Constraint Filtering & Decay Mechanism

• Constraint filtering is a technique that systematically projects or optimizes over admissible sets to enforce safety and maintain numerical constraints in dynamic systems.
• The approach integrates decay mechanisms to quantify the rate at which system dynamics move away from constraint boundaries, balancing feasibility and minimal intervention.
• Applications span safety-critical control, numerical PDE discretizations, and hybrid systems, ensuring forward invariance, convergence, and practical performance under complex conditions.

Constraint filtering is a family of methods that enforce critical system or numerical constraints—such as safety, physical admissibility, or mathematical invariants—by systematically projecting or optimizing over admissible sets. When combined with decay mechanisms, these techniques ensure the enforcement not only of hard constraints (such as set invariance or non-negativity) but also of quantitative rates at which dynamics progress toward or away from constraint boundaries. This dual structure arises across control-theoretic safety filtering, numerical PDE discretizations, and event-triggered hybrid systems.

1. Constraint Filtering via Control Barrier Functions

In modern safety-critical control for continuous-time, control-affine systems

$$\dot{x} = f(x) + g(x)u\,,\qquad x\in\mathbb{R}^n,\, u\in\mathbb{R}^m,$$

constraint filtering proceeds by imposing a barrier function $h(x)$ that defines the safe set $C = \{x \mid h(x) \geq 0\}$ and requiring forward invariance of this set. The classical Control Barrier Function (CBF) approach enforces the decay condition

$L_f h(x) + L_g h(x) u \geq -\alpha(h(x)),$

for some class-$\mathcal{K}^e$ function $\alpha$. Here, "constraint filtering" manifests as the quadratic program (QP)

$$k(x) = \arg\min_{u} \frac{1}{2} \|u - u_{\rm des}(x)\|^2_W \quad \textrm{s.t.}\quad L_f h(x) + L_g h(x) u \geq -\alpha(h(x)),$$

which selects the closest admissible control $u$ to a user-prescribed nominal control $u_{\rm des}(x)$, filtered so as to enforce the CBF constraint and, thus, safety. This safety mandate is global in the sense that it holds on the entire safe set $C$, with invariance provably guaranteed by this procedure (Ong et al., 17 Jul 2025, Zeng et al., 2021).

2. Generalized Decay Conditions: The Optimal-Decay Mechanism

The classical CBF approach places all burden on the chosen $\alpha$, which can lead to infeasibility or excessive conservatism—especially under actuator saturation or conflicting objectives. To mitigate this, the optimal-decay CBF (OD-CBF) introduces an online-optimized scalar decay parameter $\omega\geq 0$, leading to the generalized constraint

$L_f h(x) + L_g h(x) u \geq -\omega \alpha(h(x)).$

The filtering now occurs not only over $u$ but also over $\omega$, balancing minimal intervention against feasibility: $$(u, \omega) = \arg\min_{u,\,\omega} \tfrac{1}{2}\|u-u_{\rm des}(x)\|^2_W + \tfrac{1}{2} p (\omega - \omega_{\rm ref})^2 \quad \textrm{s.t.}\quad L_f h(x) + L_g h(x) u \geq -\omega \alpha(h(x)),\,\, \omega \geq \omega_{\rm ref} \geq 0.$$ Here, $p$ penalizes deviation from a nominal decay $\omega_{\rm ref}$. The resulting OD-CBF law guarantees

• Pointwise feasibility on the entire safe set, provided $(h(x)=0) \land (L_g h(x)=0) \implies L_f h(x)>0$.
• Local Lipschitz continuity and existence of closed-form control updates (Ong et al., 17 Jul 2025, Zeng et al., 2021).

This scaling mechanism ensures the constraint filter is as non-intrusive as possible while always admitting a solution except precisely at unsafe states. In high-dimensional systems, the KKT conditions provide analytic expressions for both $u$ and $\omega$, supporting real-time implementation.

3. Constraint Filtering in Numerical PDEs with Decay

In numerical PDE contexts, particularly the discretization of anisotropic diffusion–decay problems,

$$\alpha(x) c(x) - \nabla \cdot (\mathbb{D}(x)\nabla c(x)) = f(x),$$

standard discretizations (e.g., Galerkin FEM) frequently violate discrete maximum/minimum principles, leading to nonphysical negative concentrations even when the continuous system satisfies strict non-negativity (Nagarajan et al., 2010). Here, "constraint filtering" is imposed via bound-constrained quadratic programming: $$\min_{c \in \mathbb{R}^N} \frac{1}{2} c^\top K c - c^\top f \quad \textrm{s.t.} \quad 0 \leq c_i \leq \infty,$$ or more generally, $c_{\min} \leq c \leq c_{\max}$. An active-set strategy is used to enforce these constraints efficiently.

This approach restores compliance with the maximum principle and non-negativity on arbitrary (potentially highly anisotropic) meshes:

• Existence and uniqueness of the constraint-filtered solution are guaranteed.
• The algorithm converges in finitely many steps and introduces modest computational overhead (typically 2–6 extra linear solves per time step).
• The filtered solution matches the optimal energy and $L_2$/Sobolev error rates of the unconstrained Galerkin method.

Constraint filtering becomes essential for coarse or unstructured meshes, particularly for large decay coefficients or anisotropic diffusion where mesh refinement alone cannot guarantee admissibility (Nagarajan et al., 2010).

4. Event-Triggered and Hybrid Filtering with Decay Guarantees

Constraint filtering is equally crucial in event-triggered and hybrid systems, where safety filters are intermittently enabled. Two key triggers are defined: (1) activation of filtering when the nominal controller is predicted to imminently violate the safety barrier, and (2) deactivation once adequate margin is restored.

To guarantee and quantify the recovery back to the nominal controller, a decay or "constraint-improvement" mechanism is enforced while the filter is active: $L_f h(x, u) - \|\nabla h(x)\| \bar{d} \geq b$ for some fixed $b>0$, which ensures linear growth of $h(x)$ out of the dangerous region. The use of this mechanism enables proving that the safety filter is needed only for at most $T \leq c/b$ time before regaining safety margin $c$, and Zeno phenomena are excluded due to finite minimum dwell times (Ong et al., 2023).

The complete filter-switching law ensures:

• Forward invariance of the safe set under the combined nominal-filter policy.
• Guaranteed minimum dwell times between ON/OFF events due to strict increments in margin under the decay-improvement mechanism.

5. Decay Mechanism and Modal Constraint Filtering in Damped PDEs

In the context of PDE semi-discretizations, decay mechanisms interact with constraint filtering through modal projection. Standard finite-difference or finite-element approximations of boundary-damped wave equations display loss of exponential stability due to accumulation of spurious high-frequency modes as discretization is refined. By applying a direct Fourier filtering operator $P_\sigma$ — projecting state variables onto the subspace of low-frequency modes determined by a spectral cutoff parameter $\sigma$ — one enforces a spectral constraint which restores uniform decay: $$\dot{y}_h = P_\sigma \left( \mathcal{A}_h(K) y_h \right), \quad y_h(0) \in \operatorname{Ran} P_\sigma$$ where $\mathcal{A}_h(K)$ is the discrete generator and $P_\sigma$ nulls all eigenmodes with $h^2|\lambda_k|^2/(4c^2) > \sigma$ (FD) or $h^2|\lambda_k|^2/(12c^2) > \sigma$ (FEM).

A Lyapunov analysis yields uniform exponential decay

$E_h(t) \leq M e^{-\alpha t} E_h(0)$

with explicit decay rate

$$\alpha_{\max}(K,\sigma) = \delta(K) [1 - (L/c)\delta(K)] (1-\sigma).$$

Filtering is thus essential for maintaining decay guarantees as $h \to 0$, with $\sigma$ tuned according to allowable modes (Ozer et al., 2023).

6. Practical Trade-Offs, Tuning, and Implementation

Choice of the decay function $\alpha(s)$ (linear or superlinear) and penalty weight $p$ for deviation of $\omega$ influence aggressiveness and performance in OD-CBF and related filtering strategies. Linear $\alpha$ leads to exponential decay; superlinear forms deliver faster recovery for large constraint violations but diminished correction near boundaries.

When additional constraints—such as actuation bounds or numerical non-negativity—are present, further regularization (introduction of slack variables and penalization) may be required to preserve feasibility and stability (Ong et al., 17 Jul 2025, Zeng et al., 2021, Nagarajan et al., 2010).

In numerical PDEs, filtered active-set QP solvers using low-order elements (e.g., $P_1$) ensure enforceability of bounds between nodes; initialization of violations can accelerate convergence. Filtering should be employed whenever the mesh is not sufficiently fine or regular to avoid violations naturally, or when anisotropy impedes mesh-based guarantees (Nagarajan et al., 2010).

7. Illustrative Examples

• Satellite Control via OD-HOCBF: In simulation, OD-HOCBF filters maintain strict radial safety ($1.8R \leq r \leq 2.2R$) for two days, with the decay-rate multiplier $\omega(x)$ rising only as the system approaches constraint boundaries, and otherwise remaining near the baseline value. Closed-form updates for $(u, \omega)$ yield smooth and minimum-gain control compared to manually tuned alternatives (Ong et al., 17 Jul 2025).
• Adaptive Cruise Control: OD-CBF-based filtering dynamically increases $\omega$ to prevent constraint infeasibility when the classical QP with fixed decay would fail (e.g., under sudden lead-vehicle deceleration), guaranteeing safety at the expense of only temporary performance deviation (Zeng et al., 2021).
• Diffusion with Decay: For highly anisotropic diffusion-transport with decay coefficient $\alpha \gtrsim 1$, constraint filtering eliminates persistent negative nodes across mesh refinements where standard Galerkin solutions fail to recover non-negativity or the discrete maximum principle (Nagarajan et al., 2010).

In all cases, constraint filtering, coupled with explicit or online-controlled decay mechanisms, delivers provable guarantees of feasibility, set invariance, and, where applicable, uniform stabilization or non-negativity, irrespective of system or discretization peculiarities.