In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimensional. The cost functional is of a tracking type and contains a linear term in the control variables. We derive second order necessary conditions relying on the concept of alternative costates and quasi-radial critical directions. The appendix provides an example illustrating the applicability of our results.

中文翻译：

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状态约束控制仿射抛物线问题I：一阶和二阶必要最优性条件

在本文中，我们考虑由具有双线性控制状态项的半线性热方程控制并受控制和状态约束的最优控制问题。状态约束是积分类型的，积分是相对于空间变量的。该控件是多维的。成本函数为跟踪类型，在控制变量中包含线性项。我们根据替代肋骨和准径向临界方向的概念得出二阶必要条件。附录提供了一个示例，说明了我们的结果的适用性。