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

Unbounded perturbations of the generator domain

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.
Citation: Said Hadd, Rosanna Manzo, Abdelaziz Rhandi. Unbounded perturbations of the generator domain. Discrete and Continuous Dynamical Systems, 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, Boston, Basel, Berlin, 2007.

[2]

A. Chen and K. Morris, Well-posedness of boundary control systems, SIAM J. Control Optim., 42 (2003), 1244-1265. doi: 10.1137/S0363012902384916.

[3]

R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, TAM 21, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.

[4]

K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.

[5]

K. J. Engel, M. Kramar, B. Klöss, R. Nagel and E. Sikolya, Maximal controllability for boundary control problems, Appl. Math. Optim., 62 (2010), 205-227. doi: 10.1007/s00245-010-9101-1.

[6]

H. O. Fattorini, Boundary control systems, SIAM J. Control, 6 (1968), 349-385. doi: 10.1137/0306025.

[7]

G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229.

[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), 272-291. doi: 10.1007/s00498-006-0002-4.

[9]

M. Kumpf and G. Nickel, Dynamic boundary conditions and boundary control for the one-dimensional heat equation, J. Dynam. Control Systems, 10 (2004), 213-225. doi: 10.1023/B:JODS.0000024122.71407.83.

[10]

D. Salamon, Infinite-dimensional linear system with unbounded control and observation: a functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431. doi: 10.2307/2000351.

[11]

O. J. Staffans, Well-posed Linear Systems, Encyclopedia of Mathematics and its Applications, 103, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511543197.

[12]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser, Basel, Boston, Berlin, 2009. doi: 10.1007/978-3-7643-8994-9.

[13]

G. Weiss, Admissible observation operators for linear semigroups, Israel J. Math., 65 (1989), 17-43. doi: 10.1007/BF02788172.

[14]

G. Weiss, Admissibility of unbounded control operators, SIAM J. Control Optim., 27 (1989), 527-545. doi: 10.1137/0327028.

[15]

G. Weiss, Transfer functions of regular linear systems. I. Characterization of regularity, Trans. Amer. Math. Soc., 342 (1994), 827-854. doi: 10.2307/2154655.

[16]

G. Weiss, Regular linear systems with feedback, Math. Control Signals Systems, 7 (1994), 23-57. doi: 10.1007/BF01211484.

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, Boston, Basel, Berlin, 2007.

[2]

A. Chen and K. Morris, Well-posedness of boundary control systems, SIAM J. Control Optim., 42 (2003), 1244-1265. doi: 10.1137/S0363012902384916.

[3]

R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, TAM 21, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.

[4]

K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.

[5]

K. J. Engel, M. Kramar, B. Klöss, R. Nagel and E. Sikolya, Maximal controllability for boundary control problems, Appl. Math. Optim., 62 (2010), 205-227. doi: 10.1007/s00245-010-9101-1.

[6]

H. O. Fattorini, Boundary control systems, SIAM J. Control, 6 (1968), 349-385. doi: 10.1137/0306025.

[7]

G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229.

[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), 272-291. doi: 10.1007/s00498-006-0002-4.

[9]

M. Kumpf and G. Nickel, Dynamic boundary conditions and boundary control for the one-dimensional heat equation, J. Dynam. Control Systems, 10 (2004), 213-225. doi: 10.1023/B:JODS.0000024122.71407.83.

[10]

D. Salamon, Infinite-dimensional linear system with unbounded control and observation: a functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431. doi: 10.2307/2000351.

[11]

O. J. Staffans, Well-posed Linear Systems, Encyclopedia of Mathematics and its Applications, 103, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511543197.

[12]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser, Basel, Boston, Berlin, 2009. doi: 10.1007/978-3-7643-8994-9.

[13]

G. Weiss, Admissible observation operators for linear semigroups, Israel J. Math., 65 (1989), 17-43. doi: 10.1007/BF02788172.

[14]

G. Weiss, Admissibility of unbounded control operators, SIAM J. Control Optim., 27 (1989), 527-545. doi: 10.1137/0327028.

[15]

G. Weiss, Transfer functions of regular linear systems. I. Characterization of regularity, Trans. Amer. Math. Soc., 342 (1994), 827-854. doi: 10.2307/2154655.

[16]

G. Weiss, Regular linear systems with feedback, Math. Control Signals Systems, 7 (1994), 23-57. doi: 10.1007/BF01211484.

[1]

Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030

[2]

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

[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 and Continuous Dynamical Systems, 2008, 20 (3) : 577-587. doi: 10.3934/dcds.2008.20.577

[5]

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

[6]

Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control and Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026

[7]

Xun Li, Jingrui Sun, Jiongmin Yong. Mean-field stochastic linear quadratic optimal control problems: closed-loop solvability. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 2-. doi: 10.1186/s41546-016-0002-3

[8]

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

[9]

Viorel Barbu, Gabriela Marinoschi. An identification problem for a linear evolution equation in a banach space. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1429-1440. doi: 10.3934/dcdss.2020081

[10]

S. Hadd, F.Z. Lahbiri. A semigroup approach to stochastic systems with input delay at the boundary. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022004

[11]

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

[12]

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

[13]

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

[14]

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 and Management Optimization, 2015, 11 (4) : 1355-1373. doi: 10.3934/jimo.2015.11.1355

[15]

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

[16]

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

[17]

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 and Management Optimization, 2019, 15 (1) : 199-219. doi: 10.3934/jimo.2018039

[18]

Reza Lotfi, Yahia Zare Mehrjerdi, Mir Saman Pishvaee, Ahmad Sadeghieh, Gerhard-Wilhelm Weber. A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk. Numerical Algebra, Control and Optimization, 2021, 11 (2) : 221-253. doi: 10.3934/naco.2020023

[19]

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

[20]

Abdolhossein Sadrnia, Amirreza Payandeh Sani, Najme Roghani Langarudi. Sustainable closed-loop supply chain network optimization for construction machinery recovering. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2389-2414. doi: 10.3934/jimo.2020074

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (260)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]