scholarly journals Deductive Stability Proofs for Ordinary Differential Equations

2021 ◽  
pp. 181-199
Author(s):  
Yong Kiam Tan ◽  
André Platzer
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Real World ◽  
Formal Analysis ◽  
Dynamic Logic ◽  
Logical Structure ◽  
Theorem Prover ◽  
Continuous Systems ◽  
Stability Properties ◽  
Controlled Systems

AbstractStability is required for real world controlled systems as it ensures that those systems can tolerate small, real world perturbations around their desired operating states. This paper shows how stability for continuous systems modeled by ordinary differential equations (ODEs) can be formally verified in differential dynamic logic (). The key insight is to specify ODE stability by suitably nesting the dynamic modalities of with first-order logic quantifiers. Elucidating the logical structure of stability properties in this way has three key benefits: i) it provides a flexible means of formally specifying various stability properties of interest, ii) it yields rigorous proofs of those stability properties from ’s axioms with ’s ODE safety and liveness proof principles, and iii) it enables formal analysis of the relationships between various stability properties which, in turn, inform proofs of those properties. These benefits are put into practice through an implementation of stability proofs for several examples in KeYmaera X, a hybrid systems theorem prover based on .

scholarly journals Diagonally Implicit Block Backward Differentiation Formula with Optimal Stability Properties for Stiff Ordinary Differential Equations

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1342 ◽  
Author(s):  
Hazizah Mohd Ijam ◽  
Zarina Bibi Ibrahim
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Triangular Matrix ◽  
Differentiation Formula ◽  
Stability Properties ◽  
Stiff Ordinary Differential Equations ◽  
Backward Differentiation Formula ◽  
Optimal Stability ◽  
Best Value ◽  
Selection Of

This paper aims to select the best value of the parameter ρ from a general set of linear multistep formulae which have the potential for efficient implementation. The ρ -Diagonally Implicit Block Backward Differentiation Formula ( ρ -DIBBDF) was proposed to approximate the solution for stiff Ordinary Differential Equations (ODEs) to achieve the research objective. The selection of ρ for optimal stability properties in terms of zero stability, absolute stability, error constant and convergence are discussed. In the diagonally implicit formula that uses a lower triangular matrix with identical diagonal entries, allowing a maximum of one lower-upper (LU) decomposition per integration stage to be performed will result in substantial computing benefits to the user. The numerical results and plots of accuracy indicate that the ρ -DIBBDF method performs better than the existing fully and diagonally Block Backward Differentiation Formula (BBDF) methods.

scholarly journals Stability of conditionally invariant sets and controlled uncertain dynamic systems on time scales

1995 ◽  
Vol 1 (1) ◽  
pp. 1-10 ◽  
Author(s):  
V. Lakshmikantham ◽  
Z. Drici
Keyword(s):  
Time Scales ◽  
Dynamic Systems ◽  
Invariant Sets ◽  
Continuous Systems ◽  
Stability Properties ◽  
Closed Sets ◽  
Controlled Systems ◽  
Uncertain Dynamic Systems ◽  
Imperfect Knowledge ◽  
The Stability

A basic feedback control problem is that of obtaining some desired stability property from a system which contains uncertainties due to unknown inputs into the system. Despite such imperfect knowledge in the selected mathematical model, we often seek to devise controllers that will steer the system in a certain required fashion. Various classes of controllers whose design is based on the method of Lyapunov are known for both discrete [4], [10], [15], and continuous [3–9], [11] models described by difference and differential equations, respectively. Recently, a theory for what is known as dynamic systems on time scales has been built which incorporates both continuous and discrete times, namely, time as an arbitrary closed sets of reals, and allows us to handle both systems simultaneously [1], [2], [12], [13]. This theory permits one to get some insight into and better understanding of the subtle differences between discrete and continuous systems. We shall, in this paper, utilize the framework of the theory of dynamic systems on time scales to investigate the stability properties of conditionally invariant sets which are then applied to discuss controlled systems with uncertain elements. For the notion of conditionally invariant set and its stability properties, see [14]. Our results offer a new approach to the problem in question.

scholarly journals An axiomatic approach to existence and liveness for differential equations

2021 ◽  
Author(s):  
Yong Kiam Tan ◽  
André Platzer
Keyword(s):  
Differential Equations ◽  
Blow Up ◽  
Dynamic Logic ◽  
Axiomatic Approach ◽  
Theorem Prover ◽  
Important Special Case ◽  
Target Region ◽  
Invariance Properties ◽  
Liveness Properties ◽  
Verification Techniques

AbstractThis article presents an axiomatic approach for deductive verification of existence and liveness for ordinary differential equations (ODEs) with differential dynamic logic (dL). The approach yields proofs that the solution of a given ODE exists long enough to reach a given target region without leaving a given evolution domain. Numerous subtleties complicate the generalization of discrete liveness verification techniques, such as loop variants, to the continuous setting. For example, ODE solutions may blow up in finite time or their progress towards the goal may converge to zero. These subtleties are handled in dL by successively refining ODE liveness properties using ODE invariance properties which have a complete axiomatization. This approach is widely applicable: several liveness arguments from the literature are surveyed and derived as special instances of axiomatic refinement in dL. These derivations also correct several soundness errors in the surveyed literature, which further highlights the subtlety of ODE liveness reasoning and the utility of an axiomatic approach. An important special case of this approach deduces (global) existence properties of ODEs, which are a fundamental part of every ODE liveness argument. Thus, all generalizations of existence properties and their proofs immediately lead to corresponding generalizations of ODE liveness arguments. Overall, the resulting library of common refinement steps enables both the sound development and justification of new ODE existence and of liveness proof rules from dL axioms. These insights are put into practice through an implementation of ODE liveness proofs in the KeYmaera X theorem prover for hybrid systems.

scholarly journals Modeling Trajectories with Neural Ordinary Differential Equations

2021 ◽  
Author(s):  
Yuxuan Liang ◽  
Kun Ouyang ◽  
Hanshu Yan ◽  
Yiwei Wang ◽  
Zekun Tong ◽  
...  
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Real World ◽  
Continuous Time ◽  
Trajectory Data ◽  
Time Dynamics ◽  
Trajectory Classification ◽  
Novel Method ◽  
Spatial Trajectory ◽  
Hidden States

Recent advances in location-acquisition techniques have generated massive spatial trajectory data. Recurrent Neural Networks (RNNs) are modern tools for modeling such trajectory data. After revisiting RNN-based methods for trajectory modeling, we expose two common critical drawbacks in the existing uses. First, RNNs are discrete-time models that only update the hidden states upon the arrival of new observations, which makes them an awkward fit for learning real-world trajectories with continuous-time dynamics. Second, real-world trajectories are never perfectly accurate due to unexpected sensor noise. Most RNN-based approaches are deterministic and thereby vulnerable to such noise. To tackle these challenges, we devise a novel method entitled TrajODE for more natural modeling of trajectories. It combines the continuous-time characteristic of Neural Ordinary Differential Equations (ODE) with the robustness of stochastic latent spaces. Extensive experiments on the task of trajectory classification demonstrate the superiority of our framework against the RNN counterparts.

Modeling with Stochastic Fuzzy Differential Equations

2014 ◽  
pp. 150-172 ◽  
Author(s):  
Marek T. Malinowski
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Stochastic Processes ◽  
Real World ◽  
Existence And Uniqueness ◽  
Qualitative Properties ◽  
Stability Properties ◽  
Uncertain Dynamical Systems

In the chapter, the author considers an approach used in the studies of stochastic fuzzy differential equations. These equations are new mathematical tools for modeling uncertain dynamical systems. Some qualitative properties of their solutions such as existence and uniqueness are recalled, and stability properties are shown. Here, the solutions are continuous adapted fuzzy stochastic processes. The author considers some examples of applications of stochastic fuzzy differential equations in modeling real-world phenomena.

scholarly journals Gaussian quadrature rules and A-stability of Galerkin schemes for ODE

2003 ◽  
Vol 2003 (31) ◽  
pp. 1947-1959
Author(s):  
Ali Bensebah ◽  
François Dubeau ◽  
Jacques Gélinas
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
Discontinuous Galerkin ◽  
Legendre Polynomials ◽  
Discontinuous Galerkin Methods ◽  
Gaussian Quadrature ◽  
Quadrature Rules ◽  
Galerkin Methods ◽  
Stability Properties

The A-stability properties of continuous and discontinuous Galerkin methods for solving ordinary differential equations (ODEs) are established using properties of Legendre polynomials and Gaussian quadrature rules. The influence on the A-stability of the numerical integration using Gaussian quadrature rules involving a parameter is analyzed.

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