Optimal control problem with partial derivative equation(PDE) constraint is a numericalwise difficult problem because the optimality conditions lead to PDEs with mixed types of boundary values. The authors provide a n...Optimal control problem with partial derivative equation(PDE) constraint is a numericalwise difficult problem because the optimality conditions lead to PDEs with mixed types of boundary values. The authors provide a new approach to solve this type of problem by space discretization and transform it into a standard optimal control for a multi-agent system. This resulting problem is formulated from a microscopic perspective while the solution only needs limited the macroscopic measurement due to the approach of Hamilton-Jacobi-Bellman(HJB) equation approximation. For solving the problem, only an HJB equation(a PDE with only terminal boundary condition) needs to be solved, although the dimension of that PDE is increased as a drawback. A pollutant elimination problem is considered as an example and solved by this approach. A numerical method for solving the HJB equation is proposed and a simulation is carried out.展开更多
文摘Optimal control problem with partial derivative equation(PDE) constraint is a numericalwise difficult problem because the optimality conditions lead to PDEs with mixed types of boundary values. The authors provide a new approach to solve this type of problem by space discretization and transform it into a standard optimal control for a multi-agent system. This resulting problem is formulated from a microscopic perspective while the solution only needs limited the macroscopic measurement due to the approach of Hamilton-Jacobi-Bellman(HJB) equation approximation. For solving the problem, only an HJB equation(a PDE with only terminal boundary condition) needs to be solved, although the dimension of that PDE is increased as a drawback. A pollutant elimination problem is considered as an example and solved by this approach. A numerical method for solving the HJB equation is proposed and a simulation is carried out.