POD Method For Optimal Control Problems Governed By Second-order Hyperbolic Equations

The optimal control problem is widely used in many scientific and engineering fields,such as oil exploitation,temperature control,air pollution control and so on.However,due to the complexity of its physical background,its computational load and complexity are very high,and sometimes a long time of calculation will also make its results meaningless.Therefore,it is an important research field to simplify the calculation and reduce the memory requirement while maintaining the calculation accuracy.The proper orthogonal decomposition(POD)method is an efficient calculation method,which can be combined with many numerical methods.At present,it has been successfully applied to dimensionality reduction modeling of complex systems.This paper combines the POD method with the finite element method and applies it to the second-order hyperbolic optimal control problem with practical physical background.For the optimal distributed control problem of the second-order hyperbolic equation,its optimality system is first derived.On this basis,the C-N method is used in time and the finite element method is used in space to construct a fully discrete C-N finite element scheme.By using the classical optimal control theory,the priori error estimation of the finite element approximation is analyzed.Secondly,the POD method is combined with the finite element method,and the POD spaces and POD bases are constructed by selecting the numerical results of some time levels of the finite element scheme as the snapshots,and the reduceddimension C-N finite element scheme based on the POD method is established.By introducing auxiliary variables,the priori error estimates of the state,costate and control variables are analyzed,providing scientific theoretical basis for practical application.Finally,the algorithm implementation of the POD method is briefly described,and numerical solutions are constructed for the unconstrained and constrained control cases,respectively.For the optimal boundary control problem of second-order hyperbolic equation,a fully discrete C-N finite element scheme is constructed and its priori error estimate is analyzed.On this basis,the POD method is combined with the finite element method to establish a reduced-dimension C-N finite element scheme based on the POD method,and its priori error estimate is analyzed.Finally,a numerical example is constructed for numerical solution.It is proved that the POD method is feasible and effective for solving the second-order hyperbolic optimal control problem,and it can improve the calculation speed and reduce the memory requirement.In practical applications,the number of snapshot levels or POD bases can also be selected based on the acceptable degree of error,so as to achieve the goal of reasonable and effective utilization of resources.