A second order scheme for solving optimization-constrained differential equations with discontinuities

A numerical method for the resolution of a system of ordinary differential equations coupled with a mixed constrained minimization problem is presented. This coupling induces discontinuities of some time-dependent variables when inequality constraints are activated or deactivated. The ordinary differential equations are discretized in time and combined with the first order optimality conditions of the optimization problem. We use a second order multistep method based on a predictorcorrector Adams scheme to detect the discontinuities by extrapolation of the trajectories. Optimization features, namely a sensitivity analysis, are exploited to compute the derivatives of the optimization variables and track the discontinuity points. The main difficulty consists in the impossibility of defining an explicit event function to characterize the activation or deactivation of a constraint. The order of convergence of our method is proved when inequality constraints are activated and numerical results for atmospheric organic particles are presented.