Author's Summary of the dissertation for the degree of the Candidate of Science (physics and mathematics). The aim of the dissertation is to develop a generalized Kalman Duality concept applicable for linear unbounded non-invertible operators and introduce the minimax state estimation theory and algorithms for linear differential-algebraic equations. In particular, the dissertation pursues the following goals: - develop generalized duality concept for the minimax state estimation theory for DAEs with unknown but bounded model error and random observation noise with unknown but bounded correlation operator; - derive the minimax state estimation theory for linear DAEs with unknown but bounded model error and random observation noise with unknown but bounded correlation operator; - describe how the DAE model propagates uncertain parameters; - estimate the worst-case error; - construct fast estimation algorithms in the form of filters; - develop a tool for model validation, that is to assess how good the model describes observed phenomena. The dissertation contains the following new results: - generalized version of the Kalman duality principle is proposed allowing to handle unbounded linear model operators with non-trivial null-space; - new definitions of the minimax estimates for DAEs based on the generalized Kalman duality principle are proposed; - theorems of existence for minimax estimates are proved; - new minimax state estimation algorithms (in the form of filter and in the variational form) for DAE are proposed
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