Abstract
In this paper, we discuss the partial differential equation of Riccati type that describes the optimal filtering error covariance function for a linear distributed-parameter system with pointwise observations. Since this equation contains the Dirac delta function, it is impossible to apply directly the usual methods of functional analysis to prove existence and uniqueness of a bounded solution. By using properties of the fundamental solution and the classical technique of successive approximation, we prove the existence and uniqueness theorem. We then prove the comparison theorem for partial differential equations of Riccati type. Finally, we consider some applications of these theorems to the distributed-parameter optimal sensor location problem.
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References
Bensoussan, A.,Filtrage Optimal des Systemes Lineaires, Dunod, Paris, France, 1971.
Bensoussan, A.,Optimization of Sensor Location in a Linear Filtering Problem, Proceedings of the International Symposium on Stability of Dynamical Systems, Coventry, England, pp. 62–84, 1972.
Datko, R.,A Linear Control Problem in Abstract Hilbert Space, Journal of Differential Equations, Vol. 9, pp. 346–359, 1971.
Lions, J. L.,Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York, New York, 1970.
Curtain, R. F., andPritchard, A. J.,The Infinite-Dimensional Riccati Equation, Journal of Mathematical Analysis and Applications, Vol. 47, pp. 43–57, 1974.
Omatu, S., Koide, S., andSoeda, T.,Optimal Sensor Location Problem for a Linear Distributed Parameter System, IEEE Transactions on Automatic Control, Vol. AC-23, pp. 665–673, 1978.
Sakawa, Y.,Optimal Filtering in Linear Distributed-Parameter Systems, International Journal of Control, Vol. 16, pp. 115–127, 1972.
Sawaragi, Y., Soeda, T., andOmatu, S.,Modeling, Estimation, and Their Applications for Distributed Parameter Systems, Springer-Verlag, Berlin, Germany, 1978.
Ito, S.,Partial Differential Equations (in Japanese), Baifukan, Tokyo, Japan, 1966.
Yosida, K.,Functional Analysis, Springer-Verlag, New York, New York, 1968.
Omata, S., andSoeda, T.,Estimation Problems for a Linear Stochastic Differential Equation, Stochastic Differential Equations, Edited by K. Ito, Kinokuniya, Tokyo, Japan, 1978.
Omatu, S., andSoeda, T.,Optimal Sensor Location in a Linear Distributed Parameter System, Proceedings of the IFAC Symposium on Environmental System Planning, Design, and Control, Kyoto, Japan, 1977.
Omata, S., Soeda, T., andTomita, Y.,Linear Fixed-Point Smoothing by Using Functional Analysis, IEEE Transactions on Automatic Control, Vol. AC-22, pp. 9–18, 1977.
Kumar, S., andSeinfeld, J. H.,Optimal Location of Measurements for Distributed Parameter Estimation, IEEE Transactions on Automatic Control, Vol. AC-23, pp. 690–698, 1978.
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Communicated by L. Cesari
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Omatu, S., Seinfeld, J.H. Existence and comparison theorems for partial differential equations of Riccati type. J Optim Theory Appl 36, 263–276 (1982). https://doi.org/10.1007/BF00933833
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DOI: https://doi.org/10.1007/BF00933833