Calculation of positive invariant sets for nonlinear systems using efficient novel eigenvalue bounds.

提出了一种计算黑森矩阵特征值界的新方法。由于该方法故意避免了Hessian矩阵的计算，因此它所需的计算量非常低。我们将这种新方法称为特征值算法，并将其应用于非线性动力系统正不变集的数值计算中。结果表明，就特征值界的紧密性而言，本算法可以与现有方法相媲美。然而，它的计算量比现有的方法要低。因此，特征值算法是一个有吸引力的替代算法，花费相当一部分的计算时间在计算Hessian特征值边界上，例如全局优化的某些变体。接下来的项目旨在利用特征值算法中的稀疏性。虽然稀疏性通常用于减少数值算法中的操作次数，但这里可以以更基本的方式使用它来收紧特征值界限。稀疏中间矩阵自然地出现在特征值算法中，即使感兴趣的函数结果是密集的Hessian。因此，紧缩效应预计将在一大类函数中发挥作用，其中包括那些没有稀疏Hessian矩阵的函数。改进的特征值边界预计会导致扩大的正不变域。作为最后的测试，这些域将用于许多基准和技术非线性动力系统的模型预测控制的稳定性准则。

We developed a new method that calculates Hessian matrix eigenvalue bounds. Because the method deliberately avoids the calculation of Hessian matrices, it requires a surprisingly low computational effort. The new approach, which we refer to as eigenvalue arithmetic, was subsequently applied in the numerical computation of positive invariant sets of nonlinear dynamical systems. Our results show that the eigenvalue arithmetic can compete with existing methods as far as the tightness of the eigenvalue bounds is concerned. Its computational effort, however, is lower than that of existing approaches. As a result, the eigenvalue arithmetic is an attractive alternative for algorithms that spend a considerable fraction of their computational time on the calculation of Hessian eigenvalue bounds, such as certain variants of global optimization. The subsequent project aims at exploiting sparsity in the eigenvalue arithmetic. While sparsity is usually used to reduce the number of operations in numerical algorithms, it can here be used in a more fundamental way to tighten the eigenvalue bounds. Sparse intermediate matrices naturally appear in the eigenvalue arithmetic even if the function of interest results in a dense Hessian. Consequently, the tightening effect is expected to play a role for a large class of functions, among them those which do not have sparse Hessian matrices.The improved eigenvalue bounds are anticipated to result in enlarged positive invariant domains. As a final test, these domains will be used in stability criteria for model predictive control for a number of benchmark and technical nonlinear dynamical systems.