中文摘要：

研究在Dirichlet边界条件下抛物型方程的最优化问题及其弱近似解。首先给出近似解定义，利用罚函数法和Sobolev空间、变分法、偏微分方程、泛函分析等理论得出最优正则化问题解的存在性，并且以变分不等式的形式给出最优化成立的必要条件，最后构造出一个极小化序列，证明它是一弱极小化序列．从而得到弱近似解。

英文摘要：

Under Dirichlet boundary conditions, an optimal problem and weak approximate solution for a parabolic system are mainly considered. The notion of the approximate solution is given first. By using penalty method, Sobolev space,variational approach, partial differential equations, and function analysis theory, the optimal problem and the existence of the solution of the optimal regularized problem are obtained. The necessary conditions for optimality are established in the form of variational inequalities. Finally, a minimizing sequence is constructed and proved as a weak minimizing sequence. So the weak approximate solution is derived.