How to get a conservative well-posed linear system out of thin air. Part I. Well-posedness and energy balance

Weiss, George; Tucsnak, Marius

ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 247-273.

Let $A_0$ be a possibly unbounded positive operator on the Hilbert space $H$, which is boundedly invertible. Let $C_0$ be a bounded operator from $𝒟\left(A_0^{\frac{1}{2}}\right)$ to another Hilbert space $U$. We prove that the system of equations

$$\ddot{z}\left(t\right)+A_{0}z\left(t\right)+\frac{1}{2}{C}_0^{*}{C}_0\dot{z}\left(t\right)=C_0^{*}u\left(t\right)$$

$$y\left(t\right)=-C_0\dot{z}\left(t\right)+u\left(t\right),$$

determines a well-posed linear system with input $u$ and output $y$. The state of this system is

$$x\left(t\right)=\left[\begin{array}{c}z\left(t\right)\\ \dot{z}\left(t\right)\end{array}\right]\in 𝒟\left(A_0^{\frac{1}{2}}\right)\times H=X,$$

where $X$ is the state space. Moreover, we have the energy identity

$${\parallel x\left(t\right)\parallel }_X^2-{\parallel x\left(0\right)\parallel }_X^2={\int }_0^T{\parallel u\left(t\right)\parallel }_U^2\mathrm{d}t-{\int }_0^T{\parallel y\left(t\right)\parallel }_U^2\mathrm{d}t.$$

We show that the system described above is isomorphic to its dual, so that a similar energy identity holds also for the dual system and hence, the system is conservative. We derive various other properties of such systems and we give a relevant example: a wave equation on a bounded $n$-dimensional domain with boundary control and boundary observation on part of the boundary.

Classification : 93C25, 93C20, 35B37
Mots clés : well-posed linear system, operator semigroup, dual system, energy balance equation, conservative system, wave equation

[1] D.Z. Arov and M.A. Nudelman, Passive linear stationary dynamical scattering systems with continous time. Integral Equations Operator Theory 24 (1996) 1-43. | MR | Zbl

[2] J.A. Ball, Conservative dynamical systems and nonlinear Livsic-Brodskii nodes. Oper. Theory Adv. Appl. 73 (1994) 67-95. | MR | Zbl

[3] A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems, Vol. 1. Birkhäuser, Boston (1992). | MR | Zbl

[4] R.F. Curtain and G. Weiss, Well-posedness of triples of operators (in the sense of linear systems theory), Control and Estimation of Distributed Parameter Systems, edited by F. Kappel, K. Kunisch and W. Schappacher. Birkhäuser, Basel (1989) 41-59. | MR | Zbl

[5] P. Grabowski, On the spectral Lyapunov approach to parametric optimization of distributed parameter systems. IMA J. Math. Control Inform. 7 (1990) 317-338. | MR | Zbl

[6] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985). | MR | Zbl

[7] P. Grisvard, Singularities in Boundary Value Problems. Masson, Paris (1992). | MR | Zbl

[8] S. Hansen and G. Weiss, New results on the operator Carleson measure criterion. IMA J. Math. Control Inform. 14 (1997) 3-32. | MR | Zbl

[9] B. Jacob and J. Partington, The Weiss conjecture on admissibility of observation operators for contraction semigroups. Integral Equations Operator Theory (to appear). | MR | Zbl

[10] P. Lax and R. Phillips, Scattering Theory. Academic Press, New York (1967). | MR | Zbl

[11] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I. Springer-Verlag, Berlin, Grundlehren Math. Wiss. 181 (1972). | MR | Zbl

[12] B.M.J. Maschke and A.J. Van Der Schaft, Portcontrolled Hamiltonian representation of distributed parameter systems, in Proc. of the IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, edited by N.E. Leonard andR. Ortega. Princeton University (2000) 28-38.

[13] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983). | MR | Zbl

[14] A. Rodriguez-Bernal and E. Zuazua, Parabolic singular limit of a wave equation with localized boundary damping. Discrete Contin. Dynam. Systems 1 (1995) 303-346. | MR | Zbl

[15] D. Salamon, Infinite dimensional systems with unbounded control and observation: A functional analytic approach. Trans. Amer. Math. Soc. 300 (1987) 383-431. | MR | Zbl

[16] D. Salamon, Realization theory in Hilbert space. Math. Systems Theory 21 (1989) 147-164. | MR | Zbl

[17] O.J. Staffans, Quadratic optimal control of stable well-posed linear systems. Trans. Amer. Math. Soc. 349 (1997) 3679-3715. | MR | Zbl

[18] O.J. Staffans, Coprime factorizations and well-posed linear systems. SIAM J. Control Optim. 36 (1998) 1268-1292. | MR | Zbl

[19] O.J. Staffans and G. Weiss, Transfer functions of regular linear systems. Part II: The system operator and the Lax-Phillips semigroup. Trans. Amer. Math. Soc. 354 (2002) 3229-3262. | MR | Zbl

[20] O.J. Staffans and G. Weiss, Transfer functions of regular linear systems. Part III: Inversions and duality (submitted). | Zbl

[21] R. Triggiani, Wave equation on a bounded domain with boundary dissipation: An operator approach. J. Math. Anal. Appl. 137 (1989) 438-461. | MR | Zbl

[22] G. Weiss, Admissibility of unbounded control operators. SIAM J. Control Optim. 27 (1989) 527-545. | MR | Zbl

[23] G. Weiss, Admissible observation operators for linear semigroups. Israel J. Math. 65 (1989) 17-43. | MR | Zbl

[24] G. Weiss, Transfer functions of regular linear systems. Part I: Characterizations of regularity. Trans. Amer. Math. Soc. 342 (1994) 827-854. | MR | Zbl

[25] G. Weiss, Regular linear systems with feedback. Math. Control Signals Systems 7 (1994) 23-57. | MR | Zbl

[26] G. Weiss and R. Rebarber, Optimizability and estimatability for infinite-dimensional linear systems. SIAM J. Control Optim. 39 (2001) 1204-1232. | MR | Zbl

[27] G. Weiss, O.J. Staffans and M. Tucsnak, Well-posed linear systems - A survey with emphasis on conservative systems. Appl. Math. Comput. Sci. 11 (2001) 101-127. | MR | Zbl