• Previous Article
    Qualitative properties of ionic flows via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Ion size effects
  • DCDS-B Home
  • This Issue
  • Next Article
    Continuum approximations for pulses generated by impulsive initial data in binary exciton chain systems
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 & 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. doi: 10.1137/S0363012994260970. Google Scholar

[2]

A. E. Bashirov and N. I. Mahmudov, On Controllability of deterministic and stochastic systems,, SIAM Journal on Control and Optimization, 37 (1999), 1808. doi: 10.1137/S036301299732184X. Google Scholar

[3]

A. E. Bashirov, N. Mahmudov, N. Semi and H. Etikan, Partial controllability concepts,, Iternational Journal of Control, 80 (2007), 1. doi: 10.1080/00207170600885489. Google Scholar

[4]

A. E. Bashirov and N. I. Mahmudov, Partial controllability of stochastic linear systems,, International Journal of Control, 83 (2010), 2564. doi: 10.1080/00207179.2010.532570. Google Scholar

[5]

A. E. Bashirov and N. Ghahramanlou, On Partial approximate controllability of semilinear systems,, \emph{Cogent Engineering}, 1 (2014). doi: 10.1080/23311916.2014.965947. Google Scholar

[6]

A. E. Bashirov and N. Ghahramanlou, On Partial S-controllability of semilinear partially observable Systems,, International Journal of Control, 88 (2015), 969. doi: 10.1080/00207179.2014.986763. Google Scholar

[7]

R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems,, Lecture Notes in Control and Information Sciences, 8 (1978). Google Scholar

[8]

R. F. Curtain and H. J. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory,, Text in Applied Mathematics, 21 (1995). doi: 10.1007/978-1-4612-4224-6. Google Scholar

[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. doi: 10.1006/jdeq.1995.1117. Google Scholar

[10]

D. Henry, Geometry Theory of Semilinear Parabolic Equations,, Lectures Notes in Mathematics, 840 (1981). Google Scholar

[11]

H. R. Henriquez and C. Cuevas, Approximate controllability of abstract discrete-time systems,, Advances in Difference Equations, 840 (2010). doi: 10.1155/2010/695290. Google Scholar

[12]

V. Lakshmikanthan and D. Trigiante, Theory of Difference Equations: Numerical Methods and Applications,, Mathematics in Science and Engineering, (1998). Google Scholar

[13]

H. Leiva, A Lemma on $C_{0}$-semigroups and applications PDEs systems,, Quaestions Mathematicae, 26 (2003), 247. doi: 10.2989/16073600309486057. Google Scholar

[14]

H. Leiva and J. Uzcategui, Exact controlllability for semilinear difference equation and application, J. Difference Equ. Appl., 14 (2008), 671. doi: 10.1080/10236190701726170. Google Scholar

[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. doi: 10.1093/imamci/dnm027. Google Scholar

[16]

H. Leiva and J. Uzcátegui, Approximate controllability of semilinear difference equations and applications,, Journal Mathematical Control Science and Applications (JMCSA), 4 (2011), 9. Google Scholar

[17]

M. Megan, A. L. Sasu and B. Sasu, On approximate controllability of systems associated to linear skew product semiflows,, Analele Univ. I. Cuza, 47 (2001), 379. Google Scholar

[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. doi: 10.5209/rev_REMA.2002.v15.n2.16932. Google Scholar

[19]

A. L. Sasu and B. Sasu, Stability and stabilizability for linear systems of difference equations,, J. Difference Equ. Appl., 10 (2004), 1085. doi: 10.1080/10236190412331314178. Google Scholar

[20]

A. L. Sasu, Stabilizability and controllability for systems of difference equations,, J. Difference Equ. Appl., 12 (2006), 821. doi: 10.1080/10236190600734218. Google Scholar

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. doi: 10.1137/S0363012994260970. Google Scholar

[2]

A. E. Bashirov and N. I. Mahmudov, On Controllability of deterministic and stochastic systems,, SIAM Journal on Control and Optimization, 37 (1999), 1808. doi: 10.1137/S036301299732184X. Google Scholar

[3]

A. E. Bashirov, N. Mahmudov, N. Semi and H. Etikan, Partial controllability concepts,, Iternational Journal of Control, 80 (2007), 1. doi: 10.1080/00207170600885489. Google Scholar

[4]

A. E. Bashirov and N. I. Mahmudov, Partial controllability of stochastic linear systems,, International Journal of Control, 83 (2010), 2564. doi: 10.1080/00207179.2010.532570. Google Scholar

[5]

A. E. Bashirov and N. Ghahramanlou, On Partial approximate controllability of semilinear systems,, \emph{Cogent Engineering}, 1 (2014). doi: 10.1080/23311916.2014.965947. Google Scholar

[6]

A. E. Bashirov and N. Ghahramanlou, On Partial S-controllability of semilinear partially observable Systems,, International Journal of Control, 88 (2015), 969. doi: 10.1080/00207179.2014.986763. Google Scholar

[7]

R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems,, Lecture Notes in Control and Information Sciences, 8 (1978). Google Scholar

[8]

R. F. Curtain and H. J. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory,, Text in Applied Mathematics, 21 (1995). doi: 10.1007/978-1-4612-4224-6. Google Scholar

[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. doi: 10.1006/jdeq.1995.1117. Google Scholar

[10]

D. Henry, Geometry Theory of Semilinear Parabolic Equations,, Lectures Notes in Mathematics, 840 (1981). Google Scholar

[11]

H. R. Henriquez and C. Cuevas, Approximate controllability of abstract discrete-time systems,, Advances in Difference Equations, 840 (2010). doi: 10.1155/2010/695290. Google Scholar

[12]

V. Lakshmikanthan and D. Trigiante, Theory of Difference Equations: Numerical Methods and Applications,, Mathematics in Science and Engineering, (1998). Google Scholar

[13]

H. Leiva, A Lemma on $C_{0}$-semigroups and applications PDEs systems,, Quaestions Mathematicae, 26 (2003), 247. doi: 10.2989/16073600309486057. Google Scholar

[14]

H. Leiva and J. Uzcategui, Exact controlllability for semilinear difference equation and application, J. Difference Equ. Appl., 14 (2008), 671. doi: 10.1080/10236190701726170. Google Scholar

[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. doi: 10.1093/imamci/dnm027. Google Scholar

[16]

H. Leiva and J. Uzcátegui, Approximate controllability of semilinear difference equations and applications,, Journal Mathematical Control Science and Applications (JMCSA), 4 (2011), 9. Google Scholar

[17]

M. Megan, A. L. Sasu and B. Sasu, On approximate controllability of systems associated to linear skew product semiflows,, Analele Univ. I. Cuza, 47 (2001), 379. Google Scholar

[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. doi: 10.5209/rev_REMA.2002.v15.n2.16932. Google Scholar

[19]

A. L. Sasu and B. Sasu, Stability and stabilizability for linear systems of difference equations,, J. Difference Equ. Appl., 10 (2004), 1085. doi: 10.1080/10236190412331314178. Google Scholar

[20]

A. L. Sasu, Stabilizability and controllability for systems of difference equations,, J. Difference Equ. Appl., 12 (2006), 821. doi: 10.1080/10236190600734218. Google Scholar

[1]

Valentin Keyantuo, Mahamadi Warma. On the interior approximate controllability for fractional wave equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3719-3739. doi: 10.3934/dcds.2016.36.3719

[2]

Jonathan Touboul. Controllability of the heat and wave equations and their finite difference approximations by the shape of the domain. Mathematical Control & Related Fields, 2012, 2 (4) : 429-455. doi: 10.3934/mcrf.2012.2.429

[3]

Jonathan Touboul. Erratum on: Controllability of the heat and wave equations and their finite difference approximations by the shape of the domain. Mathematical Control & Related Fields, 2019, 9 (1) : 221-222. doi: 10.3934/mcrf.2019006

[4]

Hugo Leiva, Nelson Merentes, José L. Sánchez. Approximate controllability of semilinear reaction diffusion equations. Mathematical Control & Related Fields, 2012, 2 (2) : 171-182. doi: 10.3934/mcrf.2012.2.171

[5]

Lianwen Wang. Approximate controllability and approximate null controllability of semilinear systems. Communications on Pure & Applied Analysis, 2006, 5 (4) : 953-962. doi: 10.3934/cpaa.2006.5.953

[6]

Abdelaziz Bennour, Farid Ammar Khodja, Djamel Teniou. Exact and approximate controllability of coupled one-dimensional hyperbolic equations. Evolution Equations & Control Theory, 2017, 6 (4) : 487-516. doi: 10.3934/eect.2017025

[7]

Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control & Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305

[8]

Mohamed Ouzahra. Controllability of the semilinear wave equation governed by a multiplicative control. Evolution Equations & Control Theory, 2019, 8 (4) : 669-686. doi: 10.3934/eect.2019039

[9]

Louis Tebou. Simultaneous controllability of some uncoupled semilinear wave equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3721-3743. doi: 10.3934/dcds.2015.35.3721

[10]

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 & Related Models, 2009, 2 (1) : 151-179. doi: 10.3934/krm.2009.2.151

[11]

Hans Weinberger. The approximate controllability of a model for mutant selection. Evolution Equations & Control Theory, 2013, 2 (4) : 741-747. doi: 10.3934/eect.2013.2.741

[12]

Patrick Martinez, Judith Vancostenoble. Exact controllability in "arbitrarily short time" of the semilinear wave equation. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 901-924. doi: 10.3934/dcds.2003.9.901

[13]

Arnaud Heibig, Mohand Moussaoui. Exact controllability of the wave equation for domains with slits and for mixed boundary conditions. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 367-386. doi: 10.3934/dcds.1996.2.367

[14]

Irena Lasiecka, Roberto Triggiani. Global exact controllability of semilinear wave equations by a double compactness/uniqueness argument. Conference Publications, 2005, 2005 (Special) : 556-565. doi: 10.3934/proc.2005.2005.556

[15]

Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013

[16]

Guillaume Olive. Boundary approximate controllability of some linear parabolic systems. Evolution Equations & Control Theory, 2014, 3 (1) : 167-189. doi: 10.3934/eect.2014.3.167

[17]

Moncef Aouadi, Taoufik Moulahi. Approximate controllability of abstract nonsimple thermoelastic problem. Evolution Equations & Control Theory, 2015, 4 (4) : 373-389. doi: 10.3934/eect.2015.4.373

[18]

Weiguo Zhang, Yan Zhao, Xiang Li. Qualitative analysis to the traveling wave solutions of Kakutani-Kawahara equation and its approximate damped oscillatory solution. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1075-1090. doi: 10.3934/cpaa.2013.12.1075

[19]

Yacine Chitour, Guilherme Mazanti, Mario Sigalotti. Stability of non-autonomous difference equations with applications to transport and wave propagation on networks. Networks & Heterogeneous Media, 2016, 11 (4) : 563-601. doi: 10.3934/nhm.2016010

[20]

Umberto De Maio, Akamabadath K. Nandakumaran, Carmen Perugia. Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition. Evolution Equations & Control Theory, 2015, 4 (3) : 325-346. doi: 10.3934/eect.2015.4.325

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]