Unbounded perturbations of the generator domain

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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

Abstract
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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 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.

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