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Uncertainty propagation of stochastic hybrid systems: a case study for types of jump

Tejaswi K. C., William Clark, Taeyoung Lee

TL;DR

This work addresses uncertainty propagation in stochastic hybrid systems by formulating and analyzing stochastic Koopman and Frobenius-Perron operators for three one-dimensional SHS types. It reveals how the placement of randomness—in the continuous flow, in the jump mechanism, or in both—changes the FP operator form and the associated boundary conditions that govern density evolution. The contributions include a unified operator framework for deterministic, diffusion-affected, and Poisson-driven jumps, plus numerical validation via finite-difference schemes and Monte Carlo simulations, highlighting case-specific density dynamics and flux balance. The results advance understanding of density propagation in SHS and provide tools for uncertainty quantification and control in systems with intertwined continuous and discrete dynamics, with potential extension to higher dimensions.

Abstract

Stochastic hybrid systems are dynamic systems that undergo both random continuous-time flows and random discrete jumps. Depending on how randomness is introduced into the continuous dynamics, discrete transitions, or both, stochastic hybrid systems exhibit distinct characteristics. This paper investigates the role of uncertainties in the interplay between continuous flows and discrete jumps by studying probability density propagation. Specifically, we formulate stochastic Koopman/Frobenius-Perron operators for three types of one-dimensional stochastic hybrid systems to uncover their unique dynamic characteristics and verify them using Monte Carlo simulations.

Uncertainty propagation of stochastic hybrid systems: a case study for types of jump

TL;DR

This work addresses uncertainty propagation in stochastic hybrid systems by formulating and analyzing stochastic Koopman and Frobenius-Perron operators for three one-dimensional SHS types. It reveals how the placement of randomness—in the continuous flow, in the jump mechanism, or in both—changes the FP operator form and the associated boundary conditions that govern density evolution. The contributions include a unified operator framework for deterministic, diffusion-affected, and Poisson-driven jumps, plus numerical validation via finite-difference schemes and Monte Carlo simulations, highlighting case-specific density dynamics and flux balance. The results advance understanding of density propagation in SHS and provide tools for uncertainty quantification and control in systems with intertwined continuous and discrete dynamics, with potential extension to higher dimensions.

Abstract

Stochastic hybrid systems are dynamic systems that undergo both random continuous-time flows and random discrete jumps. Depending on how randomness is introduced into the continuous dynamics, discrete transitions, or both, stochastic hybrid systems exhibit distinct characteristics. This paper investigates the role of uncertainties in the interplay between continuous flows and discrete jumps by studying probability density propagation. Specifically, we formulate stochastic Koopman/Frobenius-Perron operators for three types of one-dimensional stochastic hybrid systems to uncover their unique dynamic characteristics and verify them using Monte Carlo simulations.

Paper Structure

This paper contains 13 sections, 6 theorems, 65 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Stochastic Hybrid Systems
  3. General Stochastic Hybrid Systems
  4. Stochastic Koopman/Frobenius-Perron Operator
  5. Uncertainty Propagation
  6. Deterministic Case
  7. SDE with Deterministic Jump
  8. SDE with Poisson Process
  9. Numerical Examples
  10. Deterministic Case
  11. SDE with Deterministic Jump
  12. SDE with Poisson Process
  13. Conclusions

Key Result

Theorem 1

Consider a hybrid system defined by eqn:dx1--eqn:K1. The stochastic Koopman operator $u(t, x) = \mathcal{K}_t f(x)$ on $f:\mathsf{X}\rightarrow\mathbb{R}$ satisfies with the initial condition $u(0,x) = f(x)$, where the generator $\mathcal{A}:L^\infty(\mathsf{X})\rightarrow L^\infty(X)$ is given by

Figures (6)

  • Figure 1: 1D hybrid system: the hybrid state space is $\mathsf{X}=(-\infty, b]$ (shaded by blue), and the guard is $\mathsf{G}=\partial\mathsf{X} = \{b\}$, at which the state jumps into $x=a<b$. The continuous vector field of $X(x)=-\gamma(x-c)$ for $\gamma>0$ and $c>b$ points to the right toward $x=c$.
  • Figure 2: Deterministic flow ($H = 0$)
  • Figure 3: Stochastic flow ($H = 0.5$) with deterministic jump
  • Figure 4: Stochastic flow ($H = 0.05$) with deterministic jump
  • Figure 5: Stochastic flow ($H = 0.5$) with Poisson process
  • ...and 1 more figures

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Theorem 3
  • Theorem 4
  • Remark 3
  • Remark 4
  • ...and 5 more