We study existence and approximation of optimal controls for systems governed by McKean-Vlasov stochastic differential equations. It is well known in simple examples that in the absence of convexity conditions, the strict control problem has no optimal solution. The compactification of the set of such strict admissible controls leads to measure valued controls called relaxed controls. The space of relaxed controls enjoys nice topological properties. We prove that under pathwise uniqueness of solutions of the state equation, the relaxed state process is continuous with respect to the control variable. This means that the relaxed and strict control problems have the same value function. Moreover, we show, under merely continuity of the coefficients, that an optimal control exists in the space of relaxed controls. Under additional convexity hypothesis, we show that the optimal relaxed control is a strict control. These two results extend known results to general nonlinear MVSDEs, under minimal assumptions on the coefficients.

中文翻译：

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McKean-Vlasov 随机微分方程最优控制的紧化

我们研究了由 McKean-Vlasov 随机微分方程控制的系统的最优控制的存在和近似。在简单的例子中众所周知，在没有凸性条件的情况下，严格控制问题没有最优解。这种严格的允许控制集的紧凑化导致被称为宽松控制的度量值控制。松弛控件的空间具有良好的拓扑特性。我们证明在状态方程解的路径唯一性下，松弛状态过程相对于控制变量是连续的。这意味着宽松和严格的控制问题具有相同的值函数。此外，我们表明，仅在系数的连续性下，最优控制存在于松弛控制空间中。在附加凸性假设下，我们表明最优松弛控制是严格控制。这两个结果在系数的最小假设下将已知结果扩展到一般非线性 MVSDE。