Analysis via integral equations of an identification problem for delay differential equations

Abstract

We address the problem of determining the initial function ψ(t), for εt [-t, 0], given the solution y(t) = y(; t) of the linear delay differential equation y (t) - A(t)y(t) - B(t)y(t - t) = f(t), t ε [0, T], for which y(t) = (t), t [-t, 0]. The function ψ (t) is approximated by the function ψ (t) that minimizes a certain parameter-dependent quadratic functional. The optimal function ψ * (t) is shown to satisfy a Fredholm integral equation, and the role of a regularization parameter is transparent from the form of this equation. (There is a related integral equation for ψ(t).) The convergence properties of an iterative method for finding ψ(t), using an iteration that is based upon the delay equation for y(t) and a corresponding adjoint equation, are established by considering an iteration for the solution of the Fredholm integral equation. © 2004 Rocky Mountain Mathematics Consortium.