论文标题
在期望约束下的二元性和随机最佳控制问题的近似
Duality and approximation of stochastic optimal control problems under expectation constraints
论文作者
论文摘要
我们考虑在受控过程的某些功能的期望下，在平等和不等式约束下，在平等和不平等限制下进行连续的随机最佳控制问题。在资格条件下，我们表明该问题是双重性的，涉及与约束相关的Lagrange乘法器的优化问题。然后，通过凸分析技术，我们提供了一般存在的结果，并且对双重优化器进行了一些先验估计。我们进一步为初始约束控制问题提供了必要且充分的最佳条件。对于离散的时间约束控制问题，也获得了相同的结果。此外，在额外的规律条件下，证明离散的时间控制问题可能会收敛到连续的时间问题，这可能是收敛速度的。该收敛结果可用于获取数值算法以近似连续的时间控制问题，我们通过两个简单的数值示例来说明这一点。
We consider a continuous time stochastic optimal control problem under both equality and inequality constraints on the expectation of some functionals of the controlled process. Under a qualification condition, we show that the problem is in duality with an optimization problem involving the Lagrange multiplier associated with the constraints. Then by convex analysis techniques, we provide a general existence result and some a priori estimation of the dual optimizers. We further provide a necessary and sufficient optimality condition for the initial constrained control problem. The same results are also obtained for a discrete time constrained control problem. Moreover, under additional regularity conditions, it is proved that the discrete time control problem converges to the continuous time problem, possibly with a convergence rate. This convergence result can be used to obtain numerical algorithms to approximate the continuous time control problem, which we illustrate by two simple numerical examples.
