TY - GEN

T1 - Equivalent conditions for exponential stability for a special class of conservative linear systems

AU - Weiss, G.

AU - Tucsnak, M.

N1 - Publisher Copyright:
© 2003 EUCA.

PY - 2003/4/13

Y1 - 2003/4/13

N2 - Let A0 be a possibly unbounded positive operator on the Hilbert space H, which is boundedly invertible. Let C0 be a bounded operator from D(A1/20) (with the norm ||z||21/2 = A0z, z) to another Hilbert space U. It is known that the system of equations z(t) + A0 z(t) + 1/2 C∗0 C0 Z(t) = C∗0u(t), y(t) = - C0 z(t) + u(t), determines a well-posed linear system Σ with input u and output y, input and output space U and state space X = D(A1/20) × H. Moreover, Σ is conservative, which means that a certain energy balance equation is satisfied both by the trajectories of Σ and by those of its dual system. In this paper we present various conditions which are equivalent to the exponential stability of such a systems. Among the equivalent conditions are exact controllability and exact observability. Denoting V(s) = (s2 I + s/2 C∗0 C0 + A0)-1, we also obtain that the system is exponentially stable if and only if s → A1/20V(s) is a bounded L(H)-valued function on the imaginary axis. This is also equivalent to the condition that s → sV(s) is a bounded L(H)-valued function on the imaginary axis (or equivalently, on the open right half-plane).

AB - Let A0 be a possibly unbounded positive operator on the Hilbert space H, which is boundedly invertible. Let C0 be a bounded operator from D(A1/20) (with the norm ||z||21/2 = A0z, z) to another Hilbert space U. It is known that the system of equations z(t) + A0 z(t) + 1/2 C∗0 C0 Z(t) = C∗0u(t), y(t) = - C0 z(t) + u(t), determines a well-posed linear system Σ with input u and output y, input and output space U and state space X = D(A1/20) × H. Moreover, Σ is conservative, which means that a certain energy balance equation is satisfied both by the trajectories of Σ and by those of its dual system. In this paper we present various conditions which are equivalent to the exponential stability of such a systems. Among the equivalent conditions are exact controllability and exact observability. Denoting V(s) = (s2 I + s/2 C∗0 C0 + A0)-1, we also obtain that the system is exponentially stable if and only if s → A1/20V(s) is a bounded L(H)-valued function on the imaginary axis. This is also equivalent to the condition that s → sV(s) is a bounded L(H)-valued function on the imaginary axis (or equivalently, on the open right half-plane).

KW - Conservative linear system

KW - Estimatability

KW - Exact controllability

KW - Exponential stability

KW - Optimizability

UR - http://www.scopus.com/inward/record.url?scp=84949105394&partnerID=8YFLogxK

U2 - 10.23919/ecc.2003.7084938

DO - 10.23919/ecc.2003.7084938

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AN - SCOPUS:84949105394

T3 - European Control Conference, ECC 2003

SP - 98

EP - 102

BT - European Control Conference, ECC 2003

PB - Institute of Electrical and Electronics Engineers Inc.

T2 - 2003 European Control Conference, ECC 2003

Y2 - 1 September 2003 through 4 September 2003

ER -