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Section: New Results
Optimal control of ordinary differential equations
Periodic optimal controls for the Purcell microswimmer
Participant : Pierre Martinon.
We investigate in [31] some geometric and numerical aspects related to optimal control problems for the so-called Purcell Three-link swimmer, in which the cost to minimize represents the energy consumed by the swimmer. More precisely, we focus on the periodic aspect of optimal trajectories and controls. Linearizing the control system along a reference extremal, we estimate the conjugate points, which play a crucial role for the second order optimality conditions. With techniques imported by the sub-Riemannian geometry, we also show that the nilpotent approximation of the system provides a model which is integrable, obtaining explicit expressions in terms of elliptic functions. This approximation allows to compute optimal periodic controls for small deformations of the body. Numerical simulations are presented using Hampath and Bocop codes. A first paper was submitted in october 2015.
Study of optimal health insurance policies
Participant : Pierre Martinon.
In collaboration with the Economy department of Ecole Polytechnique, we analyze the design of an optimal medical insurance contract under ex post moral hazard, i.e., when illness severity cannot be observed by insurers and policyholders may exaggerate their health expenditures. This problem is reformulated in the optimal control framework, and we study the possible existence of deductibles or bunching phenomenons in optimal contracts. A paper will be submitted in early 2016.