Controllability and stability of a second-order hyperbolic system with collocated sensor/actuator. (English) Zbl 0994.93021

Summary: A second-order hyperbolic system with collocated sensor/actuator is considered. The semigroup generation is shown for the closed-loop system under the feedback with a generic unbounded observation operator. The equivalence between the exponential stability of the closed-loop system and exact controllability of the open-loop system is established in the general framework of well-posed linear systems. Finally, the conditions are weakened for the diagonal semigroups with finite dimensional inputs. An example of a beam equation is presented to display the application.


93C20 Control/observation systems governed by partial differential equations
93B05 Controllability
93D15 Stabilization of systems by feedback
Full Text: DOI


[1] Ammari, K.; Tucsnak, M., Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM control optim. calc. var., 6, 361-386, (2001) · Zbl 0992.93039
[2] Benchimol, C.D., A note on the weak stabilizability of contraction semigroups, SIAM J. control optim., 16, 373-379, (1978) · Zbl 0384.93035
[3] Chen, G.; Delfour, M.C.; Krall, A.M.; Payre, G., Modeling, stabilization and control of serially connected beam, SIAM J. control optim., 25, 526-546, (1987) · Zbl 0621.93053
[4] Chen, G.; Krantz, S.G.; Ma, D.W.; Wayne, C.E., The euler – bernoulli beam equation with boundary energy dissipation, (), 67-96
[5] Crawley, E.F., Intelligence structures for aerospace: a technology overview and assessment, Aiaa j., 32, 1689-1699, (1994)
[6] Curtain, R.F., The salamon-Weiss class of well-posed infinite dimensional linear systems: a survey, IMA J. math. control inform., 14, 207-223, (1997) · Zbl 0880.93021
[7] Curtain, R.; Weiss, G., Well posedness of triples of operators (in the sense of linear systems theory), (), 41-59
[8] Curtain, R.F.; Zwart, H.J., An introduction to infinite dimensional linear systems theory, (1995), Springer New York · Zbl 0646.93014
[9] Ho, L.F.; Russell, D.L., Admissible input elements for systems in Hilbert space and a Carleson measure criterion, SIAM J. control optim., 21, 615-640, (1983) · Zbl 0512.93044
[10] Ingham, A.E., Some trigonometrical inequalities with applications to the theory of series, Math. Z., 41, 367-379, (1936) · Zbl 0014.21503
[11] Jocob, B.; Zwart, H., Equivalent conditions for stabilizability of infinite-dimensional systems with admissible control operators, SIAM J. control optim., 37, 1419-1455, (1999) · Zbl 0945.93018
[12] Jocob, B.; Zwart, H., Exact controllability of C0-groups with one-dimensional input operators, (), 221-242
[13] Lasiecka, I.; Triggiani, R., Control theory for partial differential equations; continuous and approximation theories, (2000), Cambridge University Press Cambridge, UK · Zbl 0983.35032
[14] J. Leblond, J.P. Marmorat, Stabilization of a vibrating beam: a regularity result, in: Proceeding of the Workshop on Stabilization of Flexible Structures, COMCON, Optimization Software Inc., 1987, pp. 162-183. · Zbl 0761.73081
[15] Luo, Z.H., Direct strain feedback control of flexible robot arms: new theoretical and experimental results, IEEE trans. automat. control, 38, 1610-1622, (1993) · Zbl 0790.93100
[16] Luo, Z.H.; Guo, B.Z., Shear force feedback control of a single link flexible robot with revolute joint, IEEE trans. automat. control, 42, 1, 53-65, (1997) · Zbl 0872.93055
[17] Lous, J.C.; Wexler, D., On exact controllability in Hilbert spaces, J. differential equations, 49, 258-269, (1983) · Zbl 0477.49022
[18] Lasiecka, I.; Triggiani, R., Differential and algebraic Riccati equations with applications to boundary/point control problems: continuous theory and approximation theory, Lecture notes in control and information sciences, Vol. 164, (1991), Springer Berlin
[19] Pazy, A., Semigroups of linear operators and applications to partial differential equations, (1983), Springer Berlin · Zbl 0516.47023
[20] Russell, D.L., Controllability and stabilizability theory for linear PDE’s: recent progress and open questions, SIAM rev., 20, 639-739, (1978) · Zbl 0397.93001
[21] Rebarber, R., Exponential stability of coupled beams with dissipative joints: a frequency domain approach, SIAM J. control optim., 33, 1-28, (1995) · Zbl 0819.93042
[22] Russell, D.L.; Weiss, G., A general necessary condition for exact observability, SIAM J. control optim., 32, 1-23, (1994) · Zbl 0795.93023
[23] Rebarber, R.; Weiss, G., Necessary conditions for exact controllability with a finite-dimensional input space, Systems and control lett., 40, 217-227, (2000) · Zbl 0985.93028
[24] Slemrod, M., A note on complete controllability and stabilizability for linear control systems in Hilbert space, SIAM J. control optim., 12, 500-508, (1974) · Zbl 0254.93006
[25] Titchmarsh, E.C., The theory of functions, (1952), Oxford University Press Oxford · Zbl 0049.18603
[26] Weiss, G., The representation of regular linear systems in Hilbert spaces, (), 401-416
[27] Weiss, G., Admissibility of unbounded control operators, SIAM J. control optim., 27, 527-545, (1989) · Zbl 0685.93043
[28] Weiss, G., Two conjectures on the admissibility of control operators, (), 367-378 · Zbl 0763.93041
[29] Weiss, G., Regular linear systems with feedback, Math. control signal systems, 7, 23-57, (1994) · Zbl 0819.93034
[30] Weiss, G.; Staffans, O.; Tucsnak, M., Well-posed linear systems-a survey with emphasis on conservative systems, Int. J. appl. math. comput. sci., 11, 7-13, (2001) · Zbl 0990.93046
[31] H.J. Zwart, Private discussion, 2001.
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