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Hamilton-Jacobi Reachability for Viability Analysis of Constrained Waste-to-Energy Systems under Adversarial Uncertainty

Achraf Bouhmady, Othman Cherkaoui Dekkaki

Abstract

This paper investigates the problem of maintaining the safe operation of Waste-to-Energy (WtE) systems under operational constraints and uncertain waste inflows. We model this as a robust viability problem, formulated as a zero-sum differential game between a control policy and an adversarial disturbance. Within a Hamilton-Jacobi framework, the viability kernel is characterized as the zero sublevel set of a value function satisfying a constrained Hamilton-Jacobi-Bellman (HJB) equation in the viscosity sense. This formulation provides formal guarantees for ensuring that system trajectories remain within prescribed operational limits under worst-case scenarios. Compared to existing viability studies, this work introduces a rigorous HJB-based characterization explicitly incorporating uncertainty, tailored to nonlinear WtE dynamics. A numerical scheme based on the Local Lax-Friedrichs method is employed to approximate the viability kernel. Numerical experiments illustrate how increasing inflow uncertainty significantly reduces the viability domain, shrinking the safe operating envelope. The proposed method is computationally tractable for systems of moderate dimension and offers a basis for synthesizing robust control policies, contributing to the design of resilient and sustainable WtE infrastructures.

Hamilton-Jacobi Reachability for Viability Analysis of Constrained Waste-to-Energy Systems under Adversarial Uncertainty

Abstract

This paper investigates the problem of maintaining the safe operation of Waste-to-Energy (WtE) systems under operational constraints and uncertain waste inflows. We model this as a robust viability problem, formulated as a zero-sum differential game between a control policy and an adversarial disturbance. Within a Hamilton-Jacobi framework, the viability kernel is characterized as the zero sublevel set of a value function satisfying a constrained Hamilton-Jacobi-Bellman (HJB) equation in the viscosity sense. This formulation provides formal guarantees for ensuring that system trajectories remain within prescribed operational limits under worst-case scenarios. Compared to existing viability studies, this work introduces a rigorous HJB-based characterization explicitly incorporating uncertainty, tailored to nonlinear WtE dynamics. A numerical scheme based on the Local Lax-Friedrichs method is employed to approximate the viability kernel. Numerical experiments illustrate how increasing inflow uncertainty significantly reduces the viability domain, shrinking the safe operating envelope. The proposed method is computationally tractable for systems of moderate dimension and offers a basis for synthesizing robust control policies, contributing to the design of resilient and sustainable WtE infrastructures.

Paper Structure

This paper contains 28 sections, 2 theorems, 44 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Mathematical Formulation
  3. State, Control, and Disturbance Variables
  4. Operational Constraints and Safe Set
  5. Control Objective and Robust Formulation
  6. Game-Theoretic Interpretation and HJBI Formulation
  7. Robust Reachability under Uncertainty: A Hamilton--Jacobi Framework
  8. Problem Setting and Assumptions
  9. Standing Assumptions.
  10. Backward Reachable Set and Viability Connection
  11. Value Function and Signed Distance Encoding
  12. Interpretation.
  13. Dynamic Programming and HJ Characterization
  14. Explicit Hamiltonian and Optimal Controls
  15. Numerical Implementation and Simulation Results
  16. ...and 13 more sections

Key Result

Proposition 3.5

Figures (9)

  • Figure 1: Backward reachable set under constant inflow, sliced at $E=50$.
  • Figure 2: Left: State trajectories $(x,K,E)$ under constant inflow. Right: $(x,K)$ projection for $(10,5,50)$.
  • Figure 3: Control evolution under constant inflow.
  • Figure 4: Backward reachable set under $\pm 10\%$ inflow uncertainty.
  • Figure 5: Left: State trajectories under three inflow profiles. Right: Control strategies.
  • ...and 4 more figures

Theorems & Definitions (11)

  • Remark 2.1: Interpretation and Novelty
  • Remark 3.1: Why Isaacs Condition Holds
  • Definition 3.2: Backward Reachable Set
  • Remark 3.3: Interpretation
  • Definition 3.4: Robust Value Function
  • Proposition 3.5: Characterization of BRS
  • proof
  • Theorem 3.6: HJ PDE for Robust Reachability
  • proof
  • Remark 3.7: Interpretation of Optimal Strategy
  • ...and 1 more