State-constrained control based on linearization of the Hamilton-Jacobi-Bellman equation

For continuous time, state constrained, stochastic control problems a method based on optimization is presented. The method applies to systems where the control signal and the disturbance both enters affinely, and it has one main tuning parameter, which determines the control activity. If the disturbance covariance is unknown, it can also be used as a tuning parameter (matrix) to adjust the control directions in an intuitive way. Optimal control problems for this type of systems result in Hamilton Jacobi Bellman (HJB) equations that are problematic to solve because of nonlinearity and infinite boundary conditions. However, by applying a logarithmic transformation we show how and when the HJB equation can be transformed into a linear eigenvalue problem for which there are sometimes analytical solutions and if not, it can readily be solved with standard numerical methods. Sufficient and necessary conditions for when the method can be applied are derived, and their physical interpretation is discussed. A MIMO buffer control problem is used as an illustration.