We consider an optimal control problem which serves as a mathematical model for several problems in economics and management.The problem is the minimization of a continuous constrained functional governed by a linear parabolic diffusion-advection equation controlled in a coefficient in advection part.The additional constraint is non-negativity of a solution of state equation.We construct and analyze several mesh schemes approximating the formulated problem using finite difference methods in space and in time.All these approximations keep the positivity of the solutions to mesh state problem,either unconditionally or under some additional constraints to mesh steps.This allows us to remove corresponding constraint from the formulation of the discrete problem to simplify its implementation.Based on theoretical estimates and numerical results,we draw conclusions about the quality of the proposed mesh schemes.
Taking into account the fact of global economic integration,this paper improves the RICE model and the MRICES model,and establishes a new integrated assessment model MRICES-2012,which takes Ramsey utility as the standard of fairness.Based on the model,schemes which meet the global emission mitigation targets as well as the interests of developing countries are simulated to assess the international fairness of emission reduction.Therefore,a new feasible scheme is proposed,which can not only reach the Copenhagen Consensus but also ensure interests of every country.Specifically speaking,the US and Japan cut emissions 80% and 70% respectively by 2050 relative to the 1990 level;the EU and other developed countries cut 80% by 2050 relative to 1990 level;high human development countries cut 50% by 2050 relative to 1990 level;all above-mentioned countries start emission reduction from 2020 and keep emission on 2050 level by 2100;China begins emission reduction from 2030 and cuts emission 15% by 2050 and 25% by 2100 relative to 2005 level;medium human development countries keep emission on 2020 level by 2100;low human development countries do not take part in reduction on emission intensity and global emission.
