This paper provides the solution of optimal filtering problems for the broad class of continuous linear systems with discrete and continuous measurements with time-varying sampling intervals and time-varying measurement delays caused by communication network, human in the loop and a priori unknown triggering of data acquisition. Using the duality principle, the dual control problem with delays in actuation is then solved. The solution of filtering and control problems is obtained using the integral model of measurements in the form of Ito-Volterra integrals with discontinuous measures.