Reduced Algorithms For Convection-Diffusion Optimal Control Problems

Convection-diffusion Optimal control problems can be found everywhere in our real life,such as air pollution control,sewage treatment,fluid control problems and so on.It is difficult to find the analytic solutions for these problems.So a numerical method is a good option.The reduced algorithms for the optimal control problems of a two-dimensional unsteady convection-diffusion boundary control and distribution control problems are mainly discussed.The specific contents are as follows:For unsteady two-dimensional convection-diffusion boundary control problem,firstly,the convection-diffusion equations are discretized by upwind finite difference method,and the discrete values at some time are selected as snapshots.Then,the matrix composed of snapshots is decomposed by singular value decomposition,and a set of orthogonal basis functions are obtained.Combining orthogonal basis function with projection method,A reduced algorithm is obtained.In the unconstrained case,applied the discrete system time linear quadratic regulator to the convection-diffusion boundary control problem,the optimal input and output of the boundary control problem are obtained.For two-dimensional convection-diffusion boundary control problem with control constraints,a quadratic programming method with one-step rolling control time domain is adopted in the reduced-order state space model and verified with the improved saturated linear quadratic regulator.Finally,a numerical example is given to verify the efficiency of the reduced-order model of the eigenvalue orthogonal decomposition method.For the distributed control of 2D-unsteady convection-diffusion equation,the optimality conditions are given firstly,and the numerical algorithm is given according to the optimality conditions.An improved upwind scheme is used to discretize the convectiondiffusion equations.This method has the advantages of diagonal dominance and secondorder implicit scheme in space.Then the snapshots can be got by making some simple computation with choosing an appropriate space and time step size.A simplified scheme can be obtained by singular value decomposition of the approximate solution.Reduced algorithm can get by applying the simplified scheme to the algorithm for optimal control problem.Finally,a numerical example is simulated,and the results show the effectiveness of the proposed method.