New method to design nonlinear control and convergence analysis of nonlinear random initial‐time real‐order control systems

Abstract Extending beyond asymptotic stability dynamics of equilibrium points of nonlinear real‐order systems remains a long‐standing open problem because it is not known how to ensure asymptotic stability for some non‐trivial bounded solutions. The design of controllers for nonlinear random initial‐time real‐order systems is known to give adequate conclusions in the absence of factual mathematical theory. We design a form of nonlinear control law strategy and introduce a novel concept to dynamic asymptotic stability that enables a deeper notion of convergence of non‐trivial bounded solutions of such real‐order systems. A governing dynamic equation associated with an external order that uses the knowledge of a given system and control function has been introduced. It is shown that under implementation of a form of nonlinear control law when the coefficient matrix of a dynamic equation satisfy matrix inequality obtained, the non‐trivial bounded solutions are dynamic asymptotically stable and dynamic Mittag‐Leffler asymptotically stable. We give three examples that include a practical model and numerical simulation demonstrations to illustrate reasonable evidence that the proposed method is effective.