PITFALLS IN PARAMETER-ESTIMATION FOR DELAY-DIFFERENTIAL EQUATIONS

Given a set of data {U(gamma(i)) approximate to u(gamma(i);p)} corres ponding to the delay differential equation u'(t;p) = f(t,u(t;p),u(alph a(t;p);p);p) for t greater than or equal to t(0)(p), u(t;p) = Psi(t;p) for t less than or equal to t(0)(p), the basic problem addressed here is that of calculating the vector p epsilon R(n). (We also consider neutral differential equations.) Most approaches to parameter estimati on calculate p by minimizing a suitable objective function that is as sumed by the minimization algorithm to be sufficiently smooth. In this paper, we use the derivative discontinuity tracking theory for delay differential equations to analyze how jumps can arise in the derivativ es of a natural objective function. These jumps can occur when estimat ing parameters in lag functions and estimating the position of the ini tial point, and as such are not expected to occur in parameter estimat ion problems for ordinary differential equations.