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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

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