Abstract
We study the problem of identification and stochastic boundary control of a class of linear partial differential equations characterized by random noise on the boundary. Such a problem does not have an analogue in the case of ordinary differential equations and the phenomena are quite different. We are mainly concerned with an abstract-theoretic formulation of the problem and computational aspects are not discussed. In the formulation, we use a white-noise theory based on Gauss measure in a Hilbert Space (in contrast to the usual Wiener process theory) initiated in [1,2]. Even though the measure is only finitely additive we have the advantage of retaining the original topology introduced for the solution of the p.d.e. Moreover, as noted in [3] the white-noise interpretation is essential to provide a meaningful model where noise on the observation is involved (as here). Another feature is the use of semigroup theory (as opposed to the Lions-Magenes variational theory as in [4]) which although limited to time-invariant systems, helps to separate out those aspects of the problem that are of a more general (if abstract) nature and not specifically tied to properties of the p.d.e. involved, and in this way also to pin-point the latter.
Research supported in part under AFOSR Grant No. 73–2492, USAF, Applied Math. Div.
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Balakrishnan, A.V. (1975). Identification and Stochastic Control of a Class of Distributed Systems with Boundary Noise. In: Bensoussan, A., Lions, J.L. (eds) Control Theory, Numerical Methods and Computer Systems Modelling. Lecture Notes in Economics and Mathematical Systems, vol 107. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46317-4_11
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DOI: https://doi.org/10.1007/978-3-642-46317-4_11
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