Opuscula Math. 37, no. 3 (2017), 403-419
http://dx.doi.org/10.7494/OpMath.2017.37.3.403

Opuscula Mathematica

# Sufficient conditions for optimality for a mathematical model of drug treatment with pharmacodynamics

Maciej Leszczyński
Elżbieta Ratajczyk
Urszula Ledzewicz
Heinz Schättler

Abstract. We consider an optimal control problem for a general mathematical model of drug treatment with a single agent. The control represents the concentration of the agent and its effect (pharmacodynamics) is modelled by a Hill function (i.e., Michaelis-Menten type kinetics). The aim is to minimize a cost functional consisting of a weighted average related to the state of the system (both at the end and during a fixed therapy horizon) and to the total amount of drugs given. The latter is an indirect measure for the side effects of treatment. It is shown that optimal controls are continuous functions of time that change between full or no dose segments with connecting pieces that take values in the interior of the control set. Sufficient conditions for the strong local optimality of an extremal controlled trajectory in terms of the existence of a solution to a piecewise defined Riccati differential equation are given.

Keywords: optimal control, sufficient conditions for optimality, method of characteristics, pharmacodynamic model.

Mathematics Subject Classification: 49K15, 93C15, 92C45.

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• Maciej Leszczyński
• Lodz University of Technology, Institute of Mathematics, 90-924 Lodz, Poland
• Elżbieta Ratajczyk
• Lodz University of Technology, Institute of Mathematics, 90-924 Lodz, Poland
• Urszula Ledzewicz
• Lodz University of Technology, Institute of Mathematics, 90-924 Lodz, Poland
• Southern Illinois University Edwardsville, Department of Mathematics and Statistics, Edwardsville, Il, 62026-1653, USA
• Heinz Schättler
• Washington University, Department of Electrical and Systems Engineering, St. Louis, Mo, 63130, USA
• Communicated by Marek Galewski.
• Revised: 2016-10-10.
• Accepted: 2016-10-10.
• Published online: 2017-01-30.