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In this article, we apply an impressive method for solving linear optimal control problem based on cubic B-spline quasi-interpolation. Hamilton-Jacobi equation are applied to linear optimal control problem convert to systems of first-order equations. The main idea of our scheme is approximation derivative with cubic B-spline quasi-interpolation. This method is straightforward, without restrictive assumptions.The results of scheme are made in pleasant agreement with analytic solutions. The accuracy of the proposed method is demonstrated by absolute error. Our scheme is simple to implement because its algorithm is easy and it's one of the advantages of the proposed method.

Control and synchronization of chaotic dynamical systems is a key issue in engineering that has numerous applications in the applied sciences. In this research, single input finite-time sliding mode (FTSMC) control algorithms are developed to synchronize and stabilize a class of three-dimensional non-integer order systems where input saturation is present. Using the non-integer version of the Lyapunov stability theory (LST) and the dynamic-free idea, techniques are devised to suppress the improper behavior of the aforementioned fractional-order (FO) chaotic systems without unpleasant chattering phenomena. The proposed FTSMC approach can be utilized to stabilize and synchronize systems that include model uncertainty, external disturbances, and input saturation. The developed single input techniques have the benefits of being model-free, robust to uncertainty, user-friendly, and establishing equilibrium in a finite amount of time. In addition, the efficacy and applicability of the FTSMC approaches are shown by synchronizing two different industrial FO chaotic systems and chaos suppressing of the PMSM chaotic system utilizing these methods.