The main concern of this article is addressing a new modified approach to design a nonlinear optimal controller. The modification focuses on proposing a new approximate solution for the Hamilton–Jacobi–Bellman nonlinear partial differential equation. The introduced solution works based on the state-dependent power series expansion presentation of the involved functions in the Hamilton–Jacobi–Bellman partial differential equation. Applying this technique results in releasing a set of free state-dependent functions in the controller structure that can be adjusted to fulfill some special control missions in addition to the optimization objectives. They are formed based on the specific formulation of the candidate Lyapunov function. The proposed approach is exemplified for an intricate biological system, immunogenic tumor-immune cell interaction in the human body, to clarify the mechanism of designing the controller and adjusting the arrays of the free matrices. The closed-loop system by presented optimal state feedback controller meets the predefined optimization objectives without getting feedback from a hard-measurable state. It is achieved by adjusting the aforementioned released functions such that an optimal output feedback controller is obtained. To have some insights into the performance of the system and the effectiveness of the controller, the positiveness of the system’s states is proved and checked numerically by applying the differential transformation method to the system’s differential equations. Finally, to highlight the abilities of the proposed approach from different aspects, some simulations are carried out.
Existence results for an impulsive abstract partial differential equation with state-dependent delay
