Stochastic optimal bounded control for a system with the Boltz cost function

A single-degree-of-freedom mass-spring system is considered under white-noise excitation. To reduce expected response energy of the system, a bounded control force can be applied and optimal control law is sought for An exact explicit analytical solution, for the Lagrange cost function, is derived for the Hamilton-Jacobi-Bellman (HJB) equation within a certain "outer" domain. This solution provides boundary conditions for numerical simulation of the HJB equation within the remaining "inner" domain. Comparison of optimal control law, with a previously obtained one for the Mayer cost function, is made. The analytical solution to the HJB equation for the "combined" Boltz cost function is derived for the outer domain as a linear combination of the solutions for the Mayer and Lagrange cost functions. Sensitivity of the Bellman functional to the above optimal control laws is analyzed.