  In this paper, we consider an infinite horizon Linear-Quadratic-Gaussian
control problem with controlled and costly measurements. A control strategy and
a measurement strategy are co-designed to optimize the trade-off among control
performance, actuating costs, and measurement costs. We address the co-design
and co-optimization problem by establishing a dynamic programming equation with
controlled lookahead. By leveraging the dynamic programming equation, we fully
characterize the optimal control strategy and the measurement strategy
analytically. The optimal control is linear in the state estimate that depends
on the measurement strategy. We prove that the optimal measurement strategy is
independent of the measured state and is periodic. And the optimal period
length is determined by the cost of measurements and system parameters. We
demonstrate the potential application of the co-design and co-optimization
problem in an optimal self-triggered control paradigm. Two examples are
provided to show the effectiveness of the optimal measurement strategy in
reducing the overhead of measurements while keeping the system performance.
