Linearization and optimization of stochastic systems with bounded control

An approach using statistical linearization to an optimization problem for stochastic systems with bounded control is presented. The cost functional to be minimized is the stationary expectation of a quadratic function of the state and control variables. The class of control laws to be considered is a nonlinear function (saturation or bang-bang) of a linear combination of the state variables. Via linearization, the problem is then reduced to a parameter optimization problem. For the class of saturation functions, a property of the linearized gain is used to establish the unique minimum of the cost functional. The solution for the class of the bang-bang control functions is obtained by a technique using a Lagrange multiplier.