analyze riccati equations

This work addresses the high computational burden of traditional finite-horizon linear quadratic regulator (LQR) control, which requires repeated online solution of differential Riccati equations and thus struggles to meet real-time demands. The paper introduces Deep Operator Networks (DeepONets) to the LQR problem for the first time, establishing an offline learning framework that directly maps time-varying system parameters to Riccati solution trajectories, thereby shifting the computational load from online execution to a one-time training phase. The proposed approach features a scalable network architecture tailored for matrix-valued time-varying signals and a progressive training strategy, accompanied by theoretical guarantees on error propagation of the approximate solution and closed-loop stability. Experiments demonstrate that the method achieves high accuracy, strong generalization, and substantial computational speedup across diverse time-varying and time-invariant LQR tasks, making it well-suited for parametric and real-time optimal control applications.

研究用符号计算方法精确求解有限维线性系统的L∞范数，比较了Sturm-Habicht序列、RUR和CAD等方法与数值法的优劣，突出符号法在参数化情况下的精度优势。

Traditional control theory neglects computational uncertainty—such as mathematical object distortion induced by finite-precision arithmetic—leading to reliability gaps between Lyapunov stability analysis and digital controller implementation. Methodologically, this paper introduces the first constructive control framework that explicitly treats computational uncertainty as an independent modeling dimension in controller synthesis and system analysis. Leveraging tools from computability theory, constructive analysis, and measurable selection, we establish a constructive Danskin theorem and provide computable reconstructions of fundamental objects—including control Lyapunov functions (CLFs), Carathéodory trajectories, and eigenvalue problems. Our primary contribution is a computationally feasible paradigm for stability and stabilization proofs: all mathematical constructs are uniformly approximable by finite-precision algorithms while rigorously preserving required properties. This ensures robustness and implementability of digital controllers under realistic computational constraints.

This work addresses indirect data-driven control of bilinear systems under finite stochastic data, specifically ensuring end-to-end closed-loop stability in the presence of unbounded-support noise. The method introduces the first prior- and data-dependent finite-sample identification error ellipsoidal bound for bilinear systems, which is directly integrated into a robust controller synthesis framework. It synergistically combines statistical learning theory, ellipsoidal-set-based robust control, and Koopman operator approximation, and rigorously establishes exponential stability of the closed-loop system. Numerical experiments quantitatively characterize the trade-off between identification accuracy and control performance. Furthermore, the framework is extended to general nonlinear systems via Koopman-based data-driven control, substantially enhancing both theoretical guarantees and practical applicability under limited data.

This work addresses the problem of computing rational function solutions to first-order algebraic ordinary differential equations (AODEs) with parameters. For equations whose coefficients are rational functions, we construct an equivalent algebraic system and establish, for the first time, necessary and sufficient conditions for the existence of rational solutions. Our method introduces a symbolic algorithmic framework based on parameter elimination and polynomial system solving, achieving full decidability for constant-coefficient cases and enabling the construction of general rational solutions—including those involving transcendental constants—for single-function coefficient cases. The main contributions are: (1) the first universal and exact criterion for deciding the existence of rational solutions to parametric first-order AODEs; (2) algorithmic resolution in two fundamental cases—constant coefficients and single-function coefficients; and (3) an open-source, executable prototype implementation for symbolic decision-making.

This study addresses the challenge of global linear modeling and control for highly nonlinear dynamical systems by leveraging Koopman operator theory. By introducing observable functions, the nonlinear dynamics are lifted into a higher-dimensional space where they admit an approximately linear representation. A data-driven surrogate model is constructed through a synergistic integration of Extended Dynamic Mode Decomposition (EDMD), kernelized EDMD, and machine learning techniques. The work innovatively extends the Koopman framework to input-affine systems, proposing a unified modeling approach and a corresponding Koopman-based Model Predictive Control (MPC) design methodology. Numerical simulations demonstrate that the proposed method achieves high-fidelity modeling accuracy and effective closed-loop control performance. Full reproducibility is supported by the accompanying open-source implementation.

This work addresses the efficient algebraic representation and factorization of linear ordinary differential operators over compatible derivation modules. By implementing differential operators as first-class objects in Scratchpad II, the approach supports standard notation and provides a unified treatment of left and right module structures. For operators with coefficients in a field or polynomial ring, it integrates Ore localization, pseudo-division, and construction of right fraction fields to enable left and right division, computation of greatest common divisors, least common multiples, and extended Euclidean algorithms. Furthermore, by combining Riccati equations with Newton polygon analysis, the method effectively characterizes the singularities of factors. This framework facilitates constructive factorization and algebraic manipulation of operators with constant, elementary, rational, and even matrix-valued coefficients.

This work addresses the lack of theoretical guarantees for the reliability of Koopman eigenpairs computed from noisy data in data-driven spectral analysis. It introduces, for the first time, shadowing trajectory theory combined with backward error analysis to interpret the residual of eigenpairs obtained via Extended Dynamic Mode Decomposition (EDMD) as an operator perturbation of the original dynamical system. The study rigorously proves that this approximate solution corresponds exactly to a pseudo-trajectory shadowed by a true system trajectory. By establishing a precise connection among residuals, operator perturbations, and system trajectories, the paper constructs a backward stability framework for assessing Koopman eigenpairs, thereby providing a novel theoretical foundation for the credibility of data-driven methods in noisy environments.