Approximate Minimax Estimation of Functionals of Solutions to the Wave Equation under Nonlinear Observations
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APPROXIMATE MINIMAX ESTIMATION OF FUNCTIONALS OF SOLUTIONS TO THE WAVE EQUATION UNDER NONLINEAR OBSERVATIONS O. A. Kapustian1† and O. G. Nakonechnyi1‡
UDC 517.9
Abstract. The authors consider 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 e > 0 , the existence of the solution of the original problem is proved using the traditional minimax approach. Transition to a homogenized parameter problem allows us to remove the nonlinearity in the observation. The main result of the paper is that the minimax 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. INTRODUCTION Development of modern technologies leads to new efficient algorithms for solving problems of estimation, forecasting, optimization, stability analysis, and analysis of systems that operate under uncertainty and incomplete data. The fundamentals of control theory for systems described by ordinary differential equations or partial differential equations are substantiated in [1–4]. However, problems of minimax estimation for infinite-dimensional systems have been studied insufficiently. A lot of problems also occur in case of generalization of estimation problems to the case of partial differential equations. To this end, the theory of minimax estimation of functionals of solutions to partial differential equations, in particular, parabolic and elliptic equations, was constructed [5, 6]. The methods of minimax estimation theory were used to solve a number of problems of predicting solutions to parabolic equations with rapidly oscillating coefficients based on measurement data, in particular [7]. The studies [8–10] propose and substantiate a procedure for constructing an approximate optimal feedback control (synthesis) for broad classes of distributed processes in microinhomogeneous media, which were studied earlier in [11, 12]. In the general case, it is impossible to find an exact optimal synthesis formula for such problems. However, transition to averaged parameters significantly simplifies the structure of the problem. Based on this approach, an approximate minimax estimate was obtained for the functionals of the solution of the parabolic problem with rapidly oscillating coefficients for nonlinear observations [13]. In the paper, we will consider the problem of minimax estimation of the functional of the solution of the wave equation with rapidly oscillating coefficients. Not the quantity that describes the phenomenon under study is measured, but some value of the solution with the operator that determines the method of measurement is observed. The problem is complicated not only by the presence of rapidly oscillating coefficients, but also by the fact that the observation has a
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