Chemotherapy and Immunotherapy for Tumors: A Study of Quadratic Optimal Control

Optimal regulation of tumour-associated neutrophils in cancer progression

“…However, its use in order to incorporate the nonlinearity of the problem has produced qualitatively significant results [73,74]. It should be noted that the quadratic controls in the cost function model the adverse effects of using too much drugs, as used in several works [73,[75][76][77][78][79]. An isoperimetric optimal control has been applied to cancer immunotherapy [80][81][82].…”

Multiobjective approach in the treatment of cancer

“…The answer is affirmative since the function f satisfies the standards hypotheses for the existence of a unique solution. Indeed, as shown in [18], the following assumptions are fulfilled by f :…”

“…Proof. According to (3), (4) and [18], we prove that for each control u ∈ U there exists a unique solution (T, E) of the system (1) defined on [0,t f ]. Now, to prove that the optimal control problem admit a solution of (7), it suffices to prove, according to [9], that the following conditions hold: A1-The dynamic field f is continuous, and Lipschitz in its first argument x uniformly bounded with respect to the second u: The continuity of f is obvious, the Lipschitz property is proved by (4) and the boundedness derives from (3) by taking the supremum over u on U sup…”

“…The present work differs from [18], since it provides two features. Firstly, we investigate the optimal control cancer problem, in which the control models the amount of chemotherapy drug, with a linear control objective function, which is more realistic and present more challenges, [5,13,14].…”

“…In a previous work [18], we gave, by means of Pontryagin maximum principle, a characterization of a quadratic optimal control problem subject a dynamical system [12], describing the interaction between cancer cells and effector cells.…”