Most recent studies have been based on the application of linear quadratic regulator control to earthquake-excited structures. In linear quadratic regulator control problems, the objective function is defined as the integral of a quadratic expression in the control interval with respect to structural states and control vectors, and the optimal regulator can be derived using Pontryagin's maximum principle or Bellman's method of dynamic programming. In traditional linear quadratic regulator control problems, the Riccati equation is obtained without considering the earthquake excitation term. To optimize control and satisfy the optimality condition, in this study, we propose a new closed/open-loop control strategy for structures under earthquake excitation. We derive an analytical solution to a linear regulator problem for structural control without neglecting unknown disturbances. The optimal regulator depends on both the state and disturbances. The solution for this closed/open-loop control requires the knowledge of the earthquake in the control interval, which is approximated based on the real-time prediction of near-future earthquake excitation using the Kalman filtering technique. Earthquake excitation is modeled as an autoregressive process. The prediction algorithm can predict seismic excitation in the near future with high accuracy, although it lacks prediction accuracy for more distant future events. Considering the measurement difficulty of all state variables, especially for some high-order systems, the proposed control strategy only requires the measurement of a partial state. In the calculation of a state transition matrix, which is required to solve a differential equation, large rounding errors may occur when the time-step size is excessively small. To overcome this limitation, we introduce a precise integration algorithm to solve the differential equation. This algorithm is always numerically stable and yields very high precision solutions for numerical integration problems. To demonstrate the effectiveness of the proposed control strategy, we investigated the undamped vibration of a three-story building subjected to horizontal seismic forces. We assumed that the columns of the building are massless and that the mass of the structure is concentrated at floor levels. We implemented control using actuators exerting forces on each story. We also assumed that floor velocities can be measured in real time by sensors installed in every story unit. We used the NS component of the 1940 El Centro earthquake ground acceleration record as the excitation source and performed calculations for its entire duration. We modeled the columns of the building as linear elastic springs and assumed the response mitigation effect of the actuators to be sufficient for the building to behave in a linear elastic manner during earthquake excitation. We did not consider the soil-structure interaction or the dynamic characteristics of the actuators. We investigated the controlled and uncontrolled behavior of the three-story undamped building and compared the relative displacement, velocity, acceleration, and inter-story displacement responses. Our numerical simulation results show that the proposed closed/open-loop sub-optimal output feedback control strategy can significantly reduce structural earthquake responses.