Theory Of Ordinary Differential Equations

By Kristian Kirsch. 2016

1,399 citations recorded by the source corpus.

Subfields: Numerical Methods & Scientific Computing

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Citations in the corpus

Neural Ordinary Differential Equations (2018, positive): “Uniqueness When do continuous dynamics have a unique solution? Picard’s existence theorem (Coddington and Levinson, 1955) states that the solution to an initial value problem exists and is unique if the differential equation is uniformly Lipschitz continuous in z and continuous in t.”
Implicit Regularization of Discrete Gradient Dynamics in Deep Linear Neural Networks (2019, methodology): “(47) In order to get a solution for w2,i(t) and w1,i(t), we will use Cauchy-Lipschitz theorem [Coddington and Levinson, 1955].”
Stability notions and Lyapunov functions for sliding mode control systems (2014, background): “Systems with discontinuous right-hand sides The classical theory of differential equations [33] introduces a solution of the ordinary differential equation (ODE)”
Reach-avoid problems with time-varying dynamics, targets and constraints (2014, background): “Then for any initial time t P r0, T s and state x, under input signals ap ̈q P At, bp ̈q P Bt, the evolution of the system is determined (see for example [8], Chapter 2, Theorems 1.”
Boundary Value Problems, Weyl Functions, and Differential Operators (2020, background): “Chapter 6 and the classics [209, 740, 741] and, e.”
A survey on stationary problems, Green’s functions and spectrum of Sturm–Liouville problem with nonlocal boundary conditions (2014, background): “The various linear NBCs (1)–(11) can be written as linear functional (for classical BC, see [42]).”
From duality to determinants for q-TASEP and ASEP (2012, background): “4 is assured since it reduces to a finite system of linear ODEs, from which the result follows from standard methods [13].”
Parametrix construction of the transition probability density of the solution to an SDE driven by $\alpha$-stable noise (2014, background): “3 and the comment in Chapter I, Section 5 from [11]), the set Υt,b = {y : θt (y) ∈ Υb} is again of zero Lebesgue measure.”
A General Safety Framework for Learning-Based Control in Uncertain Robotic Systems (2017, background): “The basic existence and uniqueness theorems [35] assume fixed control sets.”
The Gesture as an Autonomous Nonlinear Dynamical System (2016, background): “As a consequence of stationarity, no two trajectories cross [5].”
Optimal simplices and codes in projective spaces (2013, background): “Thus, by the Peano existence theorem (see Chapter 1, Sections 1–5 in [CL55]), the initial value problem (3.”
Hamilton-Jacobi reachability: A brief overview and recent advances (2017, background): “Therefore, given a(·) ∈ A and b(·) ∈ B, there exists a unique trajectory solving (1) [46].”
Reversible Architectures for Arbitrarily Deep Residual Neural Networks (2017, background): “…in deep residual networks, system of Ordinary Differential Equations (ODEs), special kinds of dynamical systems, have long been studied in ∗ Authors contributed equally mathematics and physics with rich theoretical and empirical success (Coddington and Levinson 1955; Simmons 2016; Arnold 2012).”