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Communication Dans Un Congrès Année : 2016

About minimal time crisis problems and population dynamics examples

Résumé

We study an optimal control problem where the cost functional to be minimized represents the so- called time of crisis, i.e. the time spent by a trajectory solution of a control system outside a given set K. Such a problematic finds applications in several biological models and in population dynamics (e.g. chemostat model and Lotkta-Volterra systems): typically, several species compete for the same substrate, and the selection of the species of interest requires to find a control strategy that may leave and enter the crisis domain K a number of time that increases with the time interval. One important feature of the time crisis function is that it can be expressed using the characteristic function of K that is discontinuous preventing the use of the standard Maximum Principle. We provide an approximation scheme of the problem based on the Moreau-Yosida approximation of the indicator function of K and prove the the convergence of an optimal sequence for the approximated problem to an optimal solution of the original problem when the regularization parameter goes to zero. Finally, we discuss an example illustrating the convergence property and we provide sub-optimal controls based on a periodic strategy for a biological system arising in the field of bioprocesses. Work in collaboration with T. Bayen.
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Dates et versions

hal-01270378 , version 1 (07-02-2016)

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  • HAL Id : hal-01270378 , version 1

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Alain Rapaport. About minimal time crisis problems and population dynamics examples. International Workshop on Decision Models and Population Management, Feb 2016, Paris, France. ⟨hal-01270378⟩
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