Two-mode control of Brownian Motion with quadratic loss and switching costs

We consider a simple problem in the optimal control of Brownian Motion. There are two modes of control available, each with its own drift and diffusion coefficients, and switching costs are incurred whenever the control mode is changed. Finally, holding costs are incurred according to a quadratic function of the state of the system, and all costs are continuously discounted. It is shown that there exists an optimal policy involving just two critical numbers, and formulas are given for computation of the critical numbers.