Approximate controllability of discrete semilinear systems and applications

Previous Article
Continuum approximations for pulses generated by impulsive initial data in binary exciton chain systems
DCDS-B Home
This Issue
Next Article
Qualitative properties of ionic flows via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Ion size effects

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

Approximate controllability of discrete semilinear systems and applications

Hugo Leiva ^1, and Jahnett Uzcategui ^2,

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

Abstract
Related Papers

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.

Keywords: wave equation., Difference equations, approximate controllability.

Mathematics Subject Classification: Primary: 93B05; Secondary: 93C2.

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.

show all references

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.

[1]	Valentin Keyantuo, Mahamadi Warma. On the interior approximate controllability for fractional wave equations. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3719-3739. doi: 10.3934/dcds.2016.36.3719
[2]	Jonathan Touboul. Erratum on: Controllability of the heat and wave equations and their finite difference approximations by the shape of the domain. Mathematical Control and Related Fields, 2019, 9 (1) : 221-222. doi: 10.3934/mcrf.2019006
[3]	Jonathan Touboul. Controllability of the heat and wave equations and their finite difference approximations by the shape of the domain. Mathematical Control and Related Fields, 2012, 2 (4) : 429-455. doi: 10.3934/mcrf.2012.2.429
[4]	Hugo Leiva, Nelson Merentes, José L. Sánchez. Approximate controllability of semilinear reaction diffusion equations. Mathematical Control and Related Fields, 2012, 2 (2) : 171-182. doi: 10.3934/mcrf.2012.2.171
[5]	K. Ravikumar, Manil T. Mohan, A. Anguraj. Approximate controllability of a non-autonomous evolution equation in Banach spaces. Numerical Algebra, Control and Optimization, 2021, 11 (3) : 461-485. doi: 10.3934/naco.2020038
[6]	Mohamed Ouzahra. Approximate controllability of the semilinear reaction-diffusion equation governed by a multiplicative control. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1075-1090. doi: 10.3934/dcdsb.2021081
[7]	Salah-Eddine Chorfi, Ghita El Guermai, Lahcen Maniar, Walid Zouhair. Impulse null approximate controllability for heat equation with dynamic boundary conditions. Mathematical Control and Related Fields, 2022 doi: 10.3934/mcrf.2022026
[8]	Lianwen Wang. Approximate controllability and approximate null controllability of semilinear systems. Communications on Pure and Applied Analysis, 2006, 5 (4) : 953-962. doi: 10.3934/cpaa.2006.5.953
[9]	Abdelaziz Bennour, Farid Ammar Khodja, Djamel Teniou. Exact and approximate controllability of coupled one-dimensional hyperbolic equations. Evolution Equations and Control Theory, 2017, 6 (4) : 487-516. doi: 10.3934/eect.2017025
[10]	Pengyu Chen, Xuping Zhang. Approximate controllability of nonlocal problem for non-autonomous stochastic evolution equations. Evolution Equations and Control Theory, 2021, 10 (3) : 471-489. doi: 10.3934/eect.2020076
[11]	Yassine El Gantouh, Said Hadd, Abdelaziz Rhandi. Approximate controllability of network systems. Evolution Equations and Control Theory, 2021, 10 (4) : 749-766. doi: 10.3934/eect.2020091
[12]	Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control and Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305
[13]	Mohamed Ouzahra. Controllability of the semilinear wave equation governed by a multiplicative control. Evolution Equations and Control Theory, 2019, 8 (4) : 669-686. doi: 10.3934/eect.2019039
[14]	Louis Tebou. Simultaneous controllability of some uncoupled semilinear wave equations. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3721-3743. doi: 10.3934/dcds.2015.35.3721
[15]	Bernard Ducomet, Alexander Zlotnik, Ilya Zlotnik. On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation. Kinetic and Related Models, 2009, 2 (1) : 151-179. doi: 10.3934/krm.2009.2.151
[16]	Guanggan Chen, Qin Li, Yunyun Wei. Approximate dynamics of a class of stochastic wave equations with white noise. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 73-101. doi: 10.3934/dcdsb.2021033
[17]	Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations and Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013
[18]	Hans Weinberger. The approximate controllability of a model for mutant selection. Evolution Equations and Control Theory, 2013, 2 (4) : 741-747. doi: 10.3934/eect.2013.2.741
[19]	Patrick Martinez, Judith Vancostenoble. Exact controllability in "arbitrarily short time" of the semilinear wave equation. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 901-924. doi: 10.3934/dcds.2003.9.901
[20]	Umberto Biccari, Mahamadi Warma. Null-controllability properties of a fractional wave equation with a memory term. Evolution Equations and Control Theory, 2020, 9 (2) : 399-430. doi: 10.3934/eect.2020011

2021 Impact Factor: 1.497

Tools

Metrics

Other articles
by authors

on AIMS
Hugo Leiva Jahnett Uzcategui
on Google Scholar
Hugo Leiva Jahnett Uzcategui

[Back to Top]