Necessary conditions for the optimality of discontinuous, linear, time-varying, filtering problems (Q1881896)

This mathematical note examines the problem of optimal time-varying filtering for an optimality criterion defined by an integral functional \(I\) with a discontinuous (actually, not necessarily continuous) integrand \(F\). The integral functional \(I\) has the form: \[ I= \int^{+\infty}_{-\infty} F[\Phi_1(x),\dots, \Phi_i(x),\dots, \Phi_n(x)]\,dx,\tag{1} \] while the linear integral operators \(\Phi_i(x)\) are defined as: \[ \Phi_i(x)= \int^{+\infty}_{-\infty} K(x,t)\,S_i(t)\,dt,\tag{2} \] where \(S_i\in L_p\), \(p\geq 1\), are given functions (input processes, e.g. signals or perturbations) and \(K(x,t)\) is the unknown (possibly discontinuous) kernel. The optimization problem defined by the equations (1)--(2) is solved as a variational problem and the paper obtains a nonlinear (two-dimensional) integral equation.