倒向随机系统的线性二次混合最优控制

Mixed Optimal Control of Linear Backward Stochastic Systems with Quadratic Costs

倒向随机系统的线性二次最优控制问题的状态系统是具有两个控制器的倒向随机微分方程:一个为确定的控制器,另一个为随机控制器.在适当的假设下,通过凸分析技术可以证明最优控制存在且唯一.利用Ito公式和对偶计算获得了最优控制的随机Hamiltion系统的对偶表示.随机Hamiltion系统是由状态方程、对耦方程和最优控制的对偶表示构成完全耦合的平均场类型的正倒向随机微分方程.

The state system of linear mixed quadratic optimal control problem for backward stochastic systems is a backward stochastic differential equation with two controllers: one is a determined controller and the other is a stochastic controller. Under suitable assumptions, the existence and uniqueness of optimal control is proved by the convex analysis technology. The dual representation of stochastic Hamiltonian systems with optimal control is obtained by using Ito formula and dual computation. Here the stochastic Hamiltion system is a fully-coupled forward-backward stochastic differential equation of mean-field type, which is composed of state equation, dual equation and dual representation of optimal control.