Guaranteed Estimation of Solutions of Boundary Value Problems for Linear Ordinary Differential Equations with Decomposed Boundary Data

This chapter considers the problem of minimax estimation of inner product (a, x(s)), where a is a certain vector and x(s) is an unknown solution at any fixed point s ∈ (0, T) to two-point boundary value problem (BVP) for systems of linear first‐order ordinary differential equations (ODEs) with decomposed boundary conditions at the points t = 0 and t = T. The BVP is supposed to be uniquely solvable. From observations on an interval or from point observations, the authors find general form of the minimax estimates of inner product (a, x(s)) and determine estimation errors. To do this, they reduce the guaranteed estimation problem to a certain optimal control problem of adjoint BVP for which establish its explicit form. Solving this optimal control problem, they obtain uniquely solvable system of ODEs via whose solutions the minimax estimates are expressed.