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August  2016, 21(6): 1803-1812. doi: 10.3934/dcdsb.2016023

Approximate controllability of discrete semilinear systems and applications

 1 Louisiana State University, Department of Mathematics, Baton Rouge, LA 70803, United States 2 Universidad de Los Andes, Facualtad de Ciencias, Departamento de Matematica, Merida, 5101, Venezuela

Received  May 2015 Revised  February 2016 Published  June 2016

In this paper we study the approximate controllability of the following semilinear difference equation $z(n+1)=A(n)z(n)+B(n)u(n)+f(n,z(n),u(n)), \quad n\in \mathbb{N}^*,$ $z(n)\in Z$, $u(n)\in U$, where $Z$, $U$ are Hilbert spaces, $A\in l^{\infty}(\mathbb{N},L(Z))$, $B\in l^{\infty}(\mathbb{N},L(U,Z))$, $u\in l^2(\mathbb{N},U)$ and the nonlinear term $f:\mathbb{N} \times Z\times U\longrightarrow Z$ is a suitable function. We prove that, under some conditions on the nonlinear term $f$, the approximate controllability of the linear equation is preserved. Finally, we apply this result to a discrete version of the semilinear wave equation.
Citation: Hugo Leiva, Jahnett Uzcategui. Approximate controllability of discrete semilinear systems and applications. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1803-1812. doi: 10.3934/dcdsb.2016023
References:
 [1] A. E. Bashirov and K. R. Kerimov, On controllability conception for stochastic systems, SIAM Journal on Control and Optimization, 35 (1997), 384-398. doi: 10.1137/S0363012994260970. [2] A. E. Bashirov and N. I. Mahmudov, On Controllability of deterministic and stochastic systems, SIAM Journal on Control and Optimization, 37 (1999), 1808-1821. doi: 10.1137/S036301299732184X. [3] A. E. Bashirov, N. Mahmudov, N. Semi and H. Etikan, Partial controllability concepts, Iternational Journal of Control, 80 (2007), 1-7. doi: 10.1080/00207170600885489. [4] A. E. Bashirov and N. I. Mahmudov, Partial controllability of stochastic linear systems, International Journal of Control, 83 (2010), 2564-2572. doi: 10.1080/00207179.2010.532570. [5] A. E. Bashirov and N. Ghahramanlou, On Partial approximate controllability of semilinear systems, Cogent Engineering, 1 (2014), 965947. doi: 10.1080/23311916.2014.965947. [6] A. E. Bashirov and N. Ghahramanlou, On Partial S-controllability of semilinear partially observable Systems, International Journal of Control, 88 (2015), 969-982. doi: 10.1080/00207179.2014.986763. [7] R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems, Lecture Notes in Control and Information Sciences, 8, (1978), Springer Verlag, Berlin. [8] R. F. Curtain and H. J. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory, Text in Applied Mathematics, 21 (1995), Springer Verlag, New York. doi: 10.1007/978-1-4612-4224-6. [9] S. N. Chow and H. Leiva, Existence and roughness of the exponential dichotomy for linear skew-product semiflows in Banach spaces, J. Differential Equations, 120 (1995), 429-477. doi: 10.1006/jdeq.1995.1117. [10] D. Henry, Geometry Theory of Semilinear Parabolic Equations, Lectures Notes in Mathematics, 840 (1981) Springer Verlag, Berlin. [11] H. R. Henriquez and C. Cuevas, Approximate controllability of abstract discrete-time systems, Advances in Difference Equations,840 (2010), Article ID 695290, 17 pages. doi: 10.1155/2010/695290. [12] V. Lakshmikanthan and D. Trigiante, Theory of Difference Equations: Numerical Methods and Applications, Mathematics in Science and Engineering, 1998. [13] H. Leiva, A Lemma on $C_{0}$-semigroups and applications PDEs systems, Quaestions Mathematicae, 26 (2003), 247-265. doi: 10.2989/16073600309486057. [14] H. Leiva and J. Uzcategui, Exact controlllability for semilinear difference equation and application J. Difference Equ. Appl.,14 (2008), 671-679. doi: 10.1080/10236190701726170. [15] H. Leiva and J. Uzcategui, Controllability of linear difference equations in Hilbert Spaces and applications, IMA Journal of Math. Control and Information, 25 (2008), 323-340. doi: 10.1093/imamci/dnm027. [16] H. Leiva and J. Uzcátegui, Approximate controllability of semilinear difference equations and applications, Journal Mathematical Control Science and Applications (JMCSA), 4 (2011), 9-19. [17] M. Megan, A. L. Sasu and B. Sasu, On approximate controllability of systems associated to linear skew product semiflows, Analele Univ. I. Cuza, Iasi, 47 (2001), 379-388. [18] M. Megan, A. L. Sasu and B. Sasu, Stabilizability and controllability of systems associated to linear skew product semiflows, Rev. Mat. Complut., 15 (2002), 599-618. doi: 10.5209/rev_REMA.2002.v15.n2.16932. [19] A. L. Sasu and B. Sasu, Stability and stabilizability for linear systems of difference equations, J. Difference Equ. Appl., 10 (2004), 1085-1105. doi: 10.1080/10236190412331314178. [20] A. L. Sasu, Stabilizability and controllability for systems of difference equations, J. Difference Equ. Appl.,12 (2006), 821-826. doi: 10.1080/10236190600734218.

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References:
 [1] A. E. Bashirov and K. R. Kerimov, On controllability conception for stochastic systems, SIAM Journal on Control and Optimization, 35 (1997), 384-398. doi: 10.1137/S0363012994260970. [2] A. E. Bashirov and N. I. Mahmudov, On Controllability of deterministic and stochastic systems, SIAM Journal on Control and Optimization, 37 (1999), 1808-1821. doi: 10.1137/S036301299732184X. [3] A. E. Bashirov, N. Mahmudov, N. Semi and H. Etikan, Partial controllability concepts, Iternational Journal of Control, 80 (2007), 1-7. doi: 10.1080/00207170600885489. [4] A. E. Bashirov and N. I. Mahmudov, Partial controllability of stochastic linear systems, International Journal of Control, 83 (2010), 2564-2572. doi: 10.1080/00207179.2010.532570. [5] A. E. Bashirov and N. Ghahramanlou, On Partial approximate controllability of semilinear systems, Cogent Engineering, 1 (2014), 965947. doi: 10.1080/23311916.2014.965947. [6] A. E. Bashirov and N. Ghahramanlou, On Partial S-controllability of semilinear partially observable Systems, International Journal of Control, 88 (2015), 969-982. doi: 10.1080/00207179.2014.986763. [7] R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems, Lecture Notes in Control and Information Sciences, 8, (1978), Springer Verlag, Berlin. [8] R. F. Curtain and H. J. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory, Text in Applied Mathematics, 21 (1995), Springer Verlag, New York. doi: 10.1007/978-1-4612-4224-6. [9] S. N. Chow and H. Leiva, Existence and roughness of the exponential dichotomy for linear skew-product semiflows in Banach spaces, J. Differential Equations, 120 (1995), 429-477. doi: 10.1006/jdeq.1995.1117. [10] D. Henry, Geometry Theory of Semilinear Parabolic Equations, Lectures Notes in Mathematics, 840 (1981) Springer Verlag, Berlin. [11] H. R. Henriquez and C. Cuevas, Approximate controllability of abstract discrete-time systems, Advances in Difference Equations,840 (2010), Article ID 695290, 17 pages. doi: 10.1155/2010/695290. [12] V. Lakshmikanthan and D. Trigiante, Theory of Difference Equations: Numerical Methods and Applications, Mathematics in Science and Engineering, 1998. [13] H. Leiva, A Lemma on $C_{0}$-semigroups and applications PDEs systems, Quaestions Mathematicae, 26 (2003), 247-265. doi: 10.2989/16073600309486057. [14] H. Leiva and J. Uzcategui, Exact controlllability for semilinear difference equation and application J. Difference Equ. Appl.,14 (2008), 671-679. doi: 10.1080/10236190701726170. [15] H. Leiva and J. Uzcategui, Controllability of linear difference equations in Hilbert Spaces and applications, IMA Journal of Math. Control and Information, 25 (2008), 323-340. doi: 10.1093/imamci/dnm027. [16] H. Leiva and J. Uzcátegui, Approximate controllability of semilinear difference equations and applications, Journal Mathematical Control Science and Applications (JMCSA), 4 (2011), 9-19. [17] M. Megan, A. L. Sasu and B. Sasu, On approximate controllability of systems associated to linear skew product semiflows, Analele Univ. I. Cuza, Iasi, 47 (2001), 379-388. [18] M. Megan, A. L. Sasu and B. Sasu, Stabilizability and controllability of systems associated to linear skew product semiflows, Rev. Mat. Complut., 15 (2002), 599-618. doi: 10.5209/rev_REMA.2002.v15.n2.16932. [19] A. L. Sasu and B. Sasu, Stability and stabilizability for linear systems of difference equations, J. Difference Equ. Appl., 10 (2004), 1085-1105. doi: 10.1080/10236190412331314178. [20] A. L. Sasu, Stabilizability and controllability for systems of difference equations, J. Difference Equ. Appl.,12 (2006), 821-826. doi: 10.1080/10236190600734218.
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