SIAM Journal on Optimization, Volume 30, Issue 1, Page 585-603, January 2020.
In this paper, we analyze optimal control problems governed by semilinear parabolic equations. Box constraints for the controls are imposed, and the cost functional involves the state and possibly a sparsity-promoting term, but not a Tikhonov regularization term. Unlike finite dimensional optimization or control problems involving Tikhonov regularization, second order sufficient optimality conditions for the control problems we deal with must be imposed in a cone larger than the one used to obtain necessary conditions. Different extensions of this cone have been proposed in the literature for different kinds of minima: strong or weak minimizers for optimal control problems. After a discussion on these extensions, we propose a new extended cone smaller than those considered until now. We prove that a second order condition based on this new cone is sufficient for a strong local minimum.

中文翻译：

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PDE约束优化中足够二阶条件的临界锥

SIAM优化杂志，第30卷，第1期，第585-603页，2020年1月。
在本文中，我们分析了由半线性抛物方程控制的最优控制问题。施加了对控件的框式约束，并且成本函数涉及状态，并且可能涉及稀疏性促进术语，而不涉及Tikhonov正则化术语。与涉及Tikhonov正则化的有限维优化或控制问题不同，对于我们处理的控制问题，必须在大于用于获得必要条件的圆锥中施加二阶足够的最优条件。在文献中已经针对不同种类的最小值提出了该圆锥的不同扩展：针对最佳控制问题的强或弱最小化器。在讨论了这些扩展之后，我们提出了一个新的扩展锥，它比到目前为止考虑的要小。