Optimal control problems for linear stochastic continuous time systems are considered, where the time domain is decomposed into a finite set of N disjoint random intervals of the form [t_i, t_i+1); where a complete state observation is taken at each instant . Two optimal control problems termed respectively the (piecewise) time invariant control and time variant control are considered in this framework. Concerning the observation point process, we, first, consider the general situation where the increment intervals are i.i.d.r.v.s with unspecified probabilistic distributions. Next, the problem is specialized so that the increments are exponentially distributed, and we obtain the following results: (a) For the (piecewise) time invariant case, the optimal control is made-up of a sequence of piecewise open loop controls, where the feedback gains are piecewise constant matrices integrated off-line via a sequence of matrix equations having essentially the structure of matrix Lyapunov equation. (b) For the time variant case, we show that this control problem is closely related to that of linear quadratic Gaussian regulation with an exponentially discounted cost. The optimal control is made-up again of a sequence of piecewise open loop controls corresponding, in this case, to linear feedback of the state predictor based on the most recent information on each interval. The feedback gains are time varying matrices obtained from a sequence of algebraic Riccati equations, which are also computed off-line.