Abstract
Two goals will be achieved in this chapter. The first one concerns the feedback stabilization problem for a nonautonomous linear control system: the stabilizing feedback control is determined by formulating and solving an infinite horizon linear regulator problem. The minimizing pairs for the corresponding functional will be in a one-to-one correspondence with certain solutions of a nonautonomous linear Hamiltonian system constructed from the minimizing problem. Results of the previous chapters concerning the occurrence of exponential dichotomy and the properties of the rotation number will be applied, and only some basic elements of control theory, introduced for the most part in Chap. 3, will be required. The second goal is to give information concerning the Kalman–Bucy filter in a nonautonomous setting. Once more, the concepts of exponential dichotomy and rotation number for linear Hamiltonian systems are used to produce direct proofs of some basic results, including the asymptotic limit and the stability properties of the error covariance matrix, and the Hurwitz property at \(+\infty \) of the error propagation system.
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Johnson, R., Obaya, R., Novo, S., Núñez, C., Fabbri, R. (2016). Nonautonomous Control Theory: Linear Regulator Problem and the Kalman–Bucy Filter. In: Nonautonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory and Control. Developments in Mathematics, vol 36. Springer, Cham. https://doi.org/10.1007/978-3-319-29025-6_6
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