On the characterization of properly relaxed delayed controls (Q1359572)

A new characterization of the set of relaxed controls for a system with time-dependent delays is given. It is shown, among other things, that a function \(\sigma\) belongs to this set if and only if \[ \sum^{N- 1}_{p= 0} \int f_p(\omega)\sigma(t+ ph)(d\omega)\geq 0, \] where \(f_p:\Omega^{k+ 1}\to \mathbb{R}\) are bounded upper semicontinuous functions such that \[ \sum^{N- 1}_{p= 0} f_p(x_p, x_{p- i_1},\dots, x_{p- i_k})\geq 0 \] for all \(x= (x_{-i_k}, x_{-i_k+ 1},\dots, x_0,\dots, x_{N- 1})\in \Omega^{i_k+ N}\), where \(\Omega\) is a compact metric space and \(k\) is the number of delay terms in the problem.