On the perturbations of regular linear systems and linear systems with state and output delays. (English) Zbl 1237.47011
The authors of the paper under review study the problem of regularity of linear systems under perturbations. They consider two types of perturbations of linear systems. The results are applied to linear systems with state and output delays.
Reviewer: Vladimir V. Peller (East Lansing)
MSC:
| 47A48 | Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc. |
| 47D06 | One-parameter semigroups and linear evolution equations |
| 47A55 | Perturbation theory of linear operators |
| 93C23 | Control/observation systems governed by functional-differential equations |
References:
| [1] | Bátkai A., Piazzera S.: Semigroup and linear partial differential equations with delay. J. Math. Anal. Appl. 264, 1–20 (2001) · Zbl 1003.34044 · doi:10.1006/jmaa.2001.6705 |
| [2] | Bátkai A., Piazzera S.: Semigroups for Delay Equations, vol. 10. A K Peters, Ltd, Wellesley (2005) · Zbl 1089.35001 |
| [3] | Bhat, K.P.M., Wonham, W.M.: Stabilizability and detectability for evolution systems on Banach spaces. In: Proceedings of IEEE Conference Decision Control 15th Symposium Adaptive Processes, pp. 1240–1243 (1976) |
| [4] | Curtain, R.F., Pritchard, A.J.: Infinite dimensional linear systems theory. In: Lecture Notes in Information Sciences, vol. 8. Springer-Verlag, Berlin (1978) · Zbl 0389.93001 |
| [5] | Engel K.J., Nagel R.: One Parameter Semigroups for Linear Evolutional Equations. Springer-Verlag, New York (2000) · Zbl 0952.47036 |
| [6] | Haak B., Kunstmann P.C.: Admissibility of unbounded operators and wellposedness of linear systems in Banach spaces. Integr. Equ. Oper. Theory 55(4), 497–533 (2006) · Zbl 1138.93361 · doi:10.1007/s00020-005-1416-y |
| [7] | Hadd S.: Unbounded perturbations of C 0-semigroups on Banach spaces and applications. Semigroup Forum 70, 451–465 (2005) · Zbl 1074.47017 · doi:10.1007/s00233-004-0172-7 |
| [8] | Hadd S.: Exact controllability of infinite dimensional systems persists under small perturbations. J. Evol. Equ. 5, 545–555 (2005) · Zbl 1106.93012 · doi:10.1007/s00028-005-0229-4 |
| [9] | Hadd S., Idrissi A.: On the admissibility of observation for perturbed C 0-semigroups on Banach spaces. Syst. Control Lett. 55, 1–7 (2006) · Zbl 1129.93420 · doi:10.1016/j.sysconle.2005.04.010 |
| [10] | Hadd S., Idrissi A., Rhandi A.: The regular linear systems associated to the shift semigroups and application to control delay systems. Math. Control Signals Syst. 18, 272–291 (2006) · Zbl 1105.93037 · doi:10.1007/s00498-006-0002-4 |
| [11] | Hale J.K.: Functional Differential Equations, Applied Mathematical Sciences, vol. 3. Springer-Verlag, Berlin (1971) · Zbl 0222.34003 |
| [12] | Jacob B., Partington J.R.: The Weiss conjecture on admissibility of observation operators for contraction semigroups. Integr. Equ. Oper. Theory 40(2), 231–241 (2001) · Zbl 1031.93107 · doi:10.1007/BF01301467 |
| [13] | Jacob B., Partington J.R., Pott S.: Conditions for admissibility of observation operators and boundedness of Hankel operators. Integr. Equ. Oper. Theory 47(3), 315–338 (2003) · Zbl 1046.47036 · doi:10.1007/s00020-002-1165-0 |
| [14] | Jacob B., Zwart H.: Counterexamples concerning observation operators for C 0 semigroups. SIAM J. Control Optim. 43(1), 137–153 (2004) · Zbl 1101.93042 · doi:10.1137/S0363012903423235 |
| [15] | Maciá F., Zuazua E.: On the lack of observability for wave equations: Gaussian beam approach. Asymptot. Anal. 32(1), 1–26 (2002) · Zbl 1024.35062 |
| [16] | Malinen J., Staffans O.J., Weiss G.: When is a linear system conservative?. Quart. Appl. Math. 64, 61–91 (2006) · Zbl 1125.47007 |
| [17] | Malinen J., Staffans O.J.: Conservative boundary control systems. J. Differ. Equ. 231, 290–312 (2006) · Zbl 1117.93025 · doi:10.1016/j.jde.2006.05.012 |
| [18] | Mátrai T.: On perturbations of eventually compact semigroups preserving eventual compactness. Semigroup Forum 69, 317–340 (2004) · Zbl 1088.47032 |
| [19] | Olbrot A.W.: Stabilizability, detectability, and spectrum assignment for linear autonomous systems with general time delays. IEEE Trans. Automat. Control 23(5), 887–890 (1978) · Zbl 0399.93008 · doi:10.1109/TAC.1978.1101879 |
| [20] | Rebarber R.: Conditions for the equivalence of internal and external stability for distributed parameter systems. IEEE Trans. Automat. Control 38, 994–998 (1993) · Zbl 0786.93087 · doi:10.1109/9.222318 |
| [21] | Salamon D.: Infinite-dimensional linear system with unbounded control and observation: a functional analytic approach. Trans. Am. Math. Soc. 300, 383–431 (1987) · Zbl 0623.93040 |
| [22] | Salamon D.: Realization theory in Hilbert space. Math. Syst. Theory 21, 147–164 (1989) · Zbl 0668.93018 · doi:10.1007/BF02088011 |
| [23] | Staffans O.J., Weiss G.: Transfer functions of regular linear systems. Part II: the system operator and the Lax-Phillips semigroup. Trans. Am. Math. Soc. 354, 3229–3262 (2002) · Zbl 0996.93012 · doi:10.1090/S0002-9947-02-02976-8 |
| [24] | Staffans O.J.: Well-Posed Linear Systems. Cambridge University Press, Cambridge (2005) · Zbl 1057.93001 |
| [25] | Tucsnak M., Weiss G.: Observation and Control for Operators Semigroups. Birkhäuser Verlag, Basel (2009) · Zbl 1188.93002 |
| [26] | Weiss G.: Admissibility of unbounded control operators. SIAM J. Control Optim. 27, 527–545 (1989) · Zbl 0685.93043 · doi:10.1137/0327028 |
| [27] | Weiss G.: Admissible observation operators for linear semigroups. Isr. J. Math. 65, 17–43 (1989) · Zbl 0696.47040 · doi:10.1007/BF02788172 |
| [28] | Weiss, G.: 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), pp. 401–416. Birkhäuser, Basel |
| [29] | Weiss G.: Two conjectures on the admissibility of control operators. In: Desch, W., Kappel, F. (eds) Estimation and Control of Distributed Parameter Systems, pp. 367–378. Birkhäuser, Basel (1991) · Zbl 0763.93041 |
| [30] | Weiss G.: Transfer functions of regular linear systems. Part I: characterizations of regularity. Trans. Am. Math. Soc. 342(2), 827–854 (1994) · Zbl 0798.93036 · doi:10.2307/2154655 |
| [31] | Weiss G.: Regular linear systems with feedback. Math. Control Signals Syst. 7, 23–57 (1994) · Zbl 0819.93034 · doi:10.1007/BF01211484 |
| [32] | Weiss G., Rebarber R.: Optimizability and estimatability for infinite-dimensional linear systems. SIAM J. Control Optim. 39, 1204–1232 (2000) · Zbl 0981.93032 · doi:10.1137/S036301299833519X |
| [33] | Zhong Q.C.: Robust Control of Time-Delay Systems. Springer-Verlag, London (2006) · Zbl 1119.93005 |
| [34] | Zwart H., Jacob B., Staffans O.: Weak admissibility does not imply admissibility for analytic semigroups. Syst. Control Lett. 48(3), 341–350 (2003) · Zbl 1157.93421 · doi:10.1016/S0167-6911(02)00277-3 |
| [35] | Zwart H.: Sufficient conditions for admissibility. Syst. Control Lett. 54, 973–979 (2005) · Zbl 1129.93422 · doi:10.1016/j.sysconle.2005.02.009 |
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