This work presents a framework for control theory based on constructive analysis to account for discrepancy between mathematical results and their implementation in a computer, also referred to as computational uncertainty. In control engineering, the latter is usually either neglected or considered submerged into some other type of uncertainty, such as system noise, and addressed within robust control. However, even robust control methods may be compromised when the mathematical objects involved in the respective algorithms fail to exist in exact form and subsequently fail to satisfy the required properties. For instance, in general stabilization using a control Lyapunov function, computational uncertainty may distort stability certificates or even destabilize the system despite robustness of the stabilization routine with regards to system, actuator and measurement noise. In fact, battling numerical problems in practical implementation of controllers is common among control engineers. Such observations indicate that computational uncertainty should indeed be addressed explicitly in controller synthesis and system analysis. The major contribution here is a fairly general framework for proof techniques in analysis and synthesis of control systems based on constructive analysis which explicitly states that every computation be doable only up to a finite precision thus accounting for computational uncertainty. A series of previous works is overviewed, including constructive system stability and stabilization, approximate optimal controls, eigenvalue problems, Caratheodory trajectories, measurable selectors. Additionally, a new constructive version of the Danskin's theorem, which is crucial in adversarial defense, is presented.

翻译：本文提出了一种基于构造性分析的控制理论框架，旨在解决数学结果与其在计算机中实现之间的差异，这种差异亦称为计算不确定性。在控制工程中，后者通常被忽略，或被归入其他类型的不确定性（如系统噪声）中，并在鲁棒控制框架内处理。然而，当相关算法涉及的数学对象无法以精确形式存在，从而无法满足所需性质时，即便是鲁棒控制方法也可能失效。例如，在使用控制李雅普诺夫函数进行一般镇定控制时，尽管镇定程序对系统、执行器和测量噪声具有鲁棒性，但计算不确定性仍可能扭曲稳定性证明，甚至导致系统失稳。事实上，在实际控制器实现中应对数值问题在控制工程师中十分常见。这些观察表明，计算不确定性确实应在控制器综合与系统分析中得到显式处理。本文的主要贡献在于提出了一个相当通用的框架，用于控制系统的分析与综合中的证明技术，该框架基于构造性分析，并明确声明所有计算仅能在有限精度内完成，从而将计算不确定性纳入考量。本文综述了一系列先前工作，包括构造性系统稳定性与镇定、近似最优控制、特征值问题、Caratheodory轨迹、可测选择器等。此外，文中还提出了一个对对抗防御至关重要的Danskin定理的新构造性版本。