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Optimal control, viscosity approximation and Arrhenius Law for the shallow lake problem

Angeliki Koutsimpela, Michail Loulakis

Abstract

We prove existence of optimal control for the deterministic and stochastic shallow lake problem without any restrictions on the parameter space and we establish a generalization of the Arrhenius Law in the case of noise-dependent potentials, which naturally arise in control theory problems. We also prove a result about convergence of the derivatives in the viscosity approximation of the value function and use this result to derive the Arrhenius Law for the shallow lake problem.

Optimal control, viscosity approximation and Arrhenius Law for the shallow lake problem

Abstract

We prove existence of optimal control for the deterministic and stochastic shallow lake problem without any restrictions on the parameter space and we establish a generalization of the Arrhenius Law in the case of noise-dependent potentials, which naturally arise in control theory problems. We also prove a result about convergence of the derivatives in the viscosity approximation of the value function and use this result to derive the Arrhenius Law for the shallow lake problem.
Paper Structure (17 sections, 14 theorems, 121 equations, 3 figures)

This paper contains 17 sections, 14 theorems, 121 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The general setting and the main results
  3. Existence of optimal control
  4. Generalization of the Arrhenius' law
  5. Framework
  6. Verification Principle
  7. Deterministic Shallow Lake Problem
  8. Proof of Theorem \ref{['constrained_vs_det']}
  9. Proof of Theorem \ref{['comparison']}
  10. Proof of Theorem 3
  11. Stochastic Shallow Lake Problem
  12. Generalization of the Arrhenius Law
  13. Proof of Lemma \ref{['F0']}
  14. Proof of Theorem \ref{['arrh']}
  15. Proof of Lemma \ref{['asmlemma']}
  16. ...and 2 more sections

Key Result

Theorem 1

If $\sigma=0$, the value function $V$ is a continuous, constrained viscosity solution of equation OHJB on $[0,\infty)$. Moreover, $V$ satisfies OHJB at $x=0$ in the classical sense.

Figures (3)

  • Figure 1: Part of the phase plane of the system \ref{['slakes']}. The points P, Q are the saddle steady-states, the point S is a vortex and $x_*$ is the Skiba point. The green and the blue curve form the optimal solution, which is everywhere single-valued except for the Skiba point whereat the optimal control may take two values.
  • Figure 2: Left: The paths of the optimally controlled lake (deterministic case) for different initial positions. Right: One simulated path of the optimally controlled lake (stochastic case) with two stochastic attractors.
  • Figure 3: The potential $F_0$ of the deterministic optimally controlled shallow lake when $(b,c,\rho,\sigma)=(0.65,0.512,0.03,0)$

Theorems & Definitions (30)

  • Theorem 1
  • Theorem 2
  • Remark 2.1
  • Theorem 3
  • Lemma 2.1
  • Theorem 4
  • Lemma 2.2
  • proof
  • Claim 1
  • proof
  • ...and 20 more