Fractional Optimal Control of a Riemann-Liouville Tumour Growth Model with Chemotherapy Effect

Abstract

This paper investigates a fractional optimal control problem for tumour growth dynamics governed by a Riemann-Liouville fractional differential equation. The proposed formulation incorporates chemotherapy treatment as a time-dependent control function within a nonlinear memory-dependent state system. A quadratic cost functional is introduced to balance tumour suppression and drug toxicity. The novelty of this work lies in the integration of Riemann-Liouville fractional dynamics with an optimal chemotherapy control strategy in a unified framework. Existence of admissible solutions is established using fixed-point arguments, while necessary optimality conditions are derived via a fractional Pontryagin maximum principle. Numerical simulations are presented to illustrate the effectiveness of the control strategy and to analyze the influence of the fractional order on treatment performance. The results demonstrate that fractional-order dynamics provide a more flexible framework for capturing memory-dependent tumour response under chemotherapy.