Mixed Problem Of Pollutants In Model Of Unary Water Quality

The work focuses on the theoretical study of optimal control theory in Model of Unary Water Quality, The mathematical equation is given by following equations, the initial conditions and the boundary conditionsIn the above equation, ci=ci(x,t)(i=1,2) denotes the concentration of the scalar, ci0(x)(i=1,2). is an initial concentration,f1(c1) and f2(c1,c2) is a given function describing a reaction kinetics, Ex>0 denotes the molecular diffusivity of the scalar along x-axis,Ωis a bounded domain in R3,(?) denotes the normal derivative along the boundary (?)Ω(n denotes the unit normal on the boundary), v=v(x,t) denotes the velocity of water along x-axis. The model's physical ground lies in mixing enhancement concernment in real world, i.e. how to achieve greatest homogeneity at the cost of relative low energy.There are two main difficulties of the existence of for background of nonlinear optimal control system. One is the mathematical model and building aprropriate mathematical model is the key to existence of the solutions.One is the establish requirements to reflect the objective function. Under conditions, proving that ob-jective function is weakly semi-continuous.The work focuses on the theoretical study of optimal control theory in Reaction-Convection-diffusion problem, whose physical ground lies in mixing enhancement concernment in real world, i.e. how to achieve greatest homogeneity at the cost of relative low energy. The physical models have been analyzed:the governing equa-tion is Reaction-Convection-diffusion form, taking the velocity field as our control variable, we will show the existence of an optimal control under relatively weak con-dition on velocity field, and by variational principles and optimal control theory, we prove the existence of an optimal flow.