Weakly Coupled HJB Systems

• Weakly coupled HJB systems are a class of first-order nonlinear PDEs that model optimal control problems with regime switching and off-diagonal coupling.
• The framework leverages viscosity solution theory, the structure of Aubry sets, and monotonicity to ensure uniqueness and comparison principles.
• Advanced numerical methods, including elliptic regularization and streamline-diffusion FEM, enable robust simulation of risk-aware path planning and stochastic control.

A weakly coupled Hamilton–Jacobi–Bellman (HJB) system is a system of first-order nonlinear PDEs in which several value functions, typically associated to different system regimes or modes, are linked via an off-diagonal, usually monotone, coupling structure in their zero-order terms. These systems encode optimal control problems for processes subject to random regime-switching, optimal switching problems, or path-planning tasks with stochastic or adversarial disruptions. The interaction of the Hamiltonian structure, coupling monotonicity, and degeneracy yields a rich qualitative theory drawing on viscosity solution methods, large deviations, weak KAM theory, and modern finite element (FEM) computational techniques.

1. Structural Formulation and Prototypical Systems

A general weakly coupled stationary HJB system on a bounded domain $\Omega \subset \mathbb{R}^n$ (or on a torus $\mathbb{T}^n$) for $m$ modes is

$$\begin{cases} H_i(x, D u_i(x)) + \sum_{j=1}^m D_{ij}(x)\,u_j(x) = f_i(x), & x \in \Omega,\, i=1,...,m, \ u_i(x) = \text{boundary data}, & x \in \partial\Omega,\, i=1,...,m. \end{cases}$$

Here, each $H_i$ is the Hamiltonian (convex, coercive in $Du$), and $D(x)$ is the coupling matrix. In weak coupling, the off-diagonal entries $D_{ij}(x)$ with $i\neq j$ are nonpositive, and rows sum to zero, i.e., $\sum_{j=1}^m D_{ij}(x) = 0$, so $D(x)$ is a degenerate M-matrix.

A particularly instructive concrete instance is the two-state degenerate eikonal system

$$\begin{cases} |\nabla u_1| f_1 + \phi_1(u_1 - u_2) = K_1 + \lambda R, & x \in \Omega \ |\nabla u_2| f_2 + \phi_2(u_2 - u_1) = K_2 + \phi_2 R, & x \in \Omega \ u_1=0 \text{ on } G, \quad u_2=R+u_1 \text{ on } D, \end{cases}$$

where all coupling, running cost, and transition rates are as detailed in (Chiri et al., 2023). This arises from optimal path planning under random breakdown with partial (mode 1) and total (mode 2) failure, and spatially inhomogeneous transition rates.

2. Degenerate Coupling, Monotonicity, and the Aubry Set

Crucial in the analysis of these systems is the structure of the coupling matrix $D(x)$. If the off-diagonal entries are strictly negative and the rows sum to zero (degeneracy), the system is weakly coupled and irreducible, leading to the emergence of an Aubry set — an intrinsic inner “boundary” where degeneracy inhibits propagation of information across modes. Comparison principles and uniqueness of viscosity solutions require boundary data to be specified not just on the geometric boundary $\partial\Omega$ but also on this Aubry set: $$\mathcal{A} = \left\{ x \in \Omega : K_1(x) + \lambda(x) R(x) = 0,\, K_2(x) + \phi_2(x) R(x) = 0 \right\}.$$ For weakly coupled HJB systems with (possibly nonconstant) degenerate coupling, monotonicity (increasing in the local component, nonincreasing in the others) ensures that viscosity sub- and supersolution comparison principles are valid when augmented with control on $\mathcal{A}$ (Chiri et al., 2023, Davini et al., 2012).

3. Viscosity Solution Theory: Existence, Uniqueness, and Comparison

Within the framework defined above, with continuous, positive data and nontrivial boundary/Aubry information, the system admits the following theory:

• Comparison Principle: If a bounded, upper-semicontinuous vector subsolution and a lower-semicontinuous supersolution are ordered on $\mathcal{A}\cup\partial\Omega$, then ordering persists in $\overline{\Omega}$ (Chiri et al., 2023).
• Uniqueness: There is at most one continuous viscosity solution matching the given boundary and Aubry set data.
• Existence: In classical monotone cases, existence can be established by Perron-type arguments. For degenerate coupling, existence may fail if data are not given on the Aubry set, as demonstrated by explicit 1D counterexamples (Chiri et al., 2023).

The proof of comparison involves considering a perturbed maximum of the form $\sup_{i,x}\{\delta u_i(x) - v_i(x)\}$, classical doubling-variables methods, and a careful distinction of cases depending on whether the supremum occurs on $\mathcal{A}$ or in the interior. Degeneracy necessitates explicit handling of vanishing source terms on $\mathcal{A}$.

4. Numerical Approximation: Streamline-Diffusion FEM and Artificial Viscosity

Computational strategies for weakly coupled HJB systems, especially with degenerate coupling and Hamiltonian singularities, utilize regularization and FEM technology inspired by convection-diffusion solvers. The algorithmic core follows these steps (Chiri et al., 2023):

1. Elliptic Regularization: Add artificial viscosity, i.e., an $-\varepsilon\Delta$ regularization, to each equation.
2. Linearization: At each iteration, linearize the $|\nabla u|$ nonlinearity, fixing directions from previous iterates.
3. Discretization: Employ streamline-diffusion Galerkin FEM. For each $u_{i,h}$ on a simplicial mesh $\mathcal{T}_h$, solve the linearized convection-diffusion system, stablized along approximate characteristics.
4. Stability and Convergence: By classical FEM theory (e.g., Brooks-Hughes, Morton), the method is stable (given suitable stabilization constant $\theta$), and as $\varepsilon_n \to 0$ then $h\to 0$, the discrete solutions converge in a mesh-dependent norm to the viscosity solution, provided suitable data are prescribed on $\mathcal{A}\cup\partial\Omega$.

The practical impact is the realization of robust, implementable algorithms for risk-aware path planning in stochastic environments with switching or random breakdown (Chiri et al., 2023).

5. Weakly Coupled HJB Systems in Weak KAM and Mather Theory

The qualitative structure of stationary (ergodic) weakly coupled systems is deeply intertwined with modern developments in weak KAM theory (Mitake et al., 2015, Davini et al., 2012, Terai, 2019). Key features include:

• Critical Value and Subsolution Characterization: There exists a minimal (critical) value $a_{\text{crit}}$ for which maximal subsolutions, constructed via Lagrangian minimization over Markovian path measures (the “random frame”), exist.
• Aubry Set as Uniqueness Locus: The Aubry set is the minimal closed set where uniqueness can be enforced. On this set, all critical subsolutions agree up to constants, and outside it, strictness and smoothness can be ensured.
• Mather Measures: Existence and uniqueness issues can be further analyzed via variational duality and occupation measures (generalized Mather measures), connecting large-time limits of the system to ergodic optimization in a controlled Markovian environment (Terai, 2019, Davini et al., 2012).

6. Large-Time Asymptotics and Homogenization

Time-dependent weakly coupled HJB systems, under coercivity and monotone coupling, display uniform convergence to stationary (ergodic) profiles as $t \to \infty$ (Mitake et al., 2011, Mitake et al., 2012, Nguyen, 2012, Camilli et al., 2011). For spatially periodic problems,

$u_i(x, t) + c t \to v_i(x)$

uniformly, where $c$ is the unique ergodic constant, and the $v_i$ solve the stationary cell problem. In systems subjected to fast switching (coupling rates $\sim \varepsilon^{-1}$), a single effective equation for the common limit emerges, encapsulating homogenization and averaging effects over the switching processes (Mitake et al., 2012).

7. Applications and Numerical Illustration

Applications in optimal control, path planning under switching or random failures, front propagation in layered media, and stochastic regime optimization are modeled naturally in the weakly coupled HJB framework. The approach in (Chiri et al., 2023) demonstrates practical implications:

• Path Planning Under Breakdown: In a domain $\Omega = [0,2] \times [0,1]$ with spatially localized breakdown risk, computed value functions $u_1,u_2$ exhibit detouring around stochastic obstacles reflecting optimal risk-averse policies.
• FEM Implementation: Numerical schemes based on streamline-diffusion FEM with artificial viscosity yield stable and accurate approximations that capture the correct qualitative behavior, provided boundary and Aubry data are handled correctly.

This comprehensive theory, together with the weak KAM and Mather measure perspectives, provides foundational guarantees and concrete computational methods for analysis and simulation of weakly coupled HJB systems in deterministic and stochastic contexts (Chiri et al., 2023, Davini et al., 2012, Mitake et al., 2015, Camilli et al., 2011).