论文标题
具有单调系数和应用中应用的平均场方程的最佳控制
Optimal control of mean field equations with monotone coefficients and applications in neuroscience
论文作者
论文摘要
我们对与某些二次成本函数相关的最佳控制问题感兴趣，具体取决于解决方案的$ x = x^α$，$ \ MATHBB r^d $ $ $ dx_t = b（t，x_t，x_t，\ mathcal l（x_t），x_t）dt+x__t，t，x__t，t，x_t，t，x cc， l（x_t），α_T）dw_t，$ $ x_0 \ simμ$，在封闭菲茨胡格 - nagumo神经元网络系统的假设下，在实际目的中，控制$α_t$是确定性的。为此，我们假设为满足单方面的Lipshitz条件的漂移系数，并且该动力学受到$π（x_t）\ leq0 $的（凸）级别设置约束的（凸）级别设置约束。我们提出的数学处理遵循Carmona和Delarue的最新专着的线条，以解决Lipshitz系数的类似控制问题。通过Martingale方法解决了最小化器的存在之后，我们显示了最大原理，然后数字研究梯度算法以近似最佳控制。
We are interested in the optimal control problem associated with certain quadratic cost functionals depending on the solution $X=X^α$ of the stochastic mean-field type evolution equation in $\mathbb R^d$ $dX_t=b(t,X_t,\mathcal L(X_t),α_t)dt+σ(t,X_t,\mathcal L(X_t),α_t)dW_t,$ $X_0\sim μ$ given, under assumptions that enclose a sytem of FitzHugh-Nagumo neuron networks, and where for practical purposes the control $α_t$ 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 $π(X_t)\leq0$. 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.
