×

Robustness of the controllability for the heat equation under the influence of multiple impulses and delays. (English) Zbl 1401.93030

Summary: For many control systems in real life, impulses and delays are intrinsic properties that do not modify their behavior. Thus, we conjecture that under certain conditions the abrupt changes and delays as perturbations of a system, that could model a real situation, do not modify properties such as controllability. In this regard, we prove the approximate controllability of the semilinear heat equation under the influence of multiple impulses and delays, this is done by using new techniques, avoiding fixed-point theorems, employed by A. E. Bashirov et al.

MSC:

93B05 Controllability
93C10 Nonlinear systems in control theory
93B35 Sensitivity (robustness)
35K05 Heat equation
35R12 Impulsive partial differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abada, N.; Benchohra, M.; Hammouche, H., Existence results for semilinear differential evolution equations with impulses and delay, CUBO: A Mathematical Journal, 02, 1-17, (2010) · Zbl 1214.34064 · doi:10.4067/S0719-06462010000200001
[2] Bashirov, A. E.; Ghahramanlou, Noushin, On partial approximate controllability of semilinear systems, COGENTENG-Engeneering, 1, 1, (2014) · Zbl 1316.93019
[3] Bashirov, A. E.; Jneid, M., On partial complete controllability of semilinear systems, Abstract and Applied Analysis, 8, (2013) · Zbl 1291.93034
[4] Bashirov, A. E.; Kerimov, K. R., On controllability conception for stochastic systems, SIAM Journal on Control and Optimization, 35, 2, 384-398, (1997) · Zbl 0873.93076 · doi:10.1137/S0363012994260970
[5] Bashirov, A. E.; Mahmudov, N. I., On controllability of deterministic and stochastic systems, SIAM Journal on Control and Optimization, 37, 6, 1808-1821, (1999) · Zbl 0940.93013 · doi:10.1137/S036301299732184X
[6] Bashirov, A. E.; Mahmudov, N.; Semi, N.; Etikan, H., On partial controllability concepts, International Journal of Control, 80, 1, 1-7, (2007) · Zbl 1115.93013 · doi:10.1080/00207170600885489
[7] Carrasco, A.; Leiva, Hugo; Sanchez, J. L.; Tineo M., A., Approximate controllability of the semilinear impulsive beam equation with impulses, Transaction on IoT and Cloud Computing, 2, 3, 70-88, (2014)
[8] Chalishajar, D. N., Controllability of impulsive partial neutral funcional differential equation with infinite delay, Int. Journal of Math. Analysis, 5, 8, 369-380, (2011) · Zbl 1252.34090
[9] Chen, Lizhen; Li, Gang, Approximate controllability of impulsive differential equations with nonlocal conditions, International Journal of Nonlinear Science, 10, 4, 438-446, (2010) · Zbl 1394.93019
[10] Curtain, R. F.; Pritchard, A. J., Infinite Dimensional Linear Systems, 8, (1978), Springer Verlag, Berlin
[11] Curtain, R. F.; Zwart, H. J., An Introduction to Infinite Dimensional Linear Systems Theory, 21, (1995), Springer Verlag, New York
[12] Delfour, M. C.; Mitter, S. K., Controllability, observability and optimal feedback control of Affine hereditary differential systems, SIAM Journal on Control and Optimization, 10, 298-328, (1972) · Zbl 0242.93011 · doi:10.1137/0310023
[13] Klamka, Jerzy, Constrained approximate controllability, IEEE Transactions on Automatic Control, 45, 9, 1745-1749, (2000) · Zbl 0991.93013 · doi:10.1109/9.880640
[14] Klamka, Jerzy, Constrained controllability of semilinear delayed systems, Bulletin of the Polish Academy of Sciences, Technical Sciences, Electronics and Electrotechnics, 49, 3, 505-515, (2001) · Zbl 0999.93009
[15] Klamka, Jerzy, Constrained controllability of semilinear systems, Nonlinear Analysis, 47, 2939-2949, (2001) · Zbl 1042.93504 · doi:10.1016/S0362-546X(01)00415-1
[16] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations, (1989), World Scientific, Singapore · Zbl 0719.34002
[17] Approximate Controllability of Semilinear Impulsive Evolution Equations, Abstract and Applied Analysis2015, Article ID 797439, 7 pages
[18] Leiva, Hugo, Approximate controllability of semilinear heat equation with impulses and delay on the state, Nonauton. Dyn. Syst., 2, 52-62, (2015) · Zbl 1332.93055
[19] Leiva, Hugo; Merentes, N., Approximate controllability of the impulsive semilinear heat equation, Journal of Mathematics and Applications, 38, 85-104, (2015) · Zbl 1382.93011
[20] Leiva, Hugo; Merentes, N. · Zbl 1222.93031 · doi:10.1155/2010/147195
[21] Leiva, H.; Merentes, N.; Sanchez, J. L., Mahony Equation, Journal of Mathematis and Applications, 33, 51-59, (2010) · Zbl 1382.93009
[22] Leiva, H.; Merentes, N.; Sanchez, J. L., Interior controllability of the semilinear Benjamin-Bona-Mahony equation, Journal of Mathematis and Applications, 35, 97-109, (2012) · Zbl 1382.93010
[23] Leiva, H.; Merentes, N.; Sanchez, J. L., A characterization of semilinear dense range operators and applications, Abstract and Applied Analysis, 2013, (2013) · Zbl 1287.47054 · doi:10.1155/2013/729093
[24] Larez, Hanzel; Leiva, H.; Uzcategui, Jahnett, Controllability of block diagonal system and applications, Int. J. Systems, Control and Communications, 3, 1, 64-81, (2011) · doi:10.1504/IJSCC.2011.039252
[25] Impulsive Differential Equations, World Scientific Series on Nonlinear Science Series A, Vol. 14, World Scientific Publishing Co., Singapore, 1995
[26] Selvi, S.; Mallika Arjunan, M., Controllability results for impulsive differential systems with finite delay, J. Nonlinear Sci. Appl., 5, 206-219, (2012) · Zbl 1293.93107
[27] Shikharchand Jain, R.; Baburao Dhakne, M., On mild solutions of nonlocal semilinear impulsive functional integro-differential equations, Applied Mathematics E-Notes, 13, 109-119, (2014) · Zbl 1288.45008
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.