In this paper we construct optimal, in certain sense, estimates of values of linear functionals on solutions to two-point boundary value problems (BVPs) for systems of linear first-order ordinary differential equations from observations which are linear transformations of the same solutions perturbed by additive random noises. It is assumed here that right-hand sides of equations and boundary data as well as statistical characteristics of random noises in observations are not known and belong to certain given sets in corresponding functional spaces. This leads to the necessity of introducing minimax statement of an estimation problem when optimal estimates are defined as linear, with respect to observations, estimates for which the maximum of mean square error of estimation taken over the above-mentioned sets attains minimal value. Such estimates are called minimax mean square or guaranteed estimates. We establish that the minimax mean square estimates are expressed via solutions of some systems of differential equations of special type and determine estimation errors.
[1] Kalman, R. (1960) A New Approach to Liner Filtering and Prediction Problems. Journal of Basic Engineering, 82, 35-45.
[2] Kalman, R. and Bucy, R. (1961) New Results in Liner Filtering and Prediction Theory. Transactions of the ASME-Journal of Basic Engineering, 83, 95-108. http://dx.doi.org/10.1115/1.3658902
[3] Krasovskii, N. (1968) Theory of Motion Control. Nauka, Moscow.
[4] Kurzhanskii, A. (1977) Control and Observation under Uncertainties. Nauka, Moscow.
[5] Kirichenko, N. and Nakonechnyi, O. (1977) A Minimax Approach to Recurrent Estimation of the States of Linear Dynamical Systems. Kibernetika, 4, 52-55.
[6] Nakonechnyi, O. (1979) Minimax Estimates in Systems with Distributed Parameters. Preprint 79, Acad. Sci. USSR, Inst. Cybernetics, Kyiv, 55 p.
[7] Kuntsevich, V. (2005) Accuracy of Construction of Approximating Models under Bounded Measurement Noises. Automation and Remote Control, 66, 791-798. http://dx.doi.org/10.1007/s10513-005-0123-0
[8] Kurzhanski, A. and Valyi, I. (1997) Ellipsoidal Calculus for Estimation and Control. Birkhauser Verlag, Basel.
http://dx.doi.org/10.1007/978-1-4612-0277-6
[9] Nakonechnyi, O., Podlipenko, Y. and Shestopalov, Y. (2009) Estimation of Parameters of Boundary Value Problems for Linear Ordinary Differential Equations with Uncertain Data. arXiv:0912.2872v1, 1-72.
[10] Podlipenko, Y. (2005) Minimax Estimation of Right-Hand Sides of Noetherian Equations in a Hilbert Space under Uncertainty Conditions. Reports of the National Academy of Sciences of Ukraine, 12, 36-44.
[11] Basar, T. and Bernhard, P. (1991) H-Optimal Control and Related Minimax Design Problems. Birkhauser, Basel.
[12] Fedoryuk, M. (1985) Ordinary Differential Equations. Nauka, Moscow.
[13] Naimark, M. (1969) Linear Differential Operators. Nauka, Moscow.
[14] Krein, S. (1971) Linear Equations in the Banach Space. Nauka, Moscow.
[15] Atkinson, F. (1964) Discrete and Coninuous Boundary Value Problems. Academic Press, New York.
[16] Hutson, V., Pym, J. and Cloud, M. (2005) Applications of Functional Analysis and Operator Theory. Vol. 200, 2nd Edition, Mathematics in Science and Engineering, Elsevier Science, Amsterdam.
[17] Lions, J. (1968) Controle optimal de systémes gouvernés par des équations aux dérivées partielles. Dunod, Paris.