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The reproducing kernel Hilbert spaces underlying linear SDE Estimation, Kalman filtering and their relation to optimal control
- Source :
- Pure and Applied Functional Analysis, Pure and Applied Functional Analysis, In press
- Publication Year :
- 2022
- Publisher :
- arXiv, 2022.
-
Abstract
- International audience; It is often said that control and estimation problems are in duality. Recently, in (Aubin-Frankowski,2021), we found new reproducing kernels in Linear-Quadratic optimal control by focusing on the Hilbert space of controlled trajectories, allowing for a convenient handling of state constraints and meeting points. We now extend this viewpoint to estimation problems where it is known that kernels are the covariances of stochastic processes. Here, the Markovian Gaussian processes stem from the linear stochastic differential equations describing the continuous-time dynamics and observations. Taking extensive care to require minimal invertibility requirements on the operators, we give novel explicit formulas for these covariances. We also determine their reproducing kernel Hilbert spaces, stressing the symmetries between a space of forward-time trajectories and a space of backward-time information vectors. The two spaces play an analogue role for filtering to Sobolev spaces in variational analysis, and allow to recover the Kalman estimate through a direct variational argument. For comparison, we then recover the Kalman filter and smoother formulas through more classical arguments based on the innovation process. Extension to discrete-time observations or infinite-dimensional state, tough technical, would be straightforward.
- Subjects :
- 46E22, 60G35, 62M20
Optimization and Control (math.OC)
Markovian Gaussian processes
Probability (math.PR)
FOS: Mathematics
Kalman filtering
[MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC]
2020 Mathematics Subject Classification: 46E22
60G35
62M20
[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Mathematics - Optimization and Control
Reproducing kernels
Mathematics - Probability
Optimal control
Subjects
Details
- ISSN :
- 21893756 and 21893764
- Database :
- OpenAIRE
- Journal :
- Pure and Applied Functional Analysis, Pure and Applied Functional Analysis, In press
- Accession number :
- edsair.doi.dedup.....c7a733c6c38c74d638860f2cbbc0f57a
- Full Text :
- https://doi.org/10.48550/arxiv.2208.07030