Parameter identification in classes of hereditary systems of neutral type
The authors apply the general identification framework for identifying parameters in initial value problems corresponding to neutral functional differential equations (NFDEs) of the form \[ {d\over dt}(x(t)+ q(t)x(t- \tau(t)))= f(t,x(t), x(t-\sigma(t))),\quad t\in[0, T],\tag{1} \] and initial condition \[ x(t)=\varphi(t),\quad t\in[-r,0].\tag{2} \] The goal is the identification of \(q\), \(\tau\) in (1), and the associated initial function \(\varphi\) in (2). Assuming that the parameters \(q\), \(\tau\) and \(\varphi\) are continuous functions, the parameter space is the set \(\Gamma\equiv C([0,T]; \mathbb{R})\times C([0, T]; \mathbb{R})\times C([-r,0]; \mathbb{R}^n)\). To follow the general identification method, the authors first approximate \(q\), \(\tau\) and \(\varphi\) (in supremum norm) by (finite-dimensional) functions \(q^N\), \(\tau^N\) and \(\varphi^N\). In practice linear spline approximations are used. In Section 2, the authors define several Euler-type approximation schemes for NFDEs of the form (1), using equations with piecewise constant arguments, and show that each scheme satisfies the convergence property of the general method.
neutral functional differential equations
scientific article; zbMATH DE number 1183193