Quasilinear elliptic Hamilton-Jacobi-Bellman equations and regime-switching systems

Abstract

We study a quasilinear elliptic Hamilton--Jacobi--Bellman equation in $\mathbb{R}^N$ under an admissible growth and uniqueness framework. Our contributions are as follows. First, we establish a robust comparison principle and a constructive existence theory based on monotone Dirichlet approximation on expanding balls, yielding solutions with high local regularity and providing an implementable numerical scheme. Second, we show that radial source terms generate the unique radial admissible solution, using only rotational invariance and global uniqueness, without relying on moving-plane arguments. Third, we obtain explicit quadratic solutions in the special case of polynomial data, both for the scalar equation and for the associated regime-switching HJB system, and apply these formulas to stochastic production planning models with full verification of optimality. Fourth, we extend the scalar existence and uniqueness theory to weakly coupled multi-regime systems under explicit power-growth conditions, and we formulate precise conjectures regarding the system-level generalization of the $g$-convex growth framework, together with partial comparison results. The analysis is complemented by gradient estimates, a verification theorem linking PDE solutions to stochastic control, and fully documented Python implementations consistent with the theoretical results.

Figures (6)

• Figure 1: Exact solution versus numerical approximation for the radial Hamilton--Jacobi--Bellman problem.
• Figure 2: Numerical solution of the radial HJB equation for $f(r)=r^2$. The overlap between the numerical result (Teal) and the exact solution (Orange) demonstrates the high precision of the BVP solver across different dimensions.
• Figure 3: 3D Visualization of the radial HJB solution. The solid surface (Numerical FD) and the orange wireframe (Exact) are indistinguishable, while the circular contours confirm the preserved symmetry.
• Figure 4: HJB solution for a non-radially symmetric cost function. The elliptical contour lines clearly show how the solution surface breaks rotation symmetry to align with the data $f$, while maintaining the characteristic convexity of the framework.
• Figure 5: Simulation of optimal inventory trajectories under two-regime switching. The background shading indicates the current regime (1 or 2). Observe how the inventory behavior and control intensity shift as the system transitions between different economic states.

Theorems & Definitions (55)

• Remark 2.2: The structural necessity of the convex reference $g$
• Definition 2.3: Sub/supersolutions
• Lemma 3.1: Comparison for close functions
• proof
• Lemma 3.2: Explosive supersolution approximation
• proof
• Theorem 3.3: Global comparison and uniqueness
• proof
• Lemma 4.1: Local a priori estimate
• proof