Optimal control of multiplicative control systems arising from cancer therapy

Abstract

This study deals with ways of curtailing the vapid growth of cancer cell populations. The performance functional that measures the size of the population at the terminal time as well as the control effort is devised. With use of the discrete maximum principle, the Hamiltonian for this problem is determined and the condition for optimal solutions are developed. The optimal strategy is shown to be a bang-bang control. It is shown that the optimal control for this problem must be on the vertices of an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> -dimensional cube contained in the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> -dimensional Euclidean space. An algorithm for obtaining a local minimum of the performance function in an orderly fashion is developed. Application of the algorithm to the design of antitumor drug and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</tex> -irradiation schedule is discussed.