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We are interested in the optimal control problem associated with certainquadratic cost functionals depending on the solution $X=X^\alpha$ of thestochastic mean-field type evolution equation in $\mathbb R^d$$dX_t=b(t,X_t,\mathcal L(X_t),\alpha_t)dt+\sigma(t,X_t,\mathcalL(X_t),\alpha_t)dW_t,$ $X_0\sim \mu$ given, under assumptions that enclose asytem of FitzHugh-Nagumo neuron networks, and where for practical purposes thecontrol $\alpha_t$ is deterministic. To do so, we assume that we are given adrift coefficient that satisfies a one-sided Lipshitz condition, and that thedynamics is subject to a (convex) level set constraint of the form$\pi(X_t)\leq0$. The mathematical treatment we propose follows the lines of therecent monograph of Carmona and Delarue for similar control problems withLipshitz coefficients. After addressing the existence of minimizers via amartingale approach, we show a maximum principle and then numericallyinvestigate a gradient algorithm for the approximation of the optimal control.

Optimal control of mean field equations with monotone coefficients and applications in neuroscience