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doi: 10.3934/mcrf.2021050
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Existence and uniqueness for variational data assimilation in continuous time

School of Mathematical, Physical, and Computational Sciences, University of Reading, Reading RG6 6AX, United Kingdom

Received  August 2020 Revised  May 2021 Early access October 2021

Fund Project: The author was supported by the UK Engineering and Physical Sciences Research Council under grant agreement EP/L012669/1. Fruitful discussions with Horatio Boedihardjo, Tobias Kuna, and Dan Crisan are gratefully acknowledged

A variant of the optimal control problem is considered which is nonstandard in that the performance index contains "stochastic" integrals, that is, integrals against very irregular functions. The motivation for considering such performance indices comes from dynamical estimation problems where observed time series need to be "fitted" with trajectories of dynamical models. The observations may be contaminated with white noise, which gives rise to the nonstandard performance indices. Problems of this kind appear in engineering, physics, and the geosciences where this is referred to as data assimilation. The fact that typical models in the geosciences do not satisfy linear growth nor monotonicity conditions represents an additional difficulty. Pathwise existence of minimisers is obtained, along with a maximum principle as well as preliminary results in dynamic programming. The results also extend previous work on the maximum aposteriori estimator of trajectories of diffusion processes.

Citation: Jochen Bröcker. Existence and uniqueness for variational data assimilation in continuous time. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021050
References:
[1]

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar

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F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, vol. 264 of Graduate Texts in Mathematics, Springer, London, 2013. doi: 10.1007/978-1-4471-4820-3.  Google Scholar

[3]

J. Derber, A variational continuous assimilation technique, Monthly Weather Review, 117 (1989), 2437-2446.   Google Scholar

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G. Evensen, Data Assimilation. The Ensemble Kalman Filter, 2$^{nd}$ edition, Springer-Verlag, New York, 2009. doi: 10.1007/978-3-642-03711-5.  Google Scholar

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W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, 2$^{nd}$ edition, Stochastic Modelling and Applied Probability, 25. Springer, New York, 2006.  Google Scholar

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H. Frankowska and A. Ochal, On singularities of value function for Bolza optimal control problem, J. Math. Anal. Appl., 306 (2005), 714-729.  doi: 10.1016/j.jmaa.2004.10.003.  Google Scholar

[7] P. K. Friz and N. B. Victoir, Multidimensional Stochastic Processes as Rough Paths, Theory and applications. Cambridge Studies in Advanced Mathematics, 120. Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9780511845079.  Google Scholar
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O. Hijab, Asymptotic Bayesian estimation of a first order equation with small diffusion, Ann. Probab., 12 (1984), 890-902.   Google Scholar

[9] A. H. Jazwinski, Stochastic Processes and Filtering Theory, vol. 64 of Mathematics in Science and Engineering, Academic Press, 1970.   Google Scholar
[10] E. Kalnay, Atmospheric Modeling, Data Assimilation and Predictability, 1$^{nd}$ edition, Cambridge University Press, 2001.   Google Scholar
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A. Krener, The convergence of the minimum energy estimator, In New Trends in Nonlinear Dynamics and Control and their Applications, Lect. Notes Control Inf. Sci., Springer, Berlin, 295 (2003), 187–208.  Google Scholar

[12]

R. E. Mortensen, Maximum-likelihood recursive nonlinear filtering, J. Optim. Theory Appl., 2 (1968), 386-394.  doi: 10.1007/BF00925744.  Google Scholar

[13]

G. Nakamura and R. Potthast, Inverse Modeling, IOP Expanding Physics. IOP Publishing, Bristol, 2015. doi: 10.1088/978-0-7503-1218-9.  Google Scholar

[14]

L. C. G. Rogers, Least-action Filtering, preprint, 2013, arXiv: 1301.5157. Google Scholar

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A. Sage, Optimum Systems Control, Prentice-Hall, Englewood Cliffs, NJ, 1968.  Google Scholar

[16]

E. D. Sontag, Mathematical Control Theory. Deterministic Finite-Dimensional Systems, 2$^{nd}$ edition, Texts in Applied Mathematics, 6, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0577-7.  Google Scholar

[17]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2$^{nd}$ edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[18]

Y. Tremolet, Accounting for an imperfect model in 4d-var, Quarterly Journal of the Royal Meteorological Society, 132 (2006), 2483-2504.   Google Scholar

[19]

J. Yong and X. Y. Zhou, Stochastic Controls, Hamiltonian Systems, and HJB Equations, vol. 43 of Applications of Mathematics, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

[20]

O. Zeitouni and A. Dembo, A maximum a posteriori estimator for trajectories of diffusion processes, Stochastics, 20 (1987), 221-246.  doi: 10.1080/17442508708833444.  Google Scholar

[21]

O. Zeitouni and A. Dembo, An existence theorem and some properties of maximum a posteriori estimators of trajectories of diffusions, Stochastics, 23 (1988), 197-218.  doi: 10.1080/17442508808833490.  Google Scholar

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O. Zeitouni and A. Dembo, A change of variables formula for Stratonovich integrals and existence of solutions for two-point stochastic boundary value problems, Probab. Theory Related Fields, 84 (1990), 411-425.  doi: 10.1007/BF01197893.  Google Scholar

show all references

References:
[1]

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[2]

F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, vol. 264 of Graduate Texts in Mathematics, Springer, London, 2013. doi: 10.1007/978-1-4471-4820-3.  Google Scholar

[3]

J. Derber, A variational continuous assimilation technique, Monthly Weather Review, 117 (1989), 2437-2446.   Google Scholar

[4]

G. Evensen, Data Assimilation. The Ensemble Kalman Filter, 2$^{nd}$ edition, Springer-Verlag, New York, 2009. doi: 10.1007/978-3-642-03711-5.  Google Scholar

[5]

W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, 2$^{nd}$ edition, Stochastic Modelling and Applied Probability, 25. Springer, New York, 2006.  Google Scholar

[6]

H. Frankowska and A. Ochal, On singularities of value function for Bolza optimal control problem, J. Math. Anal. Appl., 306 (2005), 714-729.  doi: 10.1016/j.jmaa.2004.10.003.  Google Scholar

[7] P. K. Friz and N. B. Victoir, Multidimensional Stochastic Processes as Rough Paths, Theory and applications. Cambridge Studies in Advanced Mathematics, 120. Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9780511845079.  Google Scholar
[8]

O. Hijab, Asymptotic Bayesian estimation of a first order equation with small diffusion, Ann. Probab., 12 (1984), 890-902.   Google Scholar

[9] A. H. Jazwinski, Stochastic Processes and Filtering Theory, vol. 64 of Mathematics in Science and Engineering, Academic Press, 1970.   Google Scholar
[10] E. Kalnay, Atmospheric Modeling, Data Assimilation and Predictability, 1$^{nd}$ edition, Cambridge University Press, 2001.   Google Scholar
[11]

A. Krener, The convergence of the minimum energy estimator, In New Trends in Nonlinear Dynamics and Control and their Applications, Lect. Notes Control Inf. Sci., Springer, Berlin, 295 (2003), 187–208.  Google Scholar

[12]

R. E. Mortensen, Maximum-likelihood recursive nonlinear filtering, J. Optim. Theory Appl., 2 (1968), 386-394.  doi: 10.1007/BF00925744.  Google Scholar

[13]

G. Nakamura and R. Potthast, Inverse Modeling, IOP Expanding Physics. IOP Publishing, Bristol, 2015. doi: 10.1088/978-0-7503-1218-9.  Google Scholar

[14]

L. C. G. Rogers, Least-action Filtering, preprint, 2013, arXiv: 1301.5157. Google Scholar

[15]

A. Sage, Optimum Systems Control, Prentice-Hall, Englewood Cliffs, NJ, 1968.  Google Scholar

[16]

E. D. Sontag, Mathematical Control Theory. Deterministic Finite-Dimensional Systems, 2$^{nd}$ edition, Texts in Applied Mathematics, 6, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0577-7.  Google Scholar

[17]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2$^{nd}$ edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[18]

Y. Tremolet, Accounting for an imperfect model in 4d-var, Quarterly Journal of the Royal Meteorological Society, 132 (2006), 2483-2504.   Google Scholar

[19]

J. Yong and X. Y. Zhou, Stochastic Controls, Hamiltonian Systems, and HJB Equations, vol. 43 of Applications of Mathematics, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

[20]

O. Zeitouni and A. Dembo, A maximum a posteriori estimator for trajectories of diffusion processes, Stochastics, 20 (1987), 221-246.  doi: 10.1080/17442508708833444.  Google Scholar

[21]

O. Zeitouni and A. Dembo, An existence theorem and some properties of maximum a posteriori estimators of trajectories of diffusions, Stochastics, 23 (1988), 197-218.  doi: 10.1080/17442508808833490.  Google Scholar

[22]

O. Zeitouni and A. Dembo, A change of variables formula for Stratonovich integrals and existence of solutions for two-point stochastic boundary value problems, Probab. Theory Related Fields, 84 (1990), 411-425.  doi: 10.1007/BF01197893.  Google Scholar

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