Hierarchical Controlled Differential Equations

• H-CDE is a family of hierarchical continuous-time control models that integrate layered strategies to tackle PDE controllability and latent degradation inference.
• It employs a Stackelberg and multiscale framework where leaders, followers, or slow-fast processes interact through best-response mappings and learned path transformations.
• The approach achieves precise target tracking and state estimation by leveraging adjoint system analysis, density principles, and numerically stable multiscale integration.

Searching arXiv for the named H-CDE papers and closely related hierarchical-control formulations. Hierarchical Controlled Differential Equation (H-CDE) denotes a class of continuous-time controlled systems in which the governing dynamics are coupled to a hierarchy rather than a single flat control law. In the PDE-control literature, the term refers to Stackelberg or Stackelberg–Nash formulations in which a leader and one or more followers act through distinct control channels and optimize different objectives under a common state equation; the canonical instance is a degenerate parabolic PDE associated with a chain of distributed subsystems (Befekadu et al., 2015). In recent machine-learning work, the same label is used for a two-level controlled differential architecture that couples a slow degradation process to a fast operational process on separate time grids in order to disentangle latent degradation from operational variability (Zhao et al., 30 Aug 2025). The common structural feature is hierarchical coupling: either between decision-makers, or between latent dynamical layers.

1. Terminological scope and defining features

The term H-CDE is not attached to a single universally standardized formalism across the literature summarized here. Instead, it appears in at least two technically distinct but structurally related settings.

In the control-theoretic PDE setting, an H-CDE is a controlled differential equation system whose state evolves according to a PDE and whose controls are organized hierarchically. The leader typically solves a controllability-type problem, while the follower or followers solve regulation or tracking problems. The hierarchy is explicit: the follower computes a best response to the leader’s announced strategy, and the leader solves its own optimization problem while anticipating that response. This viewpoint is developed for degenerate parabolic chains (Befekadu et al., 2015), one-dimensional nonlinear parabolic equations (Nuñez-Chávez et al., 2019), viscoelastic Oldroyd systems with memory (Jesus et al., 11 Jan 2025), and wave equations on moving domains (Jesus, 26 Jan 2025).

In the degradation-inference setting, H-CDE denotes a hierarchical continuous-time latent-variable model built from two coupled controlled differential equations. A slow CDE encodes degradation progression, and a fast CDE encodes operational dynamics conditioned on the slow state. The hierarchy is therefore multiscale rather than strategic: the slow layer supplies latent degradation information to the fast layer, while the slow layer itself is driven by a learned path transformation of operational history (Zhao et al., 30 Aug 2025).

These usages differ in mathematical object and research aim. The PDE literature is centered on controllability, observability, equilibrium existence, adjoint systems, and variational inequalities. The degradation-inference literature is centered on slow-fast separation, stiffness mitigation, latent disentanglement, and forecasting/tracking performance. A plausible implication is that H-CDE should be understood as a structural label for hierarchical continuous-time control architectures, not as the name of a single fixed model class.

2. Stackelberg H-CDE for a chain of distributed systems

The formulation in "On the hierarchical optimal control of a chain of distributed systems" is the foundational PDE instance of H-CDE (Befekadu et al., 2015). The state is governed on a bounded regular domain $\Omega \subset \mathbb{R}^N$, with $N = nd$, over a finite horizon $[0,T]$, by a degenerate parabolic equation with Dirichlet boundary conditions and controls acting on two disjoint open subdomains $\omega_1,\omega_2 \subset \Omega$ such that $\omega_1 \cap \omega_2 = \varnothing$. In the paper’s controlled form,

$$\partial_t y + \mathcal{L}_{t,x} y = u_1\,\mathbf{1}_{\omega_1} + u_2\,\mathbf{1}_{\omega_2}, \qquad y(0,\cdot)=0, \qquad y|_{\Sigma}=0,$$

with admissible controls

$$u_1 \in L^2\big((0,T)\times \omega_1\big), \qquad u_2 \in L^2\big((0,T)\times \omega_2\big).$$

The operator $\mathcal{L}_{t,x}$ has cascade structure:

$$\mathcal{L}_{t,x} y = \frac{1}{2}\,\mathrm{tr}\!\big(a(t,x)\,D_{x^1}^2 y\big) + f_1(t,x)\cdot D_{x^1} y + \sum_{j=2}^n f_j\!\big(t,x^{j-1}\big)\cdot D_{x^j} y.$$

Diffusion enters only through the first subsystem $x^1$, while downstream components are coupled through drift terms. The resulting PDE is therefore degenerate parabolic. The paper assumes a weak Hörmander-type condition under which the Lie algebra generated by the diffusion vector fields and the drift has full rank $N = nd$0 at every $N = nd$1. This yields hypoellipticity: regularity injected in the first block propagates through the chain by the triangular drift couplings.

The state equation is motivated by a chain of stochastic and deterministic subsystems. The first subsystem is stochastically forced, while the downstream subsystems evolve deterministically but are coupled to upstream states. The PDE thus encodes a Fokker–Planck-type generator for a cascade in which diffusion acts only in the first block and controllability or accessibility propagates through the chain.

The hierarchy is tied to two different objectives. The leader acts on $N = nd$2 and is associated with terminal controllability to a target set

$N = nd$3

where $N = nd$4 is the unit ball in $N = nd$5 and $N = nd$6 is small. The follower acts on $N = nd$7 and aims to keep the state close to a prescribed reference trajectory $N = nd$8 over the entire interval. The partition of the control domain therefore models separated actuators or authorities, while the Stackelberg structure encodes asymmetric information and decision flow.

3. Best-response maps, adjoint systems, and controllability

The H-CDE formulation in the chain-of-systems setting is a bilevel optimization problem. The follower solves a quadratic tracking problem for a fixed leader control:

$N = nd$9

The leader minimizes control effort subject to a terminal constraint:

$[0,T]$0

The follower’s decision is therefore subordinate to the leader’s choice, but the leader anticipates the follower’s optimal reaction (Befekadu et al., 2015).

For a given $[0,T]$1, the follower is characterized by a forward-backward optimality system:

$[0,T]$2

$[0,T]$3

with

$[0,T]$4

The formal adjoint is

$[0,T]$5

Stationarity gives the follower’s best-response correspondence

$[0,T]$6

The leader’s problem introduces an additional adjoint pair $[0,T]$7 and a terminal multiplier $[0,T]$8. The optimal leader control has the representation

$[0,T]$9

where $\omega_1,\omega_2 \subset \Omega$0 solve the four-equation optimality system

$\omega_1,\omega_2 \subset \Omega$1

$\omega_1,\omega_2 \subset \Omega$2

$\omega_1,\omega_2 \subset \Omega$3

$\omega_1,\omega_2 \subset \Omega$4

with

$\omega_1,\omega_2 \subset \Omega$5

and homogeneous boundary conditions on $\omega_1,\omega_2 \subset \Omega$6.

The multiplier $\omega_1,\omega_2 \subset \Omega$7 is determined by the variational inequality

$\omega_1,\omega_2 \subset \Omega$8

This terminal variational condition links the reachable target set $\omega_1,\omega_2 \subset \Omega$9 to the leader’s optimal strategy and, through the coupled state-adjoint equations, to the follower’s achievable tracking performance. The controllability-type conclusion is explicit: for every target $\omega_1 \cap \omega_2 = \varnothing$0 and every $\omega_1 \cap \omega_2 = \varnothing$1, there exists $\omega_1 \cap \omega_2 = \varnothing$2 such that

$\omega_1 \cap \omega_2 = \varnothing$3

The analysis combines hypoellipticity, unique continuation, and convex duality. The paper emphasizes weak Hörmander propagation and unique continuation of Mizohata type, rather than explicit Carleman inequalities, to establish density and controllability properties. It also remarks that the framework extends to multi-level hierarchies with one leader and $\omega_1 \cap \omega_2 = \varnothing$4 followers.

4. Related hierarchical PDE formulations

The same hierarchical-control principle has been developed for several other PDE classes. These formulations broaden the meaning of H-CDE within control theory by varying the state equation while preserving the leader–follower architecture.

For one-dimensional nonlinear parabolic equations, "Hierarchic Controllability for a nonlinear parabolic equation in one dimension" studies a Stackelberg–Nash structure with one leader and two followers (Nuñez-Chávez et al., 2019). The controlled PDE is

$\omega_1 \cap \omega_2 = \varnothing$5

with homogeneous Dirichlet boundary conditions. Each follower minimizes a quadratic observation-and-control functional, and the leader enforces null controllability, possibly to a prescribed trajectory. The follower feedback laws are

$\omega_1 \cap \omega_2 = \varnothing$6

The main results include existence of a Nash quasi-equilibrium, conditions under which the quasi-equilibrium is a true Nash equilibrium, linear null controllability via Carleman and HUM techniques, and a nonlinear local result obtained through a Right Inverse Function Theorem.

For viscoelastic fluids with memory, "Hierarchical Control for the Oldroyd Equation in Memoriam to Professor Luiz Adauto Medeiros" places one leader and several followers in a Stackelberg–Nash game over the linearized Oldroyd system (Jesus et al., 11 Jan 2025). The state equation is

$\omega_1 \cap \omega_2 = \varnothing$7

with $\omega_1 \cap \omega_2 = \varnothing$8, no-slip boundary conditions, and exponential memory kernel $\omega_1 \cap \omega_2 = \varnothing$9. The followers minimize terminal tracking functionals

$$\partial_t y + \mathcal{L}_{t,x} y = u_1\,\mathbf{1}_{\omega_1} + u_2\,\mathbf{1}_{\omega_2}, \qquad y(0,\cdot)=0, \qquad y|_{\Sigma}=0,$$0

and are characterized by the backward memory adjoint through

$$\partial_t y + \mathcal{L}_{t,x} y = u_1\,\mathbf{1}_{\omega_1} + u_2\,\mathbf{1}_{\omega_2}, \qquad y(0,\cdot)=0, \qquad y|_{\Sigma}=0,$$1

Under small $$\partial_t y + \mathcal{L}_{t,x} y = u_1\,\mathbf{1}_{\omega_1} + u_2\,\mathbf{1}_{\omega_2}, \qquad y(0,\cdot)=0, \qquad y|_{\Sigma}=0,$$2 and bounded measurable weights $$\partial_t y + \mathcal{L}_{t,x} y = u_1\,\mathbf{1}_{\omega_1} + u_2\,\mathbf{1}_{\omega_2}, \qquad y(0,\cdot)=0, \qquad y|_{\Sigma}=0,$$3, the paper proves existence and uniqueness of the followers’ Nash equilibrium, density of the reachable set $$\partial_t y + \mathcal{L}_{t,x} y = u_1\,\mathbf{1}_{\omega_1} + u_2\,\mathbf{1}_{\omega_2}, \qquad y(0,\cdot)=0, \qquad y|_{\Sigma}=0,$$4, and a leader optimality characterization

$$\partial_t y + \mathcal{L}_{t,x} y = u_1\,\mathbf{1}_{\omega_1} + u_2\,\mathbf{1}_{\omega_2}, \qquad y(0,\cdot)=0, \qquad y|_{\Sigma}=0,$$5

The analysis relies on Lax–Milgram, Fenchel–Rockafellar duality, Faedo–Galerkin well-posedness, and a unique continuation lemma for viscoelastic fluids.

For hyperbolic dynamics on non-cylindrical domains, "Hierarchical Control for the Wave Equation with a Moving Boundary" studies the one-dimensional wave equation on

$$\partial_t y + \mathcal{L}_{t,x} y = u_1\,\mathbf{1}_{\omega_1} + u_2\,\mathbf{1}_{\omega_2}, \qquad y(0,\cdot)=0, \qquad y|_{\Sigma}=0,$$6

with boundary control at the fixed endpoint split into leader and follower components over disjoint time subsets (Jesus, 26 Jan 2025). After the change of variables $$\partial_t y + \mathcal{L}_{t,x} y = u_1\,\mathbf{1}_{\omega_1} + u_2\,\mathbf{1}_{\omega_2}, \qquad y(0,\cdot)=0, \qquad y|_{\Sigma}=0,$$7, the transformed equation becomes

$$\partial_t y + \mathcal{L}_{t,x} y = u_1\,\mathbf{1}_{\omega_1} + u_2\,\mathbf{1}_{\omega_2}, \qquad y(0,\cdot)=0, \qquad y|_{\Sigma}=0,$$8

on a fixed cylinder. The follower minimizes a tracking-plus-control cost and satisfies

$$\partial_t y + \mathcal{L}_{t,x} y = u_1\,\mathbf{1}_{\omega_1} + u_2\,\mathbf{1}_{\omega_2}, \qquad y(0,\cdot)=0, \qquad y|_{\Sigma}=0,$$9

while the leader satisfies

$$u_1 \in L^2\big((0,T)\times \omega_1\big), \qquad u_2 \in L^2\big((0,T)\times \omega_2\big).$$0

When $$u_1 \in L^2\big((0,T)\times \omega_1\big), \qquad u_2 \in L^2\big((0,T)\times \omega_2\big).$$1 and $$u_1 \in L^2\big((0,T)\times \omega_1\big), \qquad u_2 \in L^2\big((0,T)\times \omega_2\big).$$2, the reachable terminal set is dense in $$u_1 \in L^2\big((0,T)\times \omega_1\big), \qquad u_2 \in L^2\big((0,T)\times \omega_2\big).$$3. The proof uses duality and Holmgren’s uniqueness theorem rather than explicit Carleman estimates.

Taken together, these works show that in PDE control the H-CDE idea is not tied to a single operator class. Degenerate parabolic chains, nonlinear parabolic equations, viscoelastic equations with memory, and moving-boundary wave equations all admit hierarchical formulations in which equilibrium maps, adjoint systems, and controllability conditions are central.

5. Slow-fast H-CDE for degradation inference

In "Disentangling Slow and Fast Temporal Dynamics in Degradation Inference with Hierarchical Differential Models", H-CDE is used in a different but mathematically explicit sense: a hierarchical pair of coupled controlled differential equations designed to separate latent degradation from fast operational dynamics in sensor streams (Zhao et al., 30 Aug 2025). The problem setting is slow-fast: operational and environmental factors generate large-amplitude short-term fluctuations, while degradation evolves gradually and is rarely directly observable. Residual-based approaches are described as entangling operational history with degradation, especially in dynamic-response systems.

The model introduces two latent states,

$$u_1 \in L^2\big((0,T)\times \omega_1\big), \qquad u_2 \in L^2\big((0,T)\times \omega_2\big).$$4

for fast operation and slow degradation, respectively. The fast layer is a degradation-aware CDE,

$$u_1 \in L^2\big((0,T)\times \omega_1\big), \qquad u_2 \in L^2\big((0,T)\times \omega_2\big).$$5

where $$u_1 \in L^2\big((0,T)\times \omega_1\big), \qquad u_2 \in L^2\big((0,T)\times \omega_2\big).$$6 is constructed from measured fast states and control inputs augmented by the interpolated slow state. The slow layer is

$$u_1 \in L^2\big((0,T)\times \omega_1\big), \qquad u_2 \in L^2\big((0,T)\times \omega_2\big).$$7

where $$u_1 \in L^2\big((0,T)\times \omega_1\big), \qquad u_2 \in L^2\big((0,T)\times \omega_2\big).$$8 is a learnable path transformation intended to extract latent degradation drivers from operational history. Initial states are produced by small MLPs $$u_1 \in L^2\big((0,T)\times \omega_1\big), \qquad u_2 \in L^2\big((0,T)\times \omega_2\big).$$9 and $\mathcal{L}_{t,x}$0, and decoding is performed by a readout from the fast latent state, with the slow state available as an auxiliary health index.

A central design element is the path transformation. At each slow step,

$\mathcal{L}_{t,x}$1

and the sequence $\mathcal{L}_{t,x}$2 is interpolated by natural cubic splines to form a differentiable slow control path $\mathcal{L}_{t,x}$3, which is identified with $\mathcal{L}_{t,x}$4. The purpose is to align the slow driver with cumulative usage, stress, fatigue, or related latent causes rather than with raw high-frequency fluctuations.

The slow operator is regularized by the monotonicity-enforcing activation

$\mathcal{L}_{t,x}$5

The slow dynamics are written as

$\mathcal{L}_{t,x}$6

According to the paper, $\mathcal{L}_{t,x}$7, $\mathcal{L}_{t,x}$8, and for large $\mathcal{L}_{t,x}$9 negative outputs are strongly suppressed, yielding approximate non-negativity when the slow driver and vector field are non-negative on the relevant domain. No explicit monotonic penalty is used in the reported experiments.

Time integration is explicitly multiscale. A slow grid $$\mathcal{L}_{t,x} y = \frac{1}{2}\,\mathrm{tr}\!\big(a(t,x)\,D_{x^1}^2 y\big) + f_1(t,x)\cdot D_{x^1} y + \sum_{j=2}^n f_j\!\big(t,x^{j-1}\big)\cdot D_{x^j} y.$$0 with coarse step $$\mathcal{L}_{t,x} y = \frac{1}{2}\,\mathrm{tr}\!\big(a(t,x)\,D_{x^1}^2 y\big) + f_1(t,x)\cdot D_{x^1} y + \sum_{j=2}^n f_j\!\big(t,x^{j-1}\big)\cdot D_{x^j} y.$$1 and a fast grid $$\mathcal{L}_{t,x} y = \frac{1}{2}\,\mathrm{tr}\!\big(a(t,x)\,D_{x^1}^2 y\big) + f_1(t,x)\cdot D_{x^1} y + \sum_{j=2}^n f_j\!\big(t,x^{j-1}\big)\cdot D_{x^j} y.$$2 with fine step $$\mathcal{L}_{t,x} y = \frac{1}{2}\,\mathrm{tr}\!\big(a(t,x)\,D_{x^1}^2 y\big) + f_1(t,x)\cdot D_{x^1} y + \sum_{j=2}^n f_j\!\big(t,x^{j-1}\big)\cdot D_{x^j} y.$$3 are used. The slow CDE is integrated first, the resulting slow state is interpolated onto fast times, and the fast CDE is then integrated with the slow state treated as quasi-static within fast intervals. Both layers use adaptive explicit Runge–Kutta Dormand–Prince (dopri5) with $$\mathcal{L}_{t,x} y = \frac{1}{2}\,\mathrm{tr}\!\big(a(t,x)\,D_{x^1}^2 y\big) + f_1(t,x)\cdot D_{x^1} y + \sum_{j=2}^n f_j\!\big(t,x^{j-1}\big)\cdot D_{x^j} y.$$4 and $$\mathcal{L}_{t,x} y = \frac{1}{2}\,\mathrm{tr}\!\big(a(t,x)\,D_{x^1}^2 y\big) + f_1(t,x)\cdot D_{x^1} y + \sum_{j=2}^n f_j\!\big(t,x^{j-1}\big)\cdot D_{x^j} y.$$5; control paths are interpolated by natural cubic splines. Training uses the forecasting loss

$$\mathcal{L}_{t,x} y = \frac{1}{2}\,\mathrm{tr}\!\big(a(t,x)\,D_{x^1}^2 y\big) + f_1(t,x)\cdot D_{x^1} y + \sum_{j=2}^n f_j\!\big(t,x^{j-1}\big)\cdot D_{x^j} y.$$6

optimized with AdamW at learning rate $$\mathcal{L}_{t,x} y = \frac{1}{2}\,\mathrm{tr}\!\big(a(t,x)\,D_{x^1}^2 y\big) + f_1(t,x)\cdot D_{x^1} y + \sum_{j=2}^n f_j\!\big(t,x^{j-1}\big)\cdot D_{x^j} y.$$7, batch size $$\mathcal{L}_{t,x} y = \frac{1}{2}\,\mathrm{tr}\!\big(a(t,x)\,D_{x^1}^2 y\big) + f_1(t,x)\cdot D_{x^1} y + \sum_{j=2}^n f_j\!\big(t,x^{j-1}\big)\cdot D_{x^j} y.$$8, and ReduceLROnPlateau with factor $$\mathcal{L}_{t,x} y = \frac{1}{2}\,\mathrm{tr}\!\big(a(t,x)\,D_{x^1}^2 y\big) + f_1(t,x)\cdot D_{x^1} y + \sum_{j=2}^n f_j\!\big(t,x^{j-1}\big)\cdot D_{x^j} y.$$9 and patience $x^1$0.

The empirical evaluation is reported on a dynamic-response bridge dataset and on N-CMAPSS DS01. The bridge setting uses a simulated damped Euler–Bernoulli beam under traffic and thermal loads, with stiffness loss as degradation. The aero-engine setting uses the HPT efficiency modifier as ground-truth degradation for short-flight post-fault segments. The main metric is an Alignment Score, defined as the $x^1$1 of a linear regressor from latent states to ground-truth degradation.

Model	Bridge Alignment (ID / OOD)	N-CMAPSS Alignment (All / PC1)
Residual	0.324±0.054 / 0.282±0.036	0.802±0.024 / 0.566±0.072
H-CDE (w/o MC)	0.969±0.010 / 0.941±0.033	0.820±0.219 / 0.691±0.374
H-CDE (w/o PT)	0.954±0.010 / 0.820±0.045	0.003±0.064 / −0.026±0.057
H-CDE (full)	0.967±0.007 / 0.941±0.018	0.964±0.010 / 0.928±0.028

The reported slow-layer NFEs in the bridge setting are $x^1$2 (ID) and $x^1$3 (OOD) for full H-CDE, $x^1$4 (ID) and $x^1$5 (OOD) for H-CDE without monotonicity constraint, and $x^1$6 (ID) and $x^1$7 (OOD) for H-CDE without path transformation. The paper interprets this as evidence that monotonicity and path transformation stabilize slow integration and improve out-of-distribution behavior. Qualitatively, the full model yields a monotonic latent trend in the bridge case and smooth cycle-consistent degradation trajectories in N-CMAPSS. The implementation uses 2-layer MLP vector fields with hidden size $x^1$8, SiLU activations, no batch normalization or dropout, latent dimensions $x^1$9 for the bridge case and $N = nd$00 for N-CMAPSS, and parameter counts of approximately $N = nd$01 and $N = nd$02, respectively.

6. Conceptual interpretation, misconceptions, and open problems

A recurring misconception is that H-CDE denotes one specific mathematical model. The surveyed literature does not support that reading. In PDE control, H-CDE refers to hierarchical optimization over a differential equation, usually of Stackelberg or Stackelberg–Nash type, with best-response maps, adjoint systems, and controllability constraints (Befekadu et al., 2015). In degradation inference, H-CDE refers to a coupled slow-fast latent CDE architecture with a learned control path and monotonic regularization (Zhao et al., 30 Aug 2025). The shared word “hierarchical” therefore refers to different organizing principles: authority hierarchy among controllers in one case, and time-scale hierarchy among latent dynamics in the other.

Despite that difference, several structural motifs recur. First, the dynamics are partitioned into interacting subsystems or channels: disjoint control regions for leaders and followers in PDE control, or fast and slow latent states in the degradation model. Second, the hierarchy is closed through auxiliary equations: adjoint PDEs and variational inequalities in the control literature, and learned path transformations plus latent-state interpolation in the machine-learning literature. Third, well-posedness of the hierarchical closure is central. In the PDE papers this appears as existence and uniqueness of Nash equilibria or best responses; in the degradation paper it appears as numerically stable multiscale integration and monotonic latent evolution.

The open problems are likewise domain-specific. For the nonlinear parabolic Stackelberg–Nash problem, the extension from one spatial dimension to higher dimensions remains open because the proof relies on the embedding $N = nd$03 and on one-dimensional Carleman constructions (Nuñez-Chávez et al., 2019). For the moving-boundary wave equation, controllability arguments require subcharacteristic boundary speed $N = nd$04 and sufficiently large time; the paper notes difficulty when $N = nd$05 or $N = nd$06 (Jesus, 26 Jan 2025). For the Oldroyd system, extension to the nonlinear equation with the convective term is left open, and the paper notes that null controllability likely fails in general for memory systems (Jesus et al., 11 Jan 2025). For the degradation-inference H-CDE, the monotonicity prior is explicitly limited when degradation is non-monotone because of healing, maintenance resets, or abrupt faults, and the experiments are restricted to a single dominant failure mechanism (Zhao et al., 30 Aug 2025).

A plausible synthesis is that H-CDE names a family of hierarchical continuous-time constructions rather than a singular theory. In one branch, the hierarchy is game-theoretic and anchored in controllability analysis. In the other, it is multiscale and anchored in representation learning for latent degradation. The term’s technical meaning is therefore determined by context, but in both branches it denotes controlled differential dynamics whose effective behavior is shaped by a higher-level structure that cannot be reduced to a single unconstrained control input.