Optimal time substitution in a control process (Q2577207)

This paper deals with optimization problems for the following aim functional: \[ \nu(f)= {\int_X g(x) f(x) \mu(dx)\over \int_X f(x) \mu(dx)}, \] where \(X\) is a set with a given probability measure \(\mu\), \(g\) is a given integrable function and \(f\), control function, belongs to a predefined class. By taking the conditional expectation \(E(f|g)x= r(g(x))\), the aim functional can be expressed as \({\int_X g(x) r(g(x))\mu(dx)\over \int_X r(g(x))\mu(dx)}\). With this motivation, the author considered the optimization problems for more general form, \(J(r)= {\int^\infty_{-\infty} tr(t) dF_g(t)\over \int_{-\infty}^\infty w(r(t)) dF_g(t)}\) where \(F_g\) is the distribution function of \(g\). By this functional, control function \(r\) is interpreted as a time substitution. Optimal control functions for certain class and a method for computing them are determined. Relay control functions and related functions are shown to be optimal. Examples illustrate the diversity of the field of applications.