TY - JOUR

T1 - General necessary condition for exact observability

AU - Russell, David L.

AU - Weiss, George

PY - 1994

Y1 - 1994

N2 - Suppose A generates an exponentially stable strongly continuous semigroup on the Hilbert space X,Y is another Hilbert space, and C:D(A)→Y is an admissible observation operator for this semigroup. (This means that to any initial state in X we can associate an output function in L2([0,∞), Y).) This paper gives a necessary condition for the exact observability of the system defined by A and C. This condition, called (E), is related to the Hautus Lemma from finite dimensional systems theory. It is an estimate in terms of the operators A and C alone (in particular, it makes no reference to the semigroup). This paper shows that (E) implies approximate observability and, if A is bounded, it implies exact observability. This paper conjectures that (E) is in fact equivalent to exact observability. The paper also shows that for diagonal semigroups, (E) takes on a very simple form, and applies the results to sequences of complex exponential functions.

AB - Suppose A generates an exponentially stable strongly continuous semigroup on the Hilbert space X,Y is another Hilbert space, and C:D(A)→Y is an admissible observation operator for this semigroup. (This means that to any initial state in X we can associate an output function in L2([0,∞), Y).) This paper gives a necessary condition for the exact observability of the system defined by A and C. This condition, called (E), is related to the Hautus Lemma from finite dimensional systems theory. It is an estimate in terms of the operators A and C alone (in particular, it makes no reference to the semigroup). This paper shows that (E) implies approximate observability and, if A is bounded, it implies exact observability. This paper conjectures that (E) is in fact equivalent to exact observability. The paper also shows that for diagonal semigroups, (E) takes on a very simple form, and applies the results to sequences of complex exponential functions.

UR - http://www.scopus.com/inward/record.url?scp=0028201945&partnerID=8YFLogxK

U2 - 10.1137/S036301299119795X

DO - 10.1137/S036301299119795X

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AN - SCOPUS:0028201945

SN - 0363-0129

VL - 32

SP - 1

EP - 23

JO - SIAM Journal on Control and Optimization

JF - SIAM Journal on Control and Optimization

IS - 1

ER -