Conservative boundary control systems. (English) Zbl 1117.93025
A linear boundary control/observation system can be abstractly described in the form
\[
u(t) = Gz(t), \quad z'(t) = Lz(t), \quad y(t) = Kz(t), \quad z(0) = z_0
\]
where \(G, L, K\) are unbounded linear operators; typically, \(L\) is the partial differential operator in the equation and \(G, K\) are boundary trace operators. The system is internally well posed if various compatibility conditions are satisfied by the operators and \(L| {\mathcal N}(G)\) generates a strongly continuous semigroup. The system is time invertible if the conditions above are satisfied reversing the direction of time and interchanging the roles of \(K\) and \(G.\) The main result in this paper is that a time invertible system gives rise to a continuous time Livšic-Brodskiĭsystem with unbounded control and observation operators, the converse being also true.
Reviewer: Hector O. Fattorini (Los Angeles)
MSC:
| 93B28 | Operator-theoretic methods |
| 93C25 | Control/observation systems in abstract spaces |
| 47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |
| 47A48 | Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc. |
| 47N70 | Applications of operator theory in systems, signals, circuits, and control theory |
| 93C05 | Linear systems in control theory |
Keywords:
conservative systems; boundary control systems; time invertibility; operator node; Cayley transformReferences:
| [1] | Adams, R. A.; Fournier, J., Sobolev Spaces (2003), Academic Press: Academic Press New York · Zbl 1098.46001 |
| [2] | Arov, D. Z., Passive linear stationary dynamic systems, Sibirsk. Mat. Zh.. Sibirsk. Mat. Zh., Sib. Math. J., 20, 149-162 (1979), translation in: · Zbl 0437.93019 |
| [3] | Arov, D. Z., Stable dissipative linear stationary dynamical scattering systems, J. Operator Theory, 1, 95-126 (1979), translation in [5] · Zbl 0461.47005 |
| [4] | Arov, D. Z., Passive linear systems and scattering theory, (Dynamical Systems, Control Coding, Computer Vision. Dynamical Systems, Control Coding, Computer Vision, Progr. Systems Control Theory, vol. 25 (1999), Birkhäuser: Birkhäuser Boston), 27-44 · Zbl 0921.93005 |
| [5] | Arov, D. Z., Stable dissipative linear stationary dynamical scattering systems, (Interpolation Theory, Systems Theory, and Related Topics. The Harry Dym Anniversary Volume. Interpolation Theory, Systems Theory, and Related Topics. The Harry Dym Anniversary Volume, Oper. Theory Adv. Appl., vol. 134 (2002), Birkhäuser: Birkhäuser Basel). (Interpolation Theory, Systems Theory, and Related Topics. The Harry Dym Anniversary Volume. Interpolation Theory, Systems Theory, and Related Topics. The Harry Dym Anniversary Volume, Oper. Theory Adv. Appl., vol. 134 (2002), Birkhäuser: Birkhäuser Basel), J. Operator Theory, 1, 95-126 (1979), English translation in: · Zbl 0461.47005 |
| [6] | Arov, D. Z.; Nudelman, M. A., Passive linear stationary dynamical scattering systems with continuous time, Integral Equations Operator Theory, 24, 1-45 (1996) · Zbl 0838.47004 |
| [7] | Bardos, C.; Halpern, L.; Lebeau, G.; Rauch, J.; Zuazua, E., Stabilisation de l’équation des ondes au moyen d’un feedback portant sur la condition aux limites de Dirichlet, Asymptot. Anal., 4, 4, 285-291 (1991) · Zbl 0764.35055 |
| [8] | Bensoussan, A.; Da Prato, G.; Delfour, M. C.; Mitter, S. K., Representation and Control of Infinite-Dimensional Systems, vols. 1 and 2 (1992), Birkhäuser: Birkhäuser Basel · Zbl 0781.93002 |
| [9] | Byrnes, C. I.; Gilliam, D. S.; Shubov, V. I.; Weiss, G., Regular linear systems governed by a boundary controlled heat equation, J. Dynam. Control Systems, 8, 3, 341-370 (2002) · Zbl 1010.93052 |
| [10] | Brodskiĭ, M. S., Triangular and Jordan Representations of Linear Operators, Transl. Math. Monogr., vol. 32 (1971), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0214.38901 |
| [11] | Brodskiĭ, M. S., Unitary operator colligations and their characteristic functions, Russian Math. Surveys, 33, 4, 159-191 (1978) · Zbl 0415.47007 |
| [12] | Brodskiĭ, V. M., On operator colligations and their characteristic functions, Soviet Math. Dokl., 12, 696-700 (1971) · Zbl 0231.47011 |
| [13] | Fattorini, H. O., Boundary control systems, SIAM J. Control, 6, 349-385 (1968) · Zbl 0164.10902 |
| [14] | Gorbachuk, V. I.; Gorbachuk, M. L., Boundary Value Problems for Operator Differential Equations, Math. Appl. (Soviet Series), vol. 48 (1991), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht, translation and revised from the 1984 Russian original · Zbl 0751.47025 |
| [15] | Gorbachuk, V. I.; Gorbachuk, M. L.; Kochubeĭ, A. N., The theory of extensions of symmetric operators, and boundary value problems for differential equations, Ukraïn. Mat. Zh.. Ukraïn. Mat. Zh., Ukrainian Math. J., 41, 10, 1117-1129 (1990), translation in: · Zbl 0706.47001 |
| [16] | Guo, B.-Z.; Luo, Y.-H., Controllability and stability of a second-order hyperbolic system with collocated sensor/actuator, Systems Control Lett., 46, 45-65 (2002) · Zbl 0994.93021 |
| [17] | Grisvard, P., Elliptic Problems in Nonsmooth Domains (1985), Pitman: Pitman Boston · Zbl 0695.35060 |
| [18] | Kreĭn, M. G., The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications I, Sb. Math. N.S. [Mat. Sb.], 20, 62, 431-495 (1947) · Zbl 0029.14103 |
| [19] | Lagnese, J. E., Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations, 50, 163-182 (1983) · Zbl 0536.35043 |
| [20] | Lasiecka, I.; Triggiani, R., Control Theory for Partial Differential Equations: I-II, Encyclopedia Math. Appl., vols. 74-75 (2000), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0983.35032 |
| [21] | Lions, J. L., Optimal Control of Systems Governed by Partial Differential Equations, Die Grundlehren der mathematishen Wissenschaften in Einzeldarstellungen, vol. 170 (1971), Springer: Springer Berlin · Zbl 0203.09001 |
| [22] | Lions, J. L.; Magenes, E., Nonhomogenous Boundary Value Problems and Applications I, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, vol. 181 (1972), Springer: Springer Berlin · Zbl 0223.35039 |
| [23] | J. Malinen, Conservativity of time-flow invertible and boundary control systems, Technical report A479, Institute of Mathematics, Helsinki University of Technology, Espoo, Finland, 2004; J. Malinen, Conservativity of time-flow invertible and boundary control systems, Technical report A479, Institute of Mathematics, Helsinki University of Technology, Espoo, Finland, 2004 |
| [24] | J. Malinen, Conservativity and time-flow invertiblity of boundary control systems, in: Proceedings of CDC-ECC’05, 2005; J. Malinen, Conservativity and time-flow invertiblity of boundary control systems, in: Proceedings of CDC-ECC’05, 2005 |
| [25] | J. Malinen, O.J. Staffans, Semigroups of impedance conservative boundary control systems, in: Proceedings of the MTNS06, Kioto, 2006; J. Malinen, O.J. Staffans, Semigroups of impedance conservative boundary control systems, in: Proceedings of the MTNS06, Kioto, 2006 · Zbl 1117.93025 |
| [26] | Malinen, J.; Staffans, O. J.; Weiss, G., When is a linear system conservative?, Quart. Appl. Math., 64, 61-91 (2006) · Zbl 1125.47007 |
| [27] | Neumark, M., Self-adjoint extensions of the second kind of a symmetric operator, Bull. Acad. Sci. URSS Ser. Math. [Izvestia Akad. Nauk SSSR], 4, 53-104 (1940) · JFM 66.0549.01 |
| [28] | Pazy, A., Semi-Groups of Linear Operators and Applications to Partial Differential Equations (1983), Springer: Springer Berlin · Zbl 0516.47023 |
| [29] | Rodríguez-Bernal, A.; Zuazua, E., Parabolic singular limit of a wave equation with localized boundary damping, Discrete Contin. Dyn. Syst., 1, 3, 303-346 (1995) · Zbl 0891.35075 |
| [30] | Russell, D. L., Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20, 639-739 (1978) · Zbl 0397.93001 |
| [31] | Salamon, D., Infinite dimensional linear systems with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300, 383-431 (1987) · Zbl 0623.93040 |
| [32] | Salamon, D., Realization theory in Hilbert space, Math. Systems Theory, 21, 147-164 (1989) · Zbl 0668.93018 |
| [33] | Y.L. Šmuljan, Invariant subspaces of semigroups and the Lax-Phillips scheme, Deposited in VINITI, No. 8009-B86, Odessa, 1986, 49 pages; Y.L. Šmuljan, Invariant subspaces of semigroups and the Lax-Phillips scheme, Deposited in VINITI, No. 8009-B86, Odessa, 1986, 49 pages |
| [34] | Staffans, O. J., \(J\)-energy preserving well-posed linear systems, Int. J. Appl. Math. Comput. Sci., 11, 1361-1378 (2001) · Zbl 1008.93024 |
| [35] | Staffans, O. J., Passive and conservative continuous-time impedance and scattering systems. Part I: Well-posed systems, Math. Control Signals Systems, 15, 291-315 (2002) · Zbl 1158.93321 |
| [36] | Staffans, O. J., Passive and conservative infinite-dimensional impedance and scattering systems (from a personal point of view), (Mathematical Systems Theory in Biology, Communication, Computation, and Finance. Mathematical Systems Theory in Biology, Communication, Computation, and Finance, IMA Vol. Math. Appl., vol. 134 (2002), Springer: Springer New York), 375-414 · Zbl 1156.93326 |
| [37] | Staffans, O. J., Stabilization by collocated feedback, (Directions in Mathematical Systems Theory and Optimization. Directions in Mathematical Systems Theory and Optimization, Lecture Notes in Control and Inform. Sci., vol. 286 (2002), Springer: Springer New York), 261-278 |
| [38] | Staffans, O. J., Well-Posed Linear Systems (2005), Cambridge Univ. Press: Cambridge Univ. Press Cambridge/New York · Zbl 1057.93001 |
| [39] | Staffans, O. J.; Weiss, G., Transfer functions of regular linear systems. Part II: The system operator and the Lax-Phillips semigroup, Trans. Amer. Math. Soc., 354, 3229-3262 (2002) · Zbl 0996.93012 |
| [40] | Staffans, O. J.; Weiss, G., Transfer functions of regular linear systems. Part III: Inversions and duality, Integral Equations Operator Theory, 49, 517-558 (2004) · Zbl 1052.93032 |
| [41] | Triggiani, R., Wave equation on a bounded domain with boundary dissipation: An operator approach, J. Math. Anal. Appl., 137, 438-461 (1989) · Zbl 0686.35067 |
| [42] | Tucsnak, M.; Weiss, G., How to get a conservative well-posed linear system out of thin air. Part II. Controllability and stability, SIAM J. Control Optim., 42, 907-935 (2003) · Zbl 1125.93383 |
| [43] | Weiss, G.; Staffans, O. J.; Tucsnak, M., Well-posed linear systems—A survey with emphasis on conservative systems, Int. J. Appl. Math. Comput. Sci., 11, 7-34 (2001) · Zbl 0990.93046 |
| [44] | Weiss, G.; Tucsnak, M., How to get a conservative well-posed linear system out of thin air. I. Well-posedness and energy balance, ESAIM Control Optim. Calc. Var., 9, 247-274 (2003) · Zbl 1063.93026 |
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