Stochastic Optimal Control Problem With Constraints Under Incomplete Information And Its Applications

The study of stochastic optimal control problem began in the mid-1960s,it has been more than half a century of development.Most of the existed literatures are based on complete information struture.Note that most of aforementioned literatures are based on complete information and state constraints are not con-sidered.However,the cost of information acquisition in analyzing engineering systems is prohibitive.We are inevitably faced with the challenge of drawing con-clusions and make decisions under incomplete information.In mathematics,we consider the stochastic optimal control problems under incomplete information.Generally speaking,the framework of incomplete information can be divided into partial information model and partial observable model.The former is obviously a special case of the latter.Partially observable stochastic optimal control prob-lem can be widely used in mathematical finance,signal processing,and various practical engineering problems.In practice,state constraints cannot be ignored either.This is because state constraints often represent the specific goals and requirements of our control system.In this paper,what we study is a representa-tive functional state constraint,that is,integral-type constraint.More common state constraints in practice are equality,inequality constraints,and interval con-straints,etc.These constraints can be regarded as a special case of integral-type constraint.One of the key tools for studying such partially observable stochas-tic control problems is the famous Girsanov’s theorem.This theorem can break the circular dependence between control and observation,so that we can con-tinue to use the calculus of variation.We use Clarke’s generalized gradient and Ekeland’s variational principle to deal with integral constraint.Compared with the gradually mature stochastic control theory,machine learning has only made great progress in recent years and has penetrated into all aspects of scientific computing.From an abstract point of view,what machine learning deals with is the optimization problem under incomplete information.And this optimiza-tion model has a very similar structure to stochastic optimal control problem.The key to training a deep learning model is to use stochastic gradient descent(SGD)to minimize the loss function,and this loss function actually corresponds to the cost functional in the stochastic control problem.We can study stochastic optimal control problems in terms of deep learning methods,which can be used to construct loss functions based on state constraints.In summary,the main research contents of this paper are as follows:·We use the SMP to study some observable forward and backward stochas-tic systems with integral constraints.With the help of Girsanov’s theorem,convex variation,duality technique,Clarke’s generalized gradient and Eke-land’s variational principle,we can prove an SMP,that is,the necessary conditions for optimal control.In the SMP,it also contains the correspond-ing transversality condition.Finally,two examples are used to explain the theoretical results.·We use state constraints to construct a loss function,and design a deep learning algorithm to solve stochastic optimal control problem.Because there is a little numerical methods to soling partially observable forward-backward stochastic systems with integral constraints,Thus,the results in this chapter is also a preliminary attempt to study the numerical algorithm in Chapter 2.The main method is to assume that the HJB equation has a classic solution,and then perform appropriate transformations to transform it into FBSDE.A deep neural network is designed through the FBSDE to approximate the optimal control.The deep learning algorithm in this chapter can solve a large class of stochastic optimal control problems.