Guaranteed recovery of unknown data from indirect noisy observations of their solutions on a finite system of points and intervals

We consider the Cauchy problem for the first-order linear systems of ordinary differential equations with unknown right-hand sides and initial conditions that are supposed to be subjected to some quadratic restrictions.  From indirect noisy observations of their solutions on a finite system of points and intervals,  we obtain the linear guaranteed mean square estimates of linear functionals on unknown data of the above-mentioned problems.  It is established that if the correlation functions of observational errors are not known and belong to special sets, such estimates are expressed via solutions to some boundary value problems for linear systems of impulsive ordinary differential equations.

guaranteed mean square estimate
noisy observations
linear functionals from unknown data
  1. Nakonechnyi O. G.  Minimax Estimates in Systems with Distributed Parameters.  Preprint 79 Acad. Sci. USSR, Inst. Cybernetics, Kyiv (1979).
  2. Kirichenko N. F., Nakonechnyi O. G.  Minimax approach to recursive estimation of states of linear dynamic systems.  Cybernetics. 13 (4), 527--531 (1977).
  3. Kurzhanskii A. B.  Control and Observation under Uncertainties.  Nauka, Moscow  (1977), (in Russian).
  4. Kurzhanski A. B., Valyi I.  Ellipsoidal Calculus for Estimation and Control.  Birkhäuser, Boston (1997).
  5. Krasovskii N. N.  Theory of Motion Control.  Nauka, Moscow  (1968), (in Russian).
  6. Nakonechnyi O. G.  Optimal Control and Estimation for Partial Differential Equations.  Kyiv University, Kyiv (2004).
  7. Podlipenko Yu., Shestopalov Yu.  Guaranteed estimation of solutions to Helmholtz transmission problems with uncertain data from their indirect noisy observations.  Radio Science. 52 (9), 1129--1139 (2017).
  8. Podlipenko Yu., Shestopalov Yu.  Mixed variational approach to finding guaranteed estimates from solutions and right-hand sides of the second-order linear elliptic equations under incomplete data.  Minimax Theory and its Applications. 1 (2), 197--244, (2016).
  9. Shestopalov Y., Podlipenko Y., Nakonechnyi O.  Estimation of Parameters of Boundary Value Problems for Linear Ordinary Differential Equations with Uncertain Data.  Advances in Pure Mathematics. 4 (4), 118--146 (2014).
  10. Zhuk S., Nakonechnii O.  Minimax State Estimates for Abstract Neumann Problems.  Minimax Theory and its Applications. 3 (1), 1--21 (2018).
  11. Bainov D. D., Simeonov P. S.  Impulsive Differential Equations: Asymptotic Properties of the Solutions.  World Scientifc (1995).
  12. Balakrishnan A. V.  Applied Functional Analysis.  Springer--Verlag; 2nd edition (1981).
  13. Badriev I. B., Karchevsky M. M.  Duality methods in applied problems.  Publ. of Kazan State University, Kazan (1987).
  14. Nakonechny O. G., Podlipenko Yu. K.  The minimax approach to the estimation of solutions to first order linear systems of ordinary differential periodic equations with inexact data.  arXiv:1810.07228V1, p. 19 (2018).