ABSTRACT In this investigation, we study an optimal control problem for a parabolic‐hyperbolic free boundary problem modeling the growth of a tumor with drug application. In this model, the heterogeneity or different types of tumor cells (caused by mutations and different values of drug and nutrient concentrations inside the tumor) are considered that are included in the model by considering a variable . It is assumed that converting from mutation state to mutation state happens with probability . This model consists of a first‐order hyperbolic equation describing the evolution of tumor cells depending on . It also includes two time‐fractional parabolic equations describing the diffusions of nutrients (e.g., oxygen and glucose) and drug concentrations. In this paper, the concentrations of drug and nutrients on the boundary of the tumor are the control variables, which diffuse inside the tumor and affect the growth of the tumor. A cost function is presented in which the radius of the tumor and the boundary values of drug and nutrients (to limit the side effects of decreasing nutrient and increasing drug) are included. The adjoint equations are presented, and the necessary conditions in terms of adjoint equations are given. The existence and uniqueness of optimal control are also proved. In order to obtain the optimal control values, we have used the direct method. Hence, the problem is discretized using a combination of spectral method and the product trapezoidal approximation for the Caputo fractional order derivative. Then, we have minimized the discretized cost function employing Trust‐region reflective algorithm. Taking into account heterogeneity and moving boundary increases the complexity of the model but provides us with a model that satisfies more properties of real tumors. Despite these details added to the model, we have proved the existence and uniqueness of the optimal control. Finally, some figures are also presented to illustrate the effects of optimal controls and noisy controls on the cost functions and radius of the tumor. Noisy controls are obtained by adding noises generated by normal distributions. The cost function and radius of tumor for noisy and optimal boundary values of nutrient and drug are plotted. It is shown the lowest cost function and radius are for optimal controls when the noisy values satisfy the constraints for the control variables considered in the optimal control problem.
Sakine Esmaili (Mon,) studied this question.