Problem 5.4

Is the conservative wave

equation regular?

George Weiss

Dept. of Electrical and Electronic Engineering

Imperial College London

Exhibition Road

London SW7 2BT

UK

[email protected]

1 DESCRIPTION OF THE PROBLEM

We consider an infinite-dimensional system described by the wave equation on an n–dimensional domain, with mixed boundary control and mixed boundary observation, which has been analyzed (as an example for a certain class of conservative linear systems) in [13]. A somewhat simpler version of this system has appeared (also as an example) in the paper [11, section 7] and a related system has been discussed in [5].

BIBLIOGRAPHY

[1] D. Z. Arov and M. A. Nudelman, “Passive linear stationary dynamical scattering systems with continous time, ” Integral Equations and Operator Theory, 24 (1996), pp. 1–43.

[2] J. A. Ball, “Conservative dynamical systems and nonlinear Livsic-Brodskii nodes, ” Operator Theory: Advances and Applications, 73 (1994), pp. 67–95.

[3] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.

[4] B. M. J. Maschke and A. J. van der Schaft, “Portcontrolled Hamil-tonian representation of distributed parameter systems, ” Proc. of the IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, N. E. Leonard and R. Ortega, eds., Princeton University Press, 2000, pp. 28–38.

[5] A. Rodriguez–Bernal and E. Zuazua, “Parabolic singular limit of a wave equation with localized boundary damping, ” Discrete and Continuous Dynamical Systems, 1 (1995), pp. 303–346.

[6] O. J. Staffans and G. Weiss, “Transfer functions of regular linear systems. Part II: The system operator and the Lax-Phillips semigroup, ” Trans. American Math. Society, 354 (2002), pp. 3229–3262.

[7] O. J. Staffans and G. Weiss, “Transfer functions of regular linear systems. Part III: Inversions and duality, ” submitted.

[8] G. Weiss, “Transfer functions of regular linear systems. Part I: Characterizations of regularity, ” Trans. American Math. Society, 342 (1994), pp. 827–854.

[9] G. Weiss, “Regular linear systems with feedback, ” Mathematics of Control, Signals and Systems, 7 (1994), pp. 23–57.

[10] G. Weiss, “Optimal control of systems with a unitary semigroup and with colocated control and observation, ” Systems and Control Letters, 48 (2003), pp. 329–340.

[11] G. Weiss and R. Rebarber, “Optimizability and estimatability for infinite-dimensional linear systems, ” SIAM J. Control and Optimization, 39 (2001), pp. 1204–1232.

[12] G. Weiss, O. J. Staffans and M. Tucsnak, “Well-posed linear systems:

A survey with emphasis on conservative systems, ” Applied Mathematics and Computer Science, 11 (2001), pp. 101–127.

[13] G. Weiss and M. Tucsnak, “How to get a conservative well-posed linear system out of thin air. Part I: well-posedness and energy balance, ”ESAIM COCV, vol. 9, pp. 247-74, 2003.