Hamilton–Jacobi Equations with Constraint

• Hamilton–Jacobi equations with constraint are defined by adding boundary, gradient, or normalization restrictions to the classical evolution.
• The formulations use advanced methodologies like viscosity solutions, Perron’s method, and ergodic analysis to address existence and uniqueness.
• Implications span various applications including control theory, optimal dividend problems, and singular stochastic control with precise regularity results.

Hamilton–Jacobi equations with constraint comprise several mathematically distinct but structurally related formulations in which the Hamilton–Jacobi evolution is coupled to an additional restriction on admissible gradients, states, controls, maxima, or phase-space variables. In current literature, the expression covers state-constraint problems posed on $\overline{\Omega}$, lower-bound gradient constraints such as $\min(-\Delta u-r,\lvert Du\rvert-1)=0$, normalization constraints of the form $\sup_x u(\cdot,t)=0$, Hamilton–Jacobi–Bellman equations with control or entry-cost constraints, and covariant or multi-time Hamilton–Jacobi systems generated by primary and secondary constraints of singular Lagrangians (Tu et al., 2023, Chang-Lara et al., 2020, Kim, 2018, Dao et al., 2019, Aguilar-Salas et al., 2023).

1. Principal meanings of “constraint”

The modern literature does not use a single canonical notion of constraint. Instead, the term denotes several mechanisms by which the Hamilton–Jacobi dynamics is restricted: boundary viability, gradient bounds, global normalization, admissible controls, or algebraic relations among polymomenta and gauge variables. Taken together, these works use “constraint” in several non-equivalent senses.

Constraint type	Representative equation	Representative source
State constraint	$H(x,Du_\lambda,\lambda u_\lambda)\le C_\lambda$ in $\Omega_\lambda$, $H(x,Du_\lambda,\lambda u_\lambda)\ge C_\lambda$ on $\overline{\Omega}_\lambda$	(Tu et al., 2023)
Gradient constraint	$\min(-\Delta u-r,\lvert Du\rvert-1)=0$	(Chang-Lara et al., 2020)
Global normalization	$u_t=H(Du)+R(x,I(t))$, $\sup_{\mathbb{R}^n}u(\cdot,t)=0$	(Kim, 2018)
Control constraint	$\min(-\Delta u-r,\lvert Du\rvert-1)=0$0	(Kundu et al., 2020)
Singular-system HJ constraint	$\min(-\Delta u-r,\lvert Du\rvert-1)=0$1	(Aguilar-Salas et al., 2023)

A second distinction concerns whether the constraint is part of the PDE itself or part of the dynamical interpretation. In state-constraint and gradient-constraint problems, the constraint is encoded directly in the viscosity inequalities. In normalization-constrained equations, an unknown scalar function $\min(-\Delta u-r,\lvert Du\rvert-1)=0$2 acts as a Lagrange multiplier chosen so that the spatial maximum remains fixed. In constrained mechanical and field-theoretic HJ formalisms, the Hamilton–Jacobi equations arise from singular Lagrangians and their integrability conditions rather than from a boundary-value problem in physical space (Kim, 2018, Aguilar-Salas et al., 2023).

2. State constraints, boundary viability, and state-constrained selection

In first-order PDE theory, a state constraint is typically encoded by requiring the subsolution inequality in the interior and the supersolution inequality on the closure. For the model $\min(-\Delta u-r,\lvert Du\rvert-1)=0$3 with $\min(-\Delta u-r,\lvert Du\rvert-1)=0$4, $\min(-\Delta u-r,\lvert Du\rvert-1)=0$5, the constrained problem is

$\min(-\Delta u-r,\lvert Du\rvert-1)=0$6

with $\min(-\Delta u-r,\lvert Du\rvert-1)=0$7 open, bounded, and connected, and with the boundary condition understood in the Soner-type viscosity sense (Han, 2022). The same closure-based formulation appears in nonlinear contact-type families

$\min(-\Delta u-r,\lvert Du\rvert-1)=0$8

where $\min(-\Delta u-r,\lvert Du\rvert-1)=0$9 and $\sup_x u(\cdot,t)=0$0 (Tu et al., 2023).

The analytic consequences are substantial. Under $\sup_x u(\cdot,t)=0$1, $\sup_x u(\cdot,t)=0$2, $\sup_x u(\cdot,t)=0$3, and $\sup_x u(\cdot,t)=0$4, a comparison principle holds for the critical state-constraint problem, and Perron’s method yields existence (Tu et al., 2023). For $\sup_x u(\cdot,t)=0$5, local semiconcavity depends on the first time a minimizing curve hits the boundary; under the conditions

$\sup_x u(\cdot,t)=0$6

every minimizing curve has infinite hitting time, and the constrained viscosity solution is globally semiconcave (Han, 2022).

The asymptotic selection problem for state-constrained equations introduces an additional ergodic layer. When $\sup_x u(\cdot,t)=0$7 is continuous, convex and coercive in $\sup_x u(\cdot,t)=0$8, monotone in $\sup_x u(\cdot,t)=0$9, and the domains $H(x,Du_\lambda,\lambda u_\lambda)\le C_\lambda$0 are star-shaped perturbations of $H(x,Du_\lambda,\lambda u_\lambda)\le C_\lambda$1, the solutions $H(x,Du_\lambda,\lambda u_\lambda)\le C_\lambda$2 converge locally uniformly to a selected critical solution $H(x,Du_\lambda,\lambda u_\lambda)\le C_\lambda$3, characterized as the maximal element of a set $H(x,Du_\lambda,\lambda u_\lambda)\le C_\lambda$4 defined through Mather measures and the linearizations $H(x,Du_\lambda,\lambda u_\lambda)\le C_\lambda$5 and $H(x,Du_\lambda,\lambda u_\lambda)\le C_\lambda$6 (Tu et al., 2023). This places state constraints in direct contact with weak KAM theory, ergodic constants, and vanishing-discount selection.

3. Global maximum constraints and unknown Lagrange multipliers

A different use of “constraint” appears in time-dependent equations where an unknown function of time is chosen to keep the spatial maximum equal to zero. The basic model is

$H(x,Du_\lambda,\lambda u_\lambda)\le C_\lambda$7

with $H(x,Du_\lambda,\lambda u_\lambda)\le C_\lambda$8 acting as a Lagrange multiplier (Kim, 2018). Under the assumptions $H(x,Du_\lambda,\lambda u_\lambda)\le C_\lambda$9, $\Omega_\lambda$0, $\Omega_\lambda$1, and $\Omega_\lambda$2 locally Lipschitz with $\Omega_\lambda$3, existence is obtained by solving a relaxed fixed-point problem for $\Omega_\lambda$4 and passing to the limit (Kim, 2018).

Uniqueness is subtler. The same paper proves uniqueness only under additional structural forms such as

$\Omega_\lambda$5

or

$\Omega_\lambda$6

with $\Omega_\lambda$7 and $\Omega_\lambda$8 (Kim, 2018). It also provides an example with infinitely many solutions when the reaction term is not strictly decreasing in $\Omega_\lambda$9. This is a recurrent misconception in the area: the maximum constraint alone does not guarantee uniqueness.

In one space dimension, the constrained equation

$H(x,Du_\lambda,\lambda u_\lambda)\ge C_\lambda$0

admits uniqueness of the pair $H(x,Du_\lambda,\lambda u_\lambda)\ge C_\lambda$1 for a class of nonseparable $H(x,Du_\lambda,\lambda u_\lambda)\ge C_\lambda$2 under assumptions including strict decrease of $H(x,Du_\lambda,\lambda u_\lambda)\ge C_\lambda$3 in $H(x,Du_\lambda,\lambda u_\lambda)\ge C_\lambda$4, monotonicity of $H(x,Du_\lambda,\lambda u_\lambda)\ge C_\lambda$5 on $H(x,Du_\lambda,\lambda u_\lambda)\ge C_\lambda$6, and strictly increasing $H(x,Du_\lambda,\lambda u_\lambda)\ge C_\lambda$7 (Kim, 2018). The proof uses the dynamic programming principle, semiconvexity of $H(x,Du_\lambda,\lambda u_\lambda)\ge C_\lambda$8, and the identity $H(x,Du_\lambda,\lambda u_\lambda)\ge C_\lambda$9 at maximizers.

Under stronger concavity assumptions, the regulator problem becomes an ODE–PDE system. For

$\overline{\Omega}_\lambda$0

with $\overline{\Omega}_\lambda$1 and $\overline{\Omega}_\lambda$2 uniformly strictly concave in $\overline{\Omega}_\lambda$3, the solution is not merely viscosity but classical,

$\overline{\Omega}_\lambda$4

and the maximizer $\overline{\Omega}_\lambda$5 satisfies

$\overline{\Omega}_\lambda$6

This classical reduction is the basis of the constructive fixed-point scheme in that paper.

4. Lower-bound gradient constraints and non-convex free boundaries

A distinct branch of the theory studies Hamilton–Jacobi equations with gradient constraints that impose a lower bound on $\overline{\Omega}_\lambda$7. The model problem is

$\overline{\Omega}_\lambda$8

and more generally

$\overline{\Omega}_\lambda$9

with $\min(-\Delta u-r,\lvert Du\rvert-1)=0$0 open and $\min(-\Delta u-r,\lvert Du\rvert-1)=0$1 of positive Lebesgue measure (Chang-Lara et al., 2020). The operator

$\min(-\Delta u-r,\lvert Du\rvert-1)=0$2

is non-convex in $\min(-\Delta u-r,\lvert Du\rvert-1)=0$3, in contrast with classical convex gradient-constraint models of the form $\min(-\Delta u-r,\lvert Du\rvert-1)=0$4.

The equation naturally separates into two regimes. On $\min(-\Delta u-r,\lvert Du\rvert-1)=0$5, the “Brownian region,” one has $\min(-\Delta u-r,\lvert Du\rvert-1)=0$6. On $\min(-\Delta u-r,\lvert Du\rvert-1)=0$7, the “eikonal region,” the constraint is active and $\min(-\Delta u-r,\lvert Du\rvert-1)=0$8 (Chang-Lara et al., 2020). This induces a free boundary between the Poisson phase and the eikonal phase. The paper develops comparison by doubling variables, constructs strict supersolutions via

$\min(-\Delta u-r,\lvert Du\rvert-1)=0$9

and obtains existence and uniqueness on bounded domains with an exterior cone condition by Perron’s method.

The central regularity result is optimal local Lipschitz continuity: $u_t=H(Du)+R(x,I(t))$0 together with continuity of the gradient norm,

$u_t=H(Du)+R(x,I(t))$1

for every viscosity solution of $u_t=H(Du)+R(x,I(t))$2 (Chang-Lara et al., 2020). The proof uses smooth approximations

$u_t=H(Du)+R(x,I(t))$3

Bernstein-type gradient bounds, an improvement-of-flatness versus diminish-of-oscillation scheme, and Savin’s flatness theory for degenerate elliptic equations.

The regularity threshold is sharp in the sense stated by the authors: Lipschitz is optimal, and higher regularity such as $u_t=H(Du)+R(x,I(t))$4 is not expected in this lower-bound, non-convex setting (Chang-Lara et al., 2020). Even though directional derivatives may jump, $u_t=H(Du)+R(x,I(t))$5 remains continuous. The model is connected to optimal dividends for multiple insurance companies and to singular stochastic control in reversible investment, where lower-bound gradient constraints arise from admissible dividend or intervention policies.

5. Control, games, junctions, and perforated geometries

In control theory, constraints often enter through the admissible control set. For deterministic and stochastic infinite-horizon HJB equations with (u(t)\