Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

On robustness of exact controllability and exact observability under cross perturbations of the generator in Banach spaces

Author(s): Zhan-Dong Mei; Ji-Gen Peng
Journal: Proc. Amer. Math. Soc. 138 (2010), 4455-4468.
MSC (2010): Primary 93C25, 93B05, 93B07
Posted: July 19, 2010
MathSciNet review: 2680070
Retrieve article in: PDF

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Abstract: This paper is concerned with the exact controllability and exact observability of linear systems in the Banach space setting. It is proved that both the admissibility of control operators and the admissibility of observation operators are invariant to cross perturbations of the generator of a -semigroup. Moreover, under the admissibility invariance premise, the robustness of the exact controllability as well as the exact observability to such cross perturbations is verified. An illustrative example is presented.

References:

1.: S. Boulite, A. Idrissi, L. Maniar, Robustness of controllability under unbounded perturbations, J. Math. Anal. Appl., 304 (2005) 409-421. MR 2124672 (2006b:93021)
2.: S. Dolecki, D.L. Russell, A general theory of observation and control, SIAM J. Control and Optim., 15 (1977) 185-220. MR 0451141 (56:9428)
3.: K.J. Engel, R. Nagel, One Parameter Semigroups for Linear Evolutional Equations, Springer-Verlag, New York, 2000. MR 1721989 (2000i:47075)
4.: B.Z. Guo, G.Q. Xu, Riesz bases and exact controllability of -groups with one-dimensional input operators, Systems Control Lett., 52 (2004) 221-232. MR 2062595 (2005c:93016)
5.: B.Z. Guo, Z.C. Shao, Regularity of a Schrödinger equation with Dirichlet control and collocated observation, Syst. Control Lett., 54 (2005) 1135-1142. MR 2170295 (2006d:35208)
6.: S. Hadd, Unbounded perturbations of -semigroups on Banach spaces and applications, Semigroup Forum, 70 (2005) 451-465. MR 2148155 (2006a:47059)
7.: S. Hadd, Exact controllability of infinite dimensional systems persists under small perturbations, J. Evolution Equations, 5 (2005) 545-555. MR 2201526 (2006k:93020)
8.: S. Hadd, A. Idrissi, On the admissibility of observation for perturbed -semigroups on Banach spaces, Systems Control Lett., 55 (2006) 1-7. MR 2185598 (2006g:93069)
9.: S. Hadd, A. Idrissi, A. Rhandi, The regular linear systems associated to the shift semigroups and application to control delay systems, Math. Control Signals Systems, 18 (2006) 272-291. MR 2272077 (2007j:34161)
10.: B. Jacob, Exact observability of diagonal systems with a finite-dimensional output operator, Systems Control Lett., 43 (2001) 101-109. MR 2007610 (2004f:93017)
11.: A.N. Kolmogorov, S.V. Fomin, Elementi di teoria delle funzioni e di analisi funzionale, Mir, Moscow, 1980.
12.: I. Lasiecka, R. Triggiani, Optimal regularity, exact controllability and uniform stabilization of Shrödinger equations with Dirichlet control, Differential Integral Equations, 5 (1992) 521-535. MR 1157485 (93d:93048)
13.: E.B. Lee, L. Markus, Foundations of Optimal Control Theory, Wiley, New York, 1967. MR 0220537 (36:3596)
14.: H. Leiva, Unbounded perturbation of the controllability for evolution equations, J. Math. Anal. Appl., 280 (2003) 1-8. MR 1972187 (2004b:93015)
15.: J.L. Lions, Contrôlabilité exacte des systèmes distribués, C. R. Acad. Sci. Paris Sér. I Math., 302 (1986) 471-475. MR 838402 (87e:93051)
16.: J.L. Lions, Contrôlabilité exacte des systèmes distribués: remarques sur la théorie générale et les applications, in Analysis and optimization of systems (Antibes, 1986), vol. 83 of Lecture Notes in Control and Inform. Sci., Springer, Berlin, 1986, pp. 3-14. MR 870385
17.: J. Malinen, O.J. Staffans, Conservative boundary control systems, J. Differential Equations, 231 (2006), 290-312. MR 2287888 (2008a:93010)
18.: J.R. Partington, S. Pott, Admissibility and exact observability of observation operators for semigroups, Irish Math. Soc. Bull., 55 (2005) 19-39. MR 2185647 (2006h:47126)
19.: R. Rebarber, G. Weiss, Necessary conditions for exact controllability with a finite-dimensional input space, Systems Control Lett., 40 (2000) 217-227. MR 1827551 (2002d:93014)
20.: D.L. Russell, G. Weiss, A general necessary condition for exact observability, SIAM J. Control Optim., 32 (1994) 1-23. MR 1255956 (94m:93009)
21.: D. Salamon, Infinite-dimensional linear system with unbounded control and observation: a functional analytic approach, Trans. Amer. Math. Soc., 300 (1987) 383-431. MR 876460 (88d:93024)
22.: O.J. Staffans, Well-Posed Linear Systems, Cambridge Univ. Press, Cambridge, UK, 2005. MR 2154892 (2006k:93003)
23.: R. Triggiani, Exact boundary controllability on $L_2(\Omega)\times H^{-1}(\Omega)$ of the wave equation with Dirichlet boundary control acting on a portion of the boundary $\partial \Omega$ , and related problems, Appl. Math. Optim., 18 (1988) 241-277. MR 945765 (89i:93011)
24.: M. Tucsnak, G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Verlag, Basel, 2009. MR 2502023 (2010d:93001)
25.: G. Weiss, Admissibility of unbounded control operators, SIAM J. Control Optimization, 27 (1989) 527-545. MR 993285 (90c:93060)
26.: G. Weiss, Admissible observation operators for linear semigroups, Israel J. Mathematics, 65 (1989) 17-43. MR 994732 (90g:47082)
27.: G. Weiss, The representation of regular linear systems on Hilbert spaces, in: Kappel, F., Kunisch, K., Schappacher, W. (eds.), Control and estimation of distributed parameter systems (Proceedings Vorau, 1988), Birkhäuser, pp. 401-416. MR 1033074 (91d:93026)
28.: G. Weiss, Transfer functions of regular linear systems. Part I. Characterizations of regularity, Trans. Amer. Math. Soc., 342 (1994) 827-854. MR 1179402 (94f:93074)
29.: G. Weiss, Regular linear systems with feedback, Math. Control Signals Systems, 7 (1994) 23-57. MR 1359020 (96i:93046)
30.: G.Q. Xu, C. Liu, S.P. Yung, Necessary conditions for the exact observability of systems on Hilbert space, Systems Control Lett., 57 (3) (2008) 222-227. MR 2386920 (2008j:93028)

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