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首页> 外文会议>IFIP TC 7 conference on system modeling and optimization >On Existence, Uniqueness, and Convergence, of Optimal Control Problems Governed by Parabolic Variational Inequalities
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On Existence, Uniqueness, and Convergence, of Optimal Control Problems Governed by Parabolic Variational Inequalities

机译：由抛物变分不等式控制的最优控制问题的存在性，唯一性和收敛性

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Ⅰ) We consider a system governed by a free boundary problem with Tresca condition on a part of the boundary of a material domain with a source term g through a parabolic variational inequality of the second kind. We prove the existence and uniqueness results to a family of distributed optimal control problems over g for each parameter h ＞ 0, associated to the Newton law (Robin boundary condition), and of another distributed optimal control problem associated to a Dirichlet boundary condition. We generalize for parabolic variational inequalities of the second kind the Mignot's inequality obtained for elliptic variational inequalities (Mignot, J. Funct. Anal., 22 (1976), 130-185), and we obtain the strictly convexity of a quadratic cost functional through the regu-larization method for the non-differentiable term in the parabolic variational inequality for each parameter h. We also prove, when h → +∞, the strong convergence of the optimal controls and states associated to this family of optimal control problems with the Newton law to that of the optimal control problem associated to a Dirichlet boundary condition. Ⅱ) Moreover, if we consider a parabolic obstacle problem as a system governed by a parabolic variational inequalities of the first kind then we can also obtain the same results of Part Ⅰ for the existence, uniqueness and convergence for the corresponding distributed optimal control problems. Ⅲ) If we consider, in the problem given in Part Ⅰ, a flux on a part of the boundary of a material domain as a control variable (Neumann boundary optimal control problem) for a system governed by a parabolic variational inequality of second kind then we can also obtain the existence and uniqueness results for Neumann boundary optimal control problems for each parameter h ＞ 0, but in this case the convergence when h →+∞ is still an open problem.
机译：Ⅰ）我们考虑了一个系统，该系统受第二类抛物线变分不等式的限制，该系统受带有Tresca条件的自由边界问题约束，该边界在材料域边界的一部分上具有源项g。我们证明了对于每个参数h＞ 0，与牛顿定律（罗宾边界条件）相关的g分布的最优控制问题族，以及与Dirichlet边界条件相关的另一个分布式最优控制问题的存在性和唯一性结果。我们将第二种抛物线变分不等式推广为椭圆变分不等式获得的Mignot不等式（Mignot，J.Funct.Anal。，22（1976），130-185），并通过每个参数h的抛物型变分不等式中不可微项的正规化方法。我们还证明，当h→+∞时，与牛顿定律相关的该最优控制问题族的最优控制和状态与与Dirichlet边界条件相关的最优控制问题的强收敛。 Ⅱ）此外，如果将抛物线障碍问题视为由第一类抛物线变分不等式支配的系统，那么对于相应的分布式最优控制问题的存在性，唯一性和收敛性，我们也可以获得与第一部分相同的结果。 Ⅲ）如果考虑在第一部分给出的问题中，由第二类抛物变分不等式控制的系统的材料域边界上的通量作为控制变量（Neumann边界最优控制问题），则我们还可以获得每个参数h＞ 0的Neumann边界最优控制问题的存在性和唯一性结果，但是在这种情况下，当h→+∞时的收敛仍然是一个开放问题。

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