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

中文翻译：

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具有单调系数的平均场方程的最优控制及其在神经科学中的应用

我们对与某些二次成本函数相关的最优控制问题感兴趣，这取决于 $\mathbb R^d$ $dX_t=b(t, X_t,\mathcal L(X_t),\alpha_t)dt+\sigma(t,X_t,\mathcal L(X_t),\alpha_t)dW_t,$ $X_0\sim \mu$ 给定，假设包含 FitzHugh 系​​统-Nagumo 神经元网络，出于实际目的，控制 $\alpha_t$ 是确定性的。为此，我们假设我们得到一个满足单边 Lipshitz 条件的漂移系数，并且动力学受到形式为 $\pi(X_t)\leq0$ 的（凸）水平集约束。我们提出的数学处理方法遵循 Carmona 和 Delarue 最近的专着中关于 Lipshitz 系数的类似控制问题的路线。