Stochastic optimal control is an important area of research in engineering systems that undergo disturbances such as earthquake excitations and blast waves. Controlling states such as positions of different parts in these systems is critical in situations in which the system has to operate within limited range of its states. The present investigation is concerned with the application of the stochastic optimal control strategy developed by To (2010) and its implementation as well as providing computed results of linear systems under nonstationary random excitations. In the strategy the feedback matrix is designed based on the achievement of the objectives for individual states in the system through the application of the Lyapunov equation of the system. Every diagonal element in the gain or associated gain matrix is related to the corresponding states. The strategy is applied to two two-degree-of-freedom (dof) systems representing buildings under earthquake excitations. Optimally controlled nonstationary random displacements were obtained by the proposed method and presented in this paper. The computed results include the time-dependent elements of the associated gain matrix. Three-dimensional (3D) graphical representations of the optimally controlled largest peaks of mean squares of displacements and velocities against elements of the feedback gain matrix were included. The latter 3D presentations are important for the design engineer who needs to choose elements of the gain matrix in order to achieve a specific objective in certain states of the system.

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