# American Institute of Mathematical Sciences

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.

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##### 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.
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