The solvability of a class of forward-backward stochastic differential differential equations(SDEs for short)over an arbitrarily prescribed time duration is studied. The authors design a stochastic relaxed control pro...The solvability of a class of forward-backward stochastic differential differential equations(SDEs for short)over an arbitrarily prescribed time duration is studied. The authors design a stochastic relaxed control problem, with both drift and diffusion all being controlled, so that the solvability problem is converted to a problem of finding the nodal set of the viscosity solution to a certain Hamilton-Jacobi-Bellman equation.This method overcomes the fatal difficulty encountered in the traditional contraction mapping approach to the existence theorem of such SDEs.展开更多
This paper studies a stochastic linear quadratic optimal control problem (LQ problem, for short), for which the coefficients are allowed to be random and the cost functional is allowed to have a negative weight on the...This paper studies a stochastic linear quadratic optimal control problem (LQ problem, for short), for which the coefficients are allowed to be random and the cost functional is allowed to have a negative weight on the square of the control variable. The authors introduce the stochastic Riccati equation for the LQ problem. This is a backward SDE with a complicated nonlinearity and a singularity. The local solvability of such a backward SDE is established, which by no means is obvious. For the case of deterministic coefficients, some further discussions on the Riccati equations have been carried out. Finally, an illustrative example is presented.展开更多
Consider the wave equation with distributed controls supported on a subdomain, calledcontrol subdomain, which is allowed to be variant in time. For any prescribed time duration,the authors work out a scheme for changi...Consider the wave equation with distributed controls supported on a subdomain, calledcontrol subdomain, which is allowed to be variant in time. For any prescribed time duration,the authors work out a scheme for changing the control subdomain such that the wave equationis exactly controllable on this time duration, where the control subdomain at any time is allowedto have arbitrarily small measure and relatively simple shape.展开更多
文摘The solvability of a class of forward-backward stochastic differential differential equations(SDEs for short)over an arbitrarily prescribed time duration is studied. The authors design a stochastic relaxed control problem, with both drift and diffusion all being controlled, so that the solvability problem is converted to a problem of finding the nodal set of the viscosity solution to a certain Hamilton-Jacobi-Bellman equation.This method overcomes the fatal difficulty encountered in the traditional contraction mapping approach to the existence theorem of such SDEs.
基金the National Natural Science Foundation of China!(No.7979D130), theNational Distinguished Youth Science Foundation of China (N
文摘This paper studies a stochastic linear quadratic optimal control problem (LQ problem, for short), for which the coefficients are allowed to be random and the cost functional is allowed to have a negative weight on the square of the control variable. The authors introduce the stochastic Riccati equation for the LQ problem. This is a backward SDE with a complicated nonlinearity and a singularity. The local solvability of such a backward SDE is established, which by no means is obvious. For the case of deterministic coefficients, some further discussions on the Riccati equations have been carried out. Finally, an illustrative example is presented.
文摘Consider the wave equation with distributed controls supported on a subdomain, calledcontrol subdomain, which is allowed to be variant in time. For any prescribed time duration,the authors work out a scheme for changing the control subdomain such that the wave equationis exactly controllable on this time duration, where the control subdomain at any time is allowedto have arbitrarily small measure and relatively simple shape.
