Abstract
We consider an evolution equationdu/dt+Au=0 in a Banach spaceX, and suppose that the system is unstable in the sense that the operator norm ∥e −tA∥X →X grows exponentially ast→∞. Then, by choosing appropriate operatorsS andT, we want to construct a new system called a feedback system of the formdu/dt+Au=TSu such that we have ∥e −t(A −TS)∥X →X ≦Me − tω (t ≧ 0) for some positive constantsM and ω. HereS:X→C N stands for the function of sensors whileT:C N→X corresponds to the effect of controllers. Our main concern is to constructT (respectively,S) for a givenS (respectively,T) so that the resulting feedback system is stable in the sense above. Actually, we shall derive certain necessary and sufficient conditions onS (respectively, onT) which enable us to choose such aT (respectively,S). These conditions are essentially equivalent to the observability ofS and to the controllability ofT, respectively.
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Suzuki, T., Yamamoto, M. Observability, controllability, and feedback stabilizability for evolution equations, I. Japan J. Appl. Math. 2, 211–228 (1985). https://doi.org/10.1007/BF03167045
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DOI: https://doi.org/10.1007/BF03167045