, an example is given to illustrate the theory.
MSC:
| 93B05 | Controllability |
| 93C15 | Control/observation systems governed by ordinary differential equations |
| 93C25 | Control/observation systems in abstract spaces |
| 93C27 | Impulsive control/observation systems |
| 93C43 | Delay control/observation systems |
| 34K50 | Stochastic functional-differential equations |
| 34K37 | Functional-differential equations with fractional derivatives |
| 34K45 | Functional-differential equations with impulses |
Keywords:
approximate controllability; state-dependent delay; impulsive stochastic differential equations; resolvent operatorReferences:
| [1] | W. Arendt, C. Batty, M. Hieber and F. Neubrander,Vector-valued Laplace transforms and Cauchy problems, Springer Basel, 2001. · Zbl 0978.34001 |
| [2] | G. Arthi, H. P. Ju and H. Y. Jung,Existence and controllability results for second-order impulsive stochastic evolution systems with state-dependent delay, Appl. Math. Comput., 2014, 248, 328-341. · Zbl 1338.34153 |
| [3] | K. Balachandran and J. H. Kim,Remarks on the paper “Controllability of second order differential inclusion in Banach spaces”[J. Math. Anal. Appl., 2003 285, 537-550], J. Math. Anal. Appl., 2006, 324(1), 746-749. · Zbl 1116.93019 |
| [4] | A. E. Bashirov and N. I. Mahmudov,On concepts of controllability for deterministic and stochastic systems, SIAM J. Control Optim., 1999, 37, 1808-1821. · Zbl 0940.93013 |
| [5] | H. Chen, C. Zhu and Y. Zhang,A note on exponential stability for impulsive neutral stochastic partial functional differential equations, Appl. Math. Com · Zbl 1364.35448 |
| [6] | S. Das, D. Pandey and N. Sukavanam,Existence of solution and approximate controllability of a second-order neutral stochastic differential equation with state-dependent delay, Acta Mathematica Scientia, 2016, 36B(5), 1509-1523. · Zbl 1374.35433 |
| [7] | H. O. Fattorini,Controllability of higher order linear systems, In Mathematical Theory of Control, Academic Press, New York, 1967, 301-312. · Zbl 0225.93003 |
| [8] | H. O. Fattorini,Second-order linear differential equations in Banach spaces, 108, Elsevier Sience, North Holland, 1985. · Zbl 0564.34063 |
| [9] | J. K. Hale and J. Kato,Phase space for retarded equations with infinite delay, Funkcialaj Ekvacioj, 1978, 21, 11-41. · Zbl 0383.34055 |
| [10] | Y. Hino, S. Murakami and T. Naito,Functional differential equations with infinite delay, In: Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin, 1991. · Zbl 0732.34051 |
| [11] | J. R. Kang, Y. C. Kwun and J. Y. Park,Controllability of the second-order differential inclusion in Banach spaces, J. Math. Anal. Appl., 2003, 285, 537- 550. · Zbl 1049.93008 |
| [12] | J. Klamka,Stochastic controllability and minimum energy control of systems with multiple delays in control, Appl. Math. Comput., 2008, 206, 704-715. · Zbl 1167.93008 |
| [13] | M. L. Li and J. L. Ma,Approximate controllability of second-order impulsive functional differential system with infinite delay in Banach spaces, J. Appl. Anal. Comput., 2016, 6(2), 492-514. · Zbl 1463.93016 |
| [14] | N. I. Mahmudov and M. A. McKibben,Approximate controllability of secondorder neutral stochastic evolution equations, Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Applications and Algorithms, 2006, 13, 619- 634. · Zbl 1121.34082 |
| [15] | M. Martelli,A Rothe type theorem for noncompact acyclic-valued map, Boll. Un. Mat. Ital., 1975, 4, 70-76. · Zbl 0314.47035 |
| [16] | P. Muthukumar and C. Rajivganthi,Approximate controllability of secondorder neutral stochastic differential equations with infinite delay and poisson jumps, J. Systems Science and Complexity, 2015, 28, 1033-1048. · Zbl 1327.93085 |
| [17] | C. Parthasarathy and M. M. Arjunan,Controllability results for second-order impulsive stochastic functional differential systems with state-dependent delay, Electronic Journal of Mathematical Analysis and Applications, 2013, 1(1), 88- 109. · Zbl 1463.35488 |
| [18] | G. D. Prato and J. Zabczyk,Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge, 1992. · Zbl 0761.60052 |
| [19] | L. J. Shen and J. T. Sun,Approximate controllability of stochastic impulsive functional systems with infinite delay, Automatica, 2012, 48, 2705-2709. · Zbl 1271.93025 |
| [20] | C. C. Travis and G. F. Webb,Compactness, regularity, and uniform continuity properties of strongly continuous cosine families, Houston Journal of Mathematics, 1977, 3, 555-567. · Zbl 0386.47024 |
| [21] | C. C. Travis and G. F. Webb,Cosine families and abstract nonlinear secondorder differential equations, Acta Mathematica Hungarica, 1978, 32, 76-96. · Zbl 0388.34039 |
| [22] | C. C. Travis and G. F. Webb,Second order differential equations in Banach space, Proceedings International Symposium on Nonlinear Equations in Abatract Spaces, Academic Press, New York, 1987, 331-361. · Zbl 0455.34044 |
| [23] | R. Triggiani,On the relationship between first and second-order controllable systems in Banach spaces, Springer-verlag, Berlin, Notes in Control and Inform, Sciences, 1978, 370-393. · Zbl 0376.93022 |
| [24] | R. Triggiani,On the relationship between first and second order controllable systems in Banach spaces, SIAM J. Control Optim., 1978, 16, 847-859. · Zbl 0395.93012 |
| [25] | Z. M. Yan and X. X. Yan, |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
