<p>In this paper, a new hybrid algorithm for numerically solving ordinary differential equations based on the k-step methods and linear robust optimal control approach is presented. The main idea of this paper is to design a sequence of optimal adaptive step sizes for k-step methods in such a way as to guarantee system stability, the absolute stability of the numerical integration, and optimal global error control under a given tolerance level. For this purpose, a new global error dynamics for numerical integration methods and an optimal adaptive global error control problem are modeled by combining two k-step methods with different orders. Also, two nonlinear constrained optimization problems are designed in such a way that the robustness of the system, the sliding behavior of the step ratio with minimum deviation, the minimization of the global error, and the maintaining of the global error below the tolerance level are guaranteed without a significant reduction in the step length. In addition, the optimal control law is obtained using the robust and optimal eigenvalue assignment approach. Also, a new system and absolute stability region (SAS region) is designed based on a proven proposition. If all the eigenvalues of the closed-loop matrix are assigned in this region, the stability of the control system and the absolute stability of the implemented integration methods will be guaranteed. Finally, three examples are simulated to demonstrate the advantages of the proposed method.</p>