热传导方程的时间最优脉冲控制问题的有限元逼近

时间最优控制问题是一类典型的最优控制问题, 受到研究者的广泛关注. 脉冲控制是一种在工程控制中被广泛应用的控制方式. 偏微分方程描述系统的最优控制问题的数值逼近的收敛性为数值求解方法的可行性提供了定性依据. 本文研究热传导方程的时间最优脉冲控制问题的有限元逼近的收敛性. 通过利用投影算子的特性和系统状态的误差估计, 证明了逼近问题的最优时间收敛到原问题的最优时间. 由此进一步利用原问题最优控制的bangbang性证明了最优控制的收敛性.

Finite element approximations of impulsive time optimal control problems for heat equations

The time optimal control problem is a kind of typical optimal control problem, which has attracted the extensive attention of researchers. Impulsive control has been widely used in engineering control. The convergence of numerical approximations of optimal control problems for the systems governed by partial differential equations provides a qualitative basis for the feasibility of the numerical method. In this paper, the convergence of finite element approximation for the impulsive time optimal control problem of heat equations is studied. By making use of the properties of the projection operator and the error estimates of the system state, we prove that the optimal time of the approximation problem converges to the optimal time of the original problem. Furthermore, the convergence of the optimal control is proved by means of the bang-bang property of the optimal control for the original problem.