UDC 517.9
APPROXIMATE MINIMAX ESTIMATION OF FUNCTIONALS OF SOLUTIONS
TO THE WAVE EQUATION UNDER NONLINEAR OBSERVATIONS
Abstract. The paper deals with the problem of minimax estimation of a functional of the solution to the wave equation with rapidly oscillating coefficients.
The observation (output signal) is nonlinear (has the operator of superposition type).
For the small parameter ε > 0, the existence of the solution of original problem is proved using the traditional minimax approach.
Transition to homogenized parameter problem allows us to remove the nonlinearity in the observation.
The main result of the paper is to prove that the estimate of the problem with homogenized parameters is an approximate minimax estimate of the original problem.
Keywords: minimax estimation, wave equation, rapidly oscillating coefficients, homogenized problem, uncertainty, approximate estimate.
FULL TEXT
REFERENCES
- A.G. Nakonechny A.G. Minimax estimation of functionals of solutions of variational equations in Hilbert spaces [in Russian]. Kiev: KSU, 1985. 83 p.
- Krasovsky N.N. Motion control theory [in Russian]. Moscow: Nauka, 1968. 476 p.
- Lyons J.-L. Optimal control of systems described by partial differential equations [Russian translation]. Moscow: Mir, 1972. 414 p.
- Лионс Ж.-Л. Некоторые методы решения нелинейных краевых задач. Москва: Мир, 1972. 588 p.
- Nakonechnyi A.G., Podlipenko Yu.K., Zaitsev Yu.A. Minimax prediction estimation of solutions of initial-boundary-value problems for parabolic equations with discontinuous coefficients based on imperfect data. Cybernetics and Systems Analysis. 2000. Vol. 36, N 6. P. 845–854.
- Podlipenko Y., Shestopalov Y. Mixed variational approach to finding guaranteed estimates for solutions and right-hand sides of the second-order linear elliptic equations under incomplete data. Minimax Theory and its Applications. 2016. Vol. 01, N 2, P. 197–244.
- Kapustyan E.A., Nakonechnyj A.G. The minimax problems of pointwise observation for a parabolic boundary value problem. Journal of Automation and Information Sciences. 2002. Vol. 34, N 5. P. 52–63.
- Kapustyan E.A., Nakonechnyj A.G. Optimal bounded control synthesis for a parabolic boundary-value problem with fast oscillatory coefficients. Journal of Automation and Information Sciences. 1999. Vol. 31, N 12. P. 33–44.
- Kapustyan O.V., Kapustyan O.A., Sukretna A.V. Approximate bounded synthesis for one weakly nonlinear boundary-value problem. Nonlinear Oscillations. 2009. Vol. 12, N 3. P. 297–304.
- Kapustian O.A., Sobchuk V.V. Approximate homogenized synthesis for distributed optimal control problem with superposition type cost functional. Statistics, Optimization and Information Computing. 2018. Vol. 6, N 2. P. 233–239.
- Zhikov V.V., Kozlov S.M., Oleinik O.A. Averaging differential operators [in Russian]. Moscow: Fizmatlit, 1993. 464 p.
- Bakhvalov N.S., Panasenko G.P. Averaging processes in periodic media [in Russian]. Moscow: Mir, 1984. 352 p.
- Kapustyan O.A., Nakonechny O.G. Approximate minimax estimation of functionals from the solution of a parabolic problem with rapidly oscillating coefficients in nonlinear observations. Systems research and information technology. 2019. N 2. P. 94–104.
- Denkiwski Z., Mortola S. Asymptotic behavior of optimal solutions to control problems for systems described by differential inclusions corresponding to partial differential equations. Journal of Optimization Theory and Applications. 1993. Vol. 78. P. 365–391.
