  In this paper we study numerically solving optimal control problems with
bang-bang control functions. We present a formal Lagrangian approach for
solving the optimal control problem, and address difficulties encountered when
numerically solving the state and adjoint equations by using the immersed
interface method. We note that our numerical approach does not approximate the
discontinuous control function with smooth functions, instead we solve the true
bang-bang optimal control problem. Our approach for solving the optimal control
problem uses an adjoint-based gradient. We use the gradient in our first-order
trust-region method to generate a local minimizing control. We present detailed
numerical results to demonstrate the effectiveness of our method.
