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May  2009, 11(3): 691-715. doi: 10.3934/dcdsb.2009.11.691

On the optimality of singular controls for a class of mathematical models for tumor anti-angiogenesis

Urszula Ledzewicz 1, and  Heinz Schättler 2,

1. 

Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL 62026
2. 

Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Missouri, 63130-4899

Received  March 2008 Revised  June 2008 Published  March 2009

Anti-angiogenesis is a novel cancer treatment that targets the vasculature of a growing tumor. In this paper a metasystem is formulated and analyzed that describes the dynamics of the primary tumor volume and its vascular support under anti-angiogenic treatment. The system is based on a biologically validated model by Hahnfeldt et al. and encompasses several versions of this model considered in the literature. The problem how to schedule an a priori given amount of angiogenic inhibitors in order to achieve the maximum tumor reduction possible is formulated as an optimal control problem with the dosage of inhibitors playing the role of the control. It is investigated how properties of the functions defining the growth of the tumor and the vasculature in the general system affect the qualitative structure of the solution of the problem. In particular, the presence and optimality of singular controls is determined for various special cases. If optimal, singular arcs are the central part of a regular synthesis of optimal trajectories providing a full solution to the problem. Two specific examples of a regular synthesis including optimal singular arcs are given.
Keywords: maximum principle, optimal control, cancer treatment, regular synthesis, singular control, Lie-bracket., anti-angiogenesis.
Mathematics Subject Classification: Primary: 49J15; Secondary: 93C5.
Citation: Urszula Ledzewicz, Heinz Schättler. On the optimality of singular controls for a class of mathematical models for tumor anti-angiogenesis. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 691-715. doi: 10.3934/dcdsb.2009.11.691
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