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The Triality of Radial Nonlinear Dynamics: Analysis of Riccati, Schrödinger, and Hamilton--Jacobi--Bellman Equations

Dragos-Patru Covei

Abstract

This study establishes a comprehensive mathematical framework for the analysis of radial differential equations, identifying a fundamental connection between three distinct classes of problems: the nonlinear Riccati equation, the linear Schrödinger equation, and the Hamilton--Jacobi--Bellman equation for stochastic control. We prove the existence and uniqueness of regular solutions on both bounded and unbounded domains, identifying sharp growth rates and asymptotic behaviors. Furthermore, we conduct a sensitivity analysis of the noise intensity parameter, characterizing the transitions between deterministic and diffusion-dominated regimes through the lens of singular perturbation theory. Our theoretical results are complemented by numerical simulations that validate the predicted feedback laws and the structural stability of the system. This unified approach provides deep insights into the duality between global wave functions and local dynamical drifts, offering a rigorous basis for analyzing multidimensional stochastic processes under central potentials.

The Triality of Radial Nonlinear Dynamics: Analysis of Riccati, Schrödinger, and Hamilton--Jacobi--Bellman Equations

Abstract

This study establishes a comprehensive mathematical framework for the analysis of radial differential equations, identifying a fundamental connection between three distinct classes of problems: the nonlinear Riccati equation, the linear Schrödinger equation, and the Hamilton--Jacobi--Bellman equation for stochastic control. We prove the existence and uniqueness of regular solutions on both bounded and unbounded domains, identifying sharp growth rates and asymptotic behaviors. Furthermore, we conduct a sensitivity analysis of the noise intensity parameter, characterizing the transitions between deterministic and diffusion-dominated regimes through the lens of singular perturbation theory. Our theoretical results are complemented by numerical simulations that validate the predicted feedback laws and the structural stability of the system. This unified approach provides deep insights into the duality between global wave functions and local dynamical drifts, offering a rigorous basis for analyzing multidimensional stochastic processes under central potentials.

Paper Structure

This paper contains 41 sections, 10 theorems, 89 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Literature Review and Novelty
  3. Summary of Main Results
  4. Organization of the Article
  5. Motivations and Physical Significance
  6. The Riccati Perspective: Nonlinearity and Control
  7. The Schrödinger Perspective: Linearity and Regularity
  8. Main Results and Detailed Proofs
  9. Transformation and the Linear Auxiliary State
  10. Existence, Uniqueness, and Regularity at the Origin
  11. Asymptotic Behavior and Phase State Stability
  12. Rigorous Analysis of the Riccati Asymptote
  13. The Bounded Domain and Global Geometric Properties
  14. Convexity of the Auxiliary Wave Function
  15. Implications for the Riccati Solution
  16. ...and 26 more sections

Key Result

Lemma 3.1

Let $b \in C(0, R)$. A function $\phi \in C^1(0, R)$ is a solution to the Riccati equation eq:ric_ode if and only if there exists a $C^2(0, R)$ solution $u$ to the linear auxiliary equation such that $u(r) \neq 0$ for all $r \in (0, R)$, and $\phi$ is given by the logarithmic-style derivative

Figures (1)

  • Figure 1: Numerical simulation of the linear and Riccati-type radial problem for $b(r)=r^2$, $N=2$, $\sigma=1$. (Left) Auxiliary function $u(r)$; (Middle-Left) Riccati solution $\phi(r)$; (Middle-Right) Transformed value function $z(r)$; (Right) Magnitude of the optimal control $|p^*(x)|$.

Theorems & Definitions (19)

  • Lemma 3.1
  • proof
  • Theorem 3.2: Local and Global Well-Posedness
  • proof
  • Proposition 4.1: General Asymptotics of $\phi$
  • proof
  • Theorem 5.1: Strict Convexity of $u$
  • proof
  • Proposition 6.1: Vanishing Noise Limit
  • Proposition 6.2: Uniform Vanishing
  • ...and 9 more