American Institute of Mathematical Sciences

Advanced
February  2015, 35(2): 703-723. doi: 10.3934/dcds.2015.35.703

Unbounded perturbations of the generator domain

Said Hadd 1, , Rosanna Manzo 2, and  Abdelaziz Rhandi 3,

1. 

Department of Mathematics, Faculty of Sciences, Ibn Zohr University, B.P. 8106, Agadir, Morocco
2. 

Department of Information Engineering and Applied Mathematics, University of Salerno, via Ponte Don Melillo, 84084 Fisciano (SA), Italy
3. 

Dept. of Information Eng., Electrical Eng. and Applied Mathematics, University of Salerno, Via Giovanni Paolo II, 132, I 84084 Fisciano (SA), Italy

Received  January 2013 Revised  January 2014 Published  September 2014

Let $X,U$ and $Z$ be Banach spaces such that $Z\subset X$ (with continuous and dense embedding), $L:Z\to X$ be a closed linear operator and consider closed linear operators $G,M:Z\to U$. Putting conditions on $G$ and $M$ we show that the operator $\mathcal{A}=L$ with domain $D(\mathcal{A})=\big\{z\in Z:Gz=Mz\big\}$ generates a $C_0$-semigroup on $X$. Moreover, we give a variation of constants formula for the solution of the following inhomogeneous problem \begin{align*} \begin{cases} \dot{z}(t)=L z(t)+f(t),& t\ge 0,\cr G z(t)=Mz(t)+g(t),& t\ge 0,\cr z(0)=z^0. \end{cases} \end{align*} Several examples will be given, in particular a heat equation with distributed unbounded delay at the boundary condition.
Keywords: unbounded perturbation, closed-loop systems, $C_0$--semigroup, inhomogeneous boundary problem., regular linear systems, Banach space.
Mathematics Subject Classification: Primary: 47D06; Secondary: 93C25, 34G10, 34K3.
Citation: Said Hadd, Rosanna Manzo, Abdelaziz Rhandi. Unbounded perturbations of the generator domain. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 703-723. doi: 10.3934/dcds.2015.35.703
References:
[1]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite-Dimensional Systems,, Birkhäuser, (2007).   Google Scholar

[2]

A. Chen and K. Morris, Well-posedness of boundary control systems,, SIAM J. Control Optim., 42 (2003), 1244.  doi: 10.1137/S0363012902384916.  Google Scholar

[3]

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

[4]

K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations,, Springer-Verlag, (2000).   Google Scholar

[5]

K. J. Engel, M. Kramar, B. Klöss, R. Nagel and E. Sikolya, Maximal controllability for boundary control problems,, Appl. Math. Optim., 62 (2010), 205.  doi: 10.1007/s00245-010-9101-1.  Google Scholar

[6]

H. O. Fattorini, Boundary control systems,, SIAM J. Control, 6 (1968), 349.  doi: 10.1137/0306025.  Google Scholar

[7]

G. Greiner, Perturbing the boundary conditions of a generator,, Houston J. Math., 13 (1987), 213.   Google Scholar

[8]

S. Hadd, A. Idrissi and A. Rhandi, The regular linear systems associated with the shift semigroups and application to control linear systems with delay,, Math. Control Signals Systems, 18 (2006), 272.  doi: 10.1007/s00498-006-0002-4.  Google Scholar

[9]

M. Kumpf and G. Nickel, Dynamic boundary conditions and boundary control for the one-dimensional heat equation,, J. Dynam. Control Systems, 10 (2004), 213.  doi: 10.1023/B:JODS.0000024122.71407.83.  Google Scholar

[10]

D. Salamon, Infinite-dimensional linear system with unbounded control and observation: a functional analytic approach,, Trans. Amer. Math. Soc., 300 (1987), 383.  doi: 10.2307/2000351.  Google Scholar

[11]

O. J. Staffans, Well-posed Linear Systems,, Encyclopedia of Mathematics and its Applications, (2005).  doi: 10.1017/CBO9780511543197.  Google Scholar

[12]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups,, Birkhäuser, (2009).  doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[13]

G. Weiss, Admissible observation operators for linear semigroups,, Israel J. Math., 65 (1989), 17.  doi: 10.1007/BF02788172.  Google Scholar

[14]

G. Weiss, Admissibility of unbounded control operators,, SIAM J. Control Optim., 27 (1989), 527.  doi: 10.1137/0327028.  Google Scholar

[15]

G. Weiss, Transfer functions of regular linear systems. I. Characterization of regularity,, Trans. Amer. Math. Soc., 342 (1994), 827.  doi: 10.2307/2154655.  Google Scholar

[16]

G. Weiss, Regular linear systems with feedback,, Math. Control Signals Systems, 7 (1994), 23.  doi: 10.1007/BF01211484.  Google Scholar

show all references

References:
[1]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite-Dimensional Systems,, Birkhäuser, (2007).   Google Scholar

[2]

A. Chen and K. Morris, Well-posedness of boundary control systems,, SIAM J. Control Optim., 42 (2003), 1244.  doi: 10.1137/S0363012902384916.  Google Scholar

[3]

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

[4]

K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations,, Springer-Verlag, (2000).   Google Scholar

[5]

K. J. Engel, M. Kramar, B. Klöss, R. Nagel and E. Sikolya, Maximal controllability for boundary control problems,, Appl. Math. Optim., 62 (2010), 205.  doi: 10.1007/s00245-010-9101-1.  Google Scholar

[6]

H. O. Fattorini, Boundary control systems,, SIAM J. Control, 6 (1968), 349.  doi: 10.1137/0306025.  Google Scholar

[7]

G. Greiner, Perturbing the boundary conditions of a generator,, Houston J. Math., 13 (1987), 213.   Google Scholar

[8]

S. Hadd, A. Idrissi and A. Rhandi, The regular linear systems associated with the shift semigroups and application to control linear systems with delay,, Math. Control Signals Systems, 18 (2006), 272.  doi: 10.1007/s00498-006-0002-4.  Google Scholar

[9]

M. Kumpf and G. Nickel, Dynamic boundary conditions and boundary control for the one-dimensional heat equation,, J. Dynam. Control Systems, 10 (2004), 213.  doi: 10.1023/B:JODS.0000024122.71407.83.  Google Scholar

[10]

D. Salamon, Infinite-dimensional linear system with unbounded control and observation: a functional analytic approach,, Trans. Amer. Math. Soc., 300 (1987), 383.  doi: 10.2307/2000351.  Google Scholar

[11]

O. J. Staffans, Well-posed Linear Systems,, Encyclopedia of Mathematics and its Applications, (2005).  doi: 10.1017/CBO9780511543197.  Google Scholar

[12]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups,, Birkhäuser, (2009).  doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[13]

G. Weiss, Admissible observation operators for linear semigroups,, Israel J. Math., 65 (1989), 17.  doi: 10.1007/BF02788172.  Google Scholar

[14]

G. Weiss, Admissibility of unbounded control operators,, SIAM J. Control Optim., 27 (1989), 527.  doi: 10.1137/0327028.  Google Scholar

[15]

G. Weiss, Transfer functions of regular linear systems. I. Characterization of regularity,, Trans. Amer. Math. Soc., 342 (1994), 827.  doi: 10.2307/2154655.  Google Scholar

[16]

G. Weiss, Regular linear systems with feedback,, Math. Control Signals Systems, 7 (1994), 23.  doi: 10.1007/BF01211484.  Google Scholar

[1]

Yu-Xia Liang, Ze-Hua Zhou. Supercyclic translation $C_0$-semigroup on complex sectors. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 361-370. doi: 10.3934/dcds.2016.36.361
[2]

Hanxiao Wang, Jingrui Sun, Jiongmin Yong. Weak closed-loop solvability of stochastic linear-quadratic optimal control problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2785-2805. doi: 10.3934/dcds.2019117
[3]

Jiří Neustupa. On $L^2$-Boundedness of a $C_0$-Semigroup generated by the perturbed oseen-type operator arising from flow around a rotating body. Conference Publications, 2007, 2007 (Special) : 758-767. doi: 10.3934/proc.2007.2007.758
[4]

Jacek Banasiak, Marcin Moszyński. Hypercyclicity and chaoticity spaces of $C_0$ semigroups. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 577-587. doi: 10.3934/dcds.2008.20.577
[5]

Justine Yasappan, Ángela Jiménez-Casas, Mario Castro. Stabilizing interplay between thermodiffusion and viscoelasticity in a closed-loop thermosyphon. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3267-3299. doi: 10.3934/dcdsb.2015.20.3267
[6]

José A. Conejero, Alfredo Peris. Hypercyclic translation $C_0$-semigroups on complex sectors. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1195-1208. doi: 10.3934/dcds.2009.25.1195
[7]

Viorel Barbu, Gabriela Marinoschi. An identification problem for a linear evolution equation in a banach space. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-12. doi: 10.3934/dcdss.2020081
[8]

Xiaochen Sun, Fei Hu, Yancong Zhou, Cheng-Chew Lim. Optimal acquisition, inventory and production decisions for a closed-loop manufacturing system with legislation constraint. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1355-1373. doi: 10.3934/jimo.2015.11.1355
[9]

Yi Jing, Wenchuan Li. Integrated recycling-integrated production - distribution planning for decentralized closed-loop supply chain. Journal of Industrial & Management Optimization, 2018, 14 (2) : 511-539. doi: 10.3934/jimo.2017058
[10]

Pasquale Palumbo, Pierdomenico Pepe, Simona Panunzi, Andrea De Gaetano. Robust closed-loop control of plasma glycemia: A discrete-delay model approach. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 455-468. doi: 10.3934/dcdsb.2009.12.455
[11]

Filippo Cacace, Valerio Cusimano, Alfredo Germani, Pasquale Palumbo, Federico Papa. Closed-loop control of tumor growth by means of anti-angiogenic administration. Mathematical Biosciences & Engineering, 2018, 15 (4) : 827-839. doi: 10.3934/mbe.2018037
[12]

Wenbin Wang, Peng Zhang, Junfei Ding, Jian Li, Hao Sun, Lingyun He. Closed-loop supply chain network equilibrium model with retailer-collection under legislation. Journal of Industrial & Management Optimization, 2019, 15 (1) : 199-219. doi: 10.3934/jimo.2018039
[13]

Alfredo Lorenzi, Ioan I. Vrabie. An identification problem for a linear evolution equation in a Banach space and applications. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 671-691. doi: 10.3934/dcdss.2011.4.671
[14]

Roberta Fabbri, Carmen Núñez, Ana M. Sanz. A perturbation theorem for linear Hamiltonian systems with bounded orbits. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 623-635. doi: 10.3934/dcds.2005.13.623
[15]

Masoud Mohammadzadeh, Alireza Arshadi Khamseh, Mohammad Mohammadi. A multi-objective integrated model for closed-loop supply chain configuration and supplier selection considering uncertain demand and different performance levels. Journal of Industrial & Management Optimization, 2017, 13 (2) : 1041-1064. doi: 10.3934/jimo.2016061
[16]

Peter I. Kogut. On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2105-2133. doi: 10.3934/dcds.2014.34.2105
[17]

Y. Peng, X. Xiang. Second order nonlinear impulsive time-variant systems with unbounded perturbation and optimal controls. Journal of Industrial & Management Optimization, 2008, 4 (1) : 17-32. doi: 10.3934/jimo.2008.4.17
[18]

Wael Bahsoun, Benoît Saussol. Linear response in the intermittent family: Differentiation in a weighted $C^0$-norm. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6657-6668. doi: 10.3934/dcds.2016089
[19]

Liping Wang, Juncheng Wei. Infinite multiplicity for an inhomogeneous supercritical problem in entire space. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1243-1257. doi: 10.3934/cpaa.2013.12.1243
[20]

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

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences