Sufficient conditions for strong local optimality in optimal control problems with $L_{2}$-type objectives and control constraints

Heinz Schättler; Urszula Ledzewicz; Helmut Maurer

doi:10.3934/dcdsb.2014.19.2657

Article Contents

2014, Volume 19, Issue 8: 2657-2679. Doi: 10.3934/dcdsb.2014.19.2657

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Sufficient conditions for strong local optimality in optimal control problems with $L_{2}$-type objectives and control constraints

1.
Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Missouri, 63130-4899
2.
Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Illinois, 62026-1653
3.
Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms Universität Münster, D-48149 Münster

Received: November 2013

Revised: January 2014

Published: October 2014

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Abstract

We consider the optimal control problem of minimizing an objective function that is quadratic in the control over a fixed interval for a multi-input bilinear dynamical system in the presence of control constraints. Such models are motivated by and applied to mathematical models for cancer chemotherapy over an a priori specified fixed therapy horizon. The necessary conditions for optimality of the Pontryagin maximum principle are easily evaluated and give a functional description of optimal controls as continuous functions of states and multipliers. However, there is no a priori guarantee that a numerically computed extremal controlled trajectory is locally optimal. In this paper, we formulate sufficient conditions for strong local optimality that are based on the existence of a bounded solution to a matrix Riccati differential equation. The theory is applied to a $3$-compartment model for multi-drug cancer chemotherapy with cytotoxic and cytostatic agents. The numerical results are compared with those for a corresponding optimal control problem when the objective is taken linear in the controls.

Keywords:

Optimal control,
sufficient conditions for local optimality,
Riccati equation,
cancer chemotherapy.

Mathematics Subject Classification: Primary: 49K15; Secondary: 49L99, 93C15.

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