Optimal control problem for a general reaction–diffusion tumor–immune system with chemotherapy

2021 ◽  
Vol 358 (1) ◽  
pp. 448-473
Author(s):  
Feng Dai ◽  
Bin Liu
Keyword(s):  
Optimal Control ◽  
Immune System ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Reaction Diffusion ◽  
General Reaction

scholarly journals Optimal Control of HIV Dynamic Using Embedding Method

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
H. Zarei ◽  
A. V. Kamyad ◽  
M. H. Farahi
Keyword(s):  
Optimal Control ◽  
Immune System ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Control Function ◽  
Treatment Protocol ◽  
Optimal Solution ◽  
Embedding Method ◽  
Piecewise Constant Control ◽  
The Cost

This present study proposes an optimal control problem, with the final goal of implementing an optimal treatment protocol which could maximize the survival time of patients and minimize the cost of drug utilizing a system of ordinary differential equations which describes the interaction of the immune system with the human immunodeficiency virus (HIV). Optimal control problem transfers into a modified problem in measure space using an embedding method in which the existence of optimal solution is guaranteed by compactness of the space. Then the metamorphosed problem is approximated by a linear programming (LP) problem, and by solving this LP problem a suboptimal piecewise constant control function, which is more practical from the clinical viewpoint, is achieved. The comparison between the immune system dynamics in treated and untreated patients is introduced. Finally, the relationships between the healthy cells and virus are shown.

scholarly journals The Dubovitskii and Milyutin Methodology Applied to an Optimal Control Problem Originating in an Ecological System

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 479
Author(s):  
Aníbal Coronel ◽  
Fernando Huancas ◽  
Esperanza Lozada ◽  
Marko Rojas-Medar
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Cost Function ◽  
Ecological Model ◽  
Control Function ◽  
Semigroup Theory ◽  
Reaction Diffusion ◽  
Functional Responses ◽  
Adjoint State

We research a control problem for an ecological model given by a reaction–diffusion system. The ecological model is given by a nonlinear parabolic PDE system of three equations modelling the interaction of three species by considering the standard Lotka-Volterra assumptions. The optimal control problem consists of the determination of a coefficient such that the population density of predator decreases. We reformulate the control problem as an optimal control problem by introducing an appropriate cost function. Then, we introduce and prove three types of results. A first contribution of the paper is the well-posedness framework of the mathematical model by considering that the interaction of the species is given by a general functional responses. Second, we study the differentiability properties of a cost function. The third result is the existence of optimal solutions, the existence of an adjoint state, and a characterization of the control function. The first result is proved by the application of semigroup theory and the second and third result are proved by the application of Dubovitskii and Milyutin formalism.

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