Trajectory Invariance in Dynamical Systems

• Trajectory invariance is a family of principles ensuring that the structural properties of system trajectories remain unchanged under various transformations across state spaces, Lie groups, and statistical ensembles.
• It enables robust control and estimation by preserving error dynamics, maintaining forward invariance in nonlinear tracking, and certifying safety in MPC through finite-horizon trajectory certificates.
• The concept spans multiple disciplines—from quantum mechanics and reaction theory to statistical physics and multiagent navigation—demonstrating its role in achieving reliable system behavior under symmetry and geometric invariance.

Trajectory invariance denotes several distinct but structurally related notions in contemporary research. In nonlinear control it can mean forward invariance of a time-varying set centered on a nonconstant reference trajectory; in controlled-invariance theory it can mean a finite-horizon certificate that induces an invariant set; in invariant filtering it appears as state trajectory independence of estimation-error dynamics; in quantum and field-theoretic settings it can denote invariance of trajectories under local scale transformations; in reaction theory it can mean preservation of local gauge invariance under replacement of a single exchange by a full Regge trajectory; and in statistical physics it can denote geometry-controlled or diffusive-limit properties of families of trajectories (Titze et al., 3 Dec 2025, Wembe et al., 8 Apr 2026, C. et al., 2024, Sen et al., 7 Jan 2026, Haberzettl et al., 2015, Binzoni et al., 2020). A plausible implication is that the phrase is best understood not as a single definition but as a family of invariance principles indexed by the space in which trajectories are compared: state space, trajectory space, Lie groups, homotopy classes, or statistical ensembles.

1. Principal meanings across current literatures

A recurrent control-theoretic meaning is invariance of a moving set attached to a reference motion. For a single-output nonlinear system in Byrnes–Isidori normal form, the relevant object is a tube

$$\mathcal T(t)=\mathcal V_{\xi_d,\mu,c_0}(t):=\Big\{\xi\in\mathbb R^{n_\xi}\,\big|\, V_N\big(\xi-\xi_d(t)\big)\le \nu_{\mu,c_0}(t)\Big\},$$

and trajectory invariance means that if the closed loop starts in $\mathcal T(t_0)$, then it remains in $\mathcal T(t)$ for all later times, even when perturbations are only locally bounded in a prescribed operating region (Titze et al., 3 Dec 2025).

A second meaning is controlled invariance encoded by finitely long trajectories. For linear discrete-time systems, a point is open-loop convex feasible if a finite trajectory remains in the constraint set and its terminal point lies strictly inside the convex hull of the earlier points,

$x_N \in Int\!\big(ch\{x_0,\dots,x_{N-1}\}\big),$

which yields a controlled invariant set obtained as the closed convex hull of the trajectory samples (Wembe et al., 8 Apr 2026).

A third meaning is autonomy of error dynamics. In invariant Kalman filtering on Lie groups, trajectory invariance is explicitly called state trajectory independence: the error dynamics depend only on the error and the inputs, not on the current estimate. For relative dynamics, this autonomy is what makes the propagated covariance robust to large estimation errors (C. et al., 2024).

Further usages are tied to symmetry. In pilot-wave quantum theory with complexified gauge coupling $e_C=e+i e_I$, Bohmian trajectories are invariant under local scale rescalings of the wavefunction, while the conserved density becomes the trajectory-dependent ratio $R^2/\mathds 1^2[\mathcal C]$ (Sen et al., 7 Jan 2026). In joint motion forecasting, trajectory invariance refers to permutation equivariance over agents together with spatial roto-translation invariance and temporal translation invariance in the encoder (Zhou et al., 2023). In reaction theory, “trajectory invariance” names preservation of the generalized Ward–Takahashi identity when a Feynman $t$-channel exchange is replaced by a Regge trajectory, provided the interaction current is reggeized consistently (Haberzettl et al., 2015).

2. Forward invariance along reference trajectories and planned trajectories

In nonlinear tracking control, trajectory invariance is developed most explicitly as invariance of a contracting tube around a nonconstant reference. For systems in Byrnes–Isidori form with output-related coordinates $z$ and internal coordinates $\eta$, feedback linearization with Lyapunov redesign yields the tracking-error dynamics

$$\dot{\bar\xi}=A\bar\xi+B\big(v_N(\bar\xi)+v_L(\bar\xi)+\Delta(\xi_d+\bar\xi,\eta,t)\big),$$

and the nominal Lyapunov function satisfies

$\mathcal T(t_0)$0

inside $\mathcal T(t_0)$1. The scalar comparison equation

$\mathcal T(t_0)$2

then generates a contracting tube along $\mathcal T(t_0)$3. If $\mathcal T(t_0)$4, the theorem in the paper guarantees

$\mathcal T(t_0)$5

for all admissible initial conditions (Titze et al., 3 Dec 2025). Relative to constant-reference Lyapunov redesign, the reference enters through $\mathcal T(t_0)$6, the tube is time-varying, and the certification condition applies to the full union of tube slices.

A discrete-time analogue appears in controlled invariant funnels. For locally Lipschitz nonlinear systems with bounded disturbances,

$\mathcal T(t_0)$7

the funnel is parameterized by ellipsoids

$\mathcal T(t_0)$8

around a nominal trajectory $\mathcal T(t_0)$9. Under the feedback $\mathcal T(t)$0, invariance means

$\mathcal T(t)$1

for all $\mathcal T(t)$2 and admissible disturbances. The paper enforces the contraction condition $\mathcal T(t)$3, $\mathcal T(t)$4, via an LMI obtained from an S-procedure, and concludes that every closed-loop trajectory starting in $\mathcal T(t)$5 remains in $\mathcal T(t)$6 for all $\mathcal T(t)$7 (Kim et al., 2022).

An explicitly trajectory-space formulation is given by Forward Invariance in Trajectory Spaces. Planned trajectories are lifted to a finite-dimensional trajectory state

$\mathcal T(t)$8

where $\mathcal T(t)$9 is a virtual input controlling the rate of change of the planned input trajectory. State-space safety constraints $x_N \in Int\!\big(ch\{x_0,\dots,x_{N-1}\}\big),$0 are converted into a trajectory-space safe set

$x_N \in Int\!\big(ch\{x_0,\dots,x_{N-1}\}\big),$1

and forward invariance of $x_N \in Int\!\big(ch\{x_0,\dots,x_{N-1}\}\big),$2 is enforced by a CBF-type QP with inequalities

$x_N \in Int\!\big(ch\{x_0,\dots,x_{N-1}\}\big),$3

together with analogous input-bound constraints (Vahs et al., 2024). This replaces purely reactive state-space safety with proactive invariance of the evolving plan.

3. Autonomous error dynamics, stratified flows, and invariant subbundles

On Lie groups, trajectory invariance is formalized through invariant errors. For a system $x_N \in Int\!\big(ch\{x_0,\dots,x_{N-1}\}\big),$4, the left-invariant error $x_N \in Int\!\big(ch\{x_0,\dots,x_{N-1}\}\big),$5 is state trajectory independent when its dynamics depend only on $x_N \in Int\!\big(ch\{x_0,\dots,x_{N-1}\}\big),$6 and $x_N \in Int\!\big(ch\{x_0,\dots,x_{N-1}\}\big),$7. The paper derives the autonomous form

$x_N \in Int\!\big(ch\{x_0,\dots,x_{N-1}\}\big),$8

and proves the equivalence of left- and right-invariant formulations. For relative dynamics $x_N \in Int\!\big(ch\{x_0,\dots,x_{N-1}\}\big),$9, closure of the relative model automatically yields trajectory-invariant relative-error dynamics, enabling an invariant EKF whose linearized covariance dynamics depend on the inputs rather than the estimated trajectory (C. et al., 2024). In the $e_C=e+i e_I$0 example, the relative-attitude error obeys

$e_C=e+i e_I$1

so the Jacobian $e_C=e+i e_I$2 is independent of $e_C=e+i e_I$3.

For discontinuous or non-Lipschitz differential inclusions on stratified domains, the relevant invariant object is not the Filippov regularization alone but the essential velocity multifunction

$e_C=e+i e_I$4

This multifunction captures only directions realized by actual trajectories. Strong and weak invariance of a closed set $e_C=e+i e_I$5 are characterized by Hamiltonian inequalities involving $e_C=e+i e_I$6 and proximal normal cones: $e_C=e+i e_I$7 for all $e_C=e+i e_I$8 and $e_C=e+i e_I$9 (Barnard et al., 2012). The stratification matters because admissible interface directions are filtered through tangent cones of closures of lower-dimensional strata.

A related but distinct mechanical construction appears in the comparison of nonholonomic and vakonomic dynamics. There, invariant affine subbundle varieties inside the pullback bundle $R^2/\mathds 1^2[\mathcal C]$0 determine the initial conditions for which constrained variational trajectories and nonholonomic trajectories coincide. If $R^2/\mathds 1^2[\mathcal C]$1 denotes the affine vector field encoding the regular vakonomic dynamics and $R^2/\mathds 1^2[\mathcal C]$2 the cogeneralized subbundle on which the Frobenius term vanishes, then the existence of a partial or total $R^2/\mathds 1^2[\mathcal C]$3-admissible defining subbundle is equivalent to coincidence of regular vakonomic and nonholonomic trajectories (Lewis et al., 28 Apr 2025). The paper analyzes the underlying invariance PDE using Spencer cohomology and gives iterative Lie-derivative formulas for constructing the largest invariant affine subbundle.

A plausible synthesis is that these works replace invariance of points by invariance of geometrically structured objects: error fibers on Lie groups, tangent-compatible velocity sets on stratified domains, or affine subbundles in constrained mechanics.

4. Finite-trajectory certificates and data-driven controlled invariance

In linear discrete-time control, a central development is a trajectory-based characterization of controlled invariance. For

$R^2/\mathds 1^2[\mathcal C]$4

a set $R^2/\mathds 1^2[\mathcal C]$5 is controlled invariant if

$R^2/\mathds 1^2[\mathcal C]$6

The paper introduces convex feasible points through finite trajectories. If a state $R^2/\mathds 1^2[\mathcal C]$7 admits a trajectory with

$R^2/\mathds 1^2[\mathcal C]$8

and

$R^2/\mathds 1^2[\mathcal C]$9

then the convex hull of the sampled states is a controlled invariant set (Wembe et al., 8 Apr 2026). With strict feasibility, initializing the backward iteration

$t$0

yields a sequence of controlled invariant sets converging in the Hausdorff metric to the maximal controlled invariant set.

The same work uses this certificate inside MPC. Replacing a terminal invariant set by the nonconvex condition

$t$1

produces an MPC scheme that is recursively feasible without relying on precomputed terminal sets (Wembe et al., 8 Apr 2026). This does not eliminate nonconvexity, but it shifts the construction of invariance from recursive polyhedral set propagation to finite-horizon trajectory optimization.

A more algebraic trajectory-based invariance appears in behavioral system theory. For a discrete-time LTI system with a minimal realization and a measured input $t$2 that is persistently exciting of order $t$3, Willems’ Fundamental Lemma implies that every admissible length-$t$4 input-output trajectory $t$5 can be represented as

$t$6

for some coefficient vector $t$7 (Berberich et al., 2019). Here the space of all admissible trajectories is spanned by time-shifts of one measured trajectory. The paper extends this to woven trajectories and to Hammerstein, Wiener, and Hammerstein–Wiener systems that are linear in lifted coordinates, and then uses kernel methods to realize the same spanning principle in feature spaces. This suggests a different invariance concept: invariance of the trajectory space under time-shift-generated linear combinations of data windows.

5. Symmetry-preserving trajectories in physics, estimation, learning, and representation

One broad family of usages concerns invariance of trajectories under transformations that preserve the physically meaningful content of the model. In a non-Hermitian pilot-wave formulation with complexified coupling $t$8, the Schrödinger guidance law remains

$t$9

while the conserved density becomes

$z$0

Under the local transformation $z$1, both $z$2 and $z$3 acquire the same scale factor, so the ratio and the trajectories are invariant (Sen et al., 7 Jan 2026).

In hadronic reaction theory, a different symmetry problem arises when a standard $z$4-channel meson exchange is replaced by a Regge trajectory. Reggeization changes the divergence of the $z$5-channel current by introducing a term $z$6. “Trajectory invariance” in this setting means preserving the generalized Ward–Takahashi identity by reggeizing the interaction current simultaneously, so that

$z$7

The combined Reggeized current then remains locally gauge invariant (Haberzettl et al., 2015).

In learned trajectory representations, invariance is usually geometric. QCNeXt states that its query-centric encoder is equipped with permutation equivariance on set elements, spatial roto-translation invariance, and translation invariance in time through relative spacetime embeddings (Zhou et al., 2023). For rigid-body trajectory segmentation, a screw-based representation builds a geometric progress rate

$z$8

and a third-order descriptor

$z$9

achieving time-invariance and invariance to the choice of body reference point through screw-theoretic quantities (Verduyn et al., 2023). In privacy-preserving synthesis, vector translation invariance means

$\eta$0

for the Euclidean trajectory relationship, so pairwise-distance aggregates and their sensitivity are independent of the global origin (Liu et al., 2023).

A further, optimization-oriented usage appears in large-scale language-model pre-training. There, trajectory invariance refers to the observation that validation loss $\eta$1, gradient noise, and preconditioned gradient norm closely overlap when either the learning rate $\eta$2 is fixed early in training or the effective learning rate $\eta$3 is fixed later in training, under AdamW with decoupled weight decay (Li et al., 29 Sep 2025). The update

$\eta$4

makes the role of $\eta$5 explicit, and the empirical collapse of training curves reduces a two-dimensional $\eta$6 search to a one-dimensional tuning direction.

6. Topological, geometric, and statistical invariance of trajectory families

In multiagent navigation, trajectory invariance can be topological rather than metric. At uncontrolled intersections, two executions are assigned the same mode if they have the same start sides, destination sides, and signs of all pairwise winding numbers

$\eta$7

The mode $\eta$8 is invariant under continuous, collision-free deformations of the trajectories with endpoints fixed (Roh et al., 2020). This collapses a continuous space of multiagent trajectories into finitely many topological modes that can be predicted and planned over.

A different invariance concerns statistics of random curves in bounded domains. For isotropic and stationary ensembles of rectifiable curves, the mean in-domain path length $\eta$9, mean total curve length $$\dot{\bar\xi}=A\bar\xi+B\big(v_N(\bar\xi)+v_L(\bar\xi)+\Delta(\xi_d+\bar\xi,\eta,t)\big),$$0, and mean chord length $$\dot{\bar\xi}=A\bar\xi+B\big(v_N(\bar\xi)+v_L(\bar\xi)+\Delta(\xi_d+\bar\xi,\eta,t)\big),$$1 satisfy the exact universal law

$$\dot{\bar\xi}=A\bar\xi+B\big(v_N(\bar\xi)+v_L(\bar\xi)+\Delta(\xi_d+\bar\xi,\eta,t)\big),$$2

In the limit $$\dot{\bar\xi}=A\bar\xi+B\big(v_N(\bar\xi)+v_L(\bar\xi)+\Delta(\xi_d+\bar\xi,\eta,t)\big),$$3, one recovers the classical invariance property $$\dot{\bar\xi}=A\bar\xi+B\big(v_N(\bar\xi)+v_L(\bar\xi)+\Delta(\xi_d+\bar\xi,\eta,t)\big),$$4, which in $$\dot{\bar\xi}=A\bar\xi+B\big(v_N(\bar\xi)+v_L(\bar\xi)+\Delta(\xi_d+\bar\xi,\eta,t)\big),$$5D gives $$\dot{\bar\xi}=A\bar\xi+B\big(v_N(\bar\xi)+v_L(\bar\xi)+\Delta(\xi_d+\bar\xi,\eta,t)\big),$$6 and in $$\dot{\bar\xi}=A\bar\xi+B\big(v_N(\bar\xi)+v_L(\bar\xi)+\Delta(\xi_d+\bar\xi,\eta,t)\big),$$7D gives $$\dot{\bar\xi}=A\bar\xi+B\big(v_N(\bar\xi)+v_L(\bar\xi)+\Delta(\xi_d+\bar\xi,\eta,t)\big),$$8 (Binzoni et al., 2020). The wave-scattering generalization identifies the corresponding invariant mean length of wave trajectories in open media, with mean dwell time given by DOS through

$$\dot{\bar\xi}=A\bar\xi+B\big(v_N(\bar\xi)+v_L(\bar\xi)+\Delta(\xi_d+\bar\xi,\eta,t)\big),$$9

and mean length controlled by geometry alone in ballistic, chaotic, resonant, and Anderson-localized regimes (Pierrat et al., 2014).

A neighboring usage in statistical mechanics is the invariance principle for mechanical trajectories. In the three-dimensional random Lorentz gas, under the simultaneous Boltzmann–Grad and diffusive scalings

$\mathcal T(t_0)$00

the rescaled mechanical trajectory converges to Brownian motion. The key coupling statement is

$\mathcal T(t_0)$01

where $\mathcal T(t_0)$02 is the Markovian random flight process (Lutsko et al., 2018). This is not the same notion as forward invariance of sets, but it belongs to the same lexical family in which the large-scale law of trajectories is invariant under microscopic details.

Taken together, these literatures show that trajectory invariance is a domain-dependent technical term whose content is fixed by the structure one wishes to preserve: containment in a moving tube, autonomy of error dynamics, invariance of a defining subbundle, symmetry under gauge or coordinate transformations, preservation of topological class, or geometry-only statistics of path ensembles. The common thread is that the trajectory is not treated merely as a sequence of states, but as an object on which a specific invariance principle can be formulated and exploited.