Evolution equations: input functions & stability

演化方程描述了由初始状态和给定输入驱动的动力系统的时间发展。在这个项目中，我们研究了具有以下两个特征的无限维状态空间上的演化方程：我们允许输入算子的相当大的无界性和输入函数的非标准空间。对于这样的空间，标准方法是有限的，一个自然的例子是由本质上有界的函数给出的。我们的设置包含了许多具有边界输入和控制的常见偏微分方程。后者是本研究的主要动机。相关控制问题的最新技术需要相对较好的边界算子和输入函数的空间，例如平方可积函数，这可能很难检查。此外，输入规范的自然选择可能不属于这些类别。当试图将已知结果提升到这些更一般的空间时，人们很快就会面临严重的理论困难，例如，平移算子可能不再是强连续的事实。令人惊讶的是，这甚至揭示了（抽象）线性理论中的几个开放问题，例如，具有本质上有界输入的线性边界控制问题的温和解是否总是连续的问题。这些开放的问题显然是项目的一部分。这些仍然存在的困难也表明，关于非标准函数空间的无限维演化方程的控制方法是最近才出现的。相比之下，潜在的有限维理论在本质上有界输入的情况下是众所周知的，可以追溯到爱德华多·桑塔格，是众所周知的。这个项目的目的是进一步发展的适位性以及内部/外部稳定性，都是关于非标准类的输入函数。主要的焦点在于有界函数的空间，这在应用上有很高的相关性，但在数学上相当复杂。本课题包括两个子课题：抛物方程和双曲方程。在这两个子项目中，由于大部分问题都是针对线性案例开放的，所以我们首先并主要关注这个案例。然而，我们的长期目标是非线性的，特别是我们处理双线性双曲问题和半线性抛物进化方程。本项目的一个主要新颖之处在于，我们结合了算子理论、无限维系统理论、有界解析函数的泛函演算和函数空间的工具，以研究具有无界输入算子的无限维演化方程的适定性和输入到状态稳定性。

Evolution equations describe the temporal development of a dynamical system driven by an initial state and a given input. In this project, we investigate evolution equations on an infinite-dimensional state space with the following two features: We allow for a considerable unboundedness of the input operator and for non- standard spaces of input functions. A natural example for such a space, where standard methods are limited, is given by the essentially bounded functions. Our setting encompasses many of the common partial differential equations with boundary input and control. The latter being the main motivation for the proposed research. The state- of-the art for related control problems requires relatively nice boundary operators and spaces of input functions, such as square integrable functions, that may be hard to check. Moreover, the natural choice of norm for the inputs may not fall into these classes. When trying to lift the known results to these more general spaces, one is soon faced with severe theoretical difficulties, e.g., the fact that the translation operator might not be strongly continuous anymore. Surprisingly, this even reveals several open problems in the (abstract) linear theory, as for instance the question whether mild solutions of an linear boundary control problem with essentially bounded inputs are always continuous. These open problems are explicitly part of the project. These remaining difficulties also indicate that the approach to the control of infinite-dimensional evolution equations with respect to non-standard function spaces has only emerged recently. In contrast, the underlying finite-dimensional theory is well-known in the case of essentially bounded inputs, going back to Eduardo Sontag, is well- known. The aim of this project is to develop further the well- posedness as well as the internal/external stability, both with respect to non-standard classes of input functions. The main focus lies on the space of bounded functions, which are of high relevance in application, but mathematically quite intricate. The project consists of two subprojects: parabolic equations and hyperbolic equations. In both subprojects, as most of questions are even open for the linear case, we first of all and mainly focus on this case. However, our long term goal are nonlinearities, in particular, we treat bilinear hyperbolic problems and semilinear parabolic evolution equations. A major novelty of this project is that we combine tools from operator theory, infinite-dimensional systems theory, functional calculus for bounded analytic functions and function spaces in order to investigate well- posedness and input-to-state stability of infinite-dimensional evolution equations with unbounded input operators.