Ellipsoidal filtering over discrete-continuous observations with a vector measure (Q1285445)

The problem of estimation of a nonobservable state of the finite-dimensional dynamic system \[ \dot x(t)= a(t)+ b(t) x(t)+ e(t),\quad x(t_0)= x_0 \] based on discrete-continuous observations \[ dy_i(t)= (f_i(t), x(t)) du_i(t)+ \xi_i(t)du_i(t),\;i= 1,\dots, m,\;y(t_0-)= y_0 \] is considered. Here \(x(t)\in \mathbb{R}^n\); \(e(t)\in \mathbb{R}^n\) and \(\xi_i(t)\in \mathbb{R}\) are unknown perturbations; \(u_i(t)\) is a nondecreasing impulse control function. The estimation procedure consists of two steps: a) the ellipsoidal filtering equations are written down for the absolutely continuous control functions; b) then in these equations it is allowed that \(\dot u_i(t)\) be a singular distribution. Solutions of the ellipsoidal filtering equations are proved to be optimal both in the case of a unique solution and of nonunique solutions. In the latter case the solutions are additionally optimized by a criterion of minimal volume of the ellipsoid. No numerical examples are given.