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| We consider optimal control problems over a fixed interval for multi-input bilinear dynamical systems with both linear and quadratic objectives on the controls in the presence of control constraints. Problems of this type arise as mathematical models for cancer chemotherapy over an a priori specified fixed therapy horizon. Conditions are given that allow to embed extremals (controlled trajectories that satisfy the conditions of the Pontryagin maximum principle) into a field of broken extremals leading to easily verifiable sufficient conditions for strong local optimality. For a linear objective, flows of extremal bang-bang trajectories arise and a simple algorithm will be formulated that allows us to determine their local optimality. For a quadratic objective, sufficient conditions are based on the existence of a bounded solution to a matrix Riccati differential equation will be formulated. Upper and lower bounds on the solution will be formulated that for some systems allow us to guarantee the existence of such a solution. The theory will be illustrated using a 3-compartment model for multi-drug cancer chemotherapy with cytotoxic and cytostatic agents. |
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