Optimal bounded response control for a second-order system under a white-noise excitation

A single-degree-of-freedom system is excited by a white-noise random force. The system's response can be reduced by a control force of limited magnitude R, and the problem is to minimize the expected response energy at a given time instant T under this constraint. A "hybrid" solution to the corresponding Hamilton-Jacobi-Bellman (or HJB) equation is obtained for the case of a linear controlled system. Specifically, an exact analytical solution is obtained within a certain outer domain with respect to a "strip" with switching lines, indicating optimality of a "dry-friction," or the simplest version of the "bang-bang" control law within this domain. This explicit solution is matched by a numerical solution within an inner domain, where switching lines are illustrated. In the limiting case of a weak control, or small R, the hybrid solution leads to a simple asymptotically suboptimal "dry-friction" control law, which is well-known for deterministic optimal control problems; more precisely, the difference in expected response energies between cases of optimal and suboptimal control is shown to be proportional to a small parameter Numerical results are presented, which illustrate the optimal control law and evolution of the minimized functional. They are used in particular to evaluate convergence rate to the derived analytical results for the suboptimal weak control case. A special case of a nonlinear controlled system is considered also, one with a rigid barrier at the system's equilibrium position. The resulting vibroimpact system is studied for the case of perfectly elastic impacts/rebounds by using special piecewise-linear transformation of state variables, which reduces the system to the nonimpacting one. The solution to the HJB equation is shown to be valid for the transformed system as well, resulting in the optimal control law for the vibroimpact system.