Controlled-Jump Rule in Hybrid & Quantum Systems

• Controlled-jump rule is a method for specifying admissible discrete updates that ensure stability or optimality across hybrid, stochastic, sampled-data, and quantum systems.
• It employs strategies like minimum-norm selection, threshold acceptance, and feedback control to guarantee a controlled decrease in Lyapunov functions or optimal performance under constraints.
• Applications span diverse areas including control theory, legged robotics, and quantum information, where the rule governs state resets, mode selections, or hierarchy level adjustments.

“Controlled-jump rule” denotes a family of constructions in which a discrete update is not merely allowed by the model but is regulated by an explicit control principle. Across the cited literature, the phrase appears in several technically distinct senses: a pointwise minimum-norm jump input that enforces decrease of a control Lyapunov function in hybrid systems; an acceptance or thinning rule for stochastic jumps driven by an intensity; a jump-size feedback law in jump-diffusion control; a min-jumping mode-selection rule in sampled-data impulsive systems; and, in quantum information, a rule that determines how coherent control of a Clifford operation raises the level of the Clifford hierarchy (Sanfelice, 2020, Khabou et al., 15 Jul 2025, Briat, 2017, Xu et al., 25 Feb 2026).

1. Cross-domain meanings

Representative usages span deterministic hybrid control, stochastic control, sampled-data switching, robotics, and quantum information. This suggests that the common semantic core is not a single formalism, but a rule specifying how a discontinuous update is selected, admitted, or interpreted under control.

Setting	Controlled quantity	Representative rule
Hybrid CLF control	Jump input $u_d$	$$\rho_d(x)=\arg\min\{|u_d|:\ u_d\in \mathcal T_d(x)\}$$
Hawkes jump-diffusion	Accepted Poisson points	$\mathbf 1_{\theta \le \lambda_s}$
Regime-switching LQ	Jump size at mark $z$	$$\Delta X_t(z) = E_{\alpha_t,t}(z)X_{t-} + F_{\alpha_t,t}(z)u_{2,t}(z)$$
Sampled-data impulsive systems	Next jump mode	$$\sigma(t_k^+)=\arg\min_{i}\{\chi(t_k)^T P_i \chi(t_k)\}$$
Clifford hierarchy	Hierarchy level of $CU$	$$CU\in \mathcal{C}_{m+2}^{(n+1)}\setminus \mathcal{C}_{m+1}^{(n+1)}$$

In some settings the rule chooses a control value at a jump, in others it shapes an intensity or a jump amplitude, and in still others it selects a mode or proves a hierarchy-level “jump.” The distinction is essential because the controlled object can be the state reset, the jump acceptance event, the post-jump distribution, the co-state discontinuity, or the algebraic level occupied by a controlled unitary (Shi et al., 2024, Zhou et al., 2022, Rondoni, 2019, Xu et al., 25 Feb 2026).

2. Pointwise minimum-norm jump control in hybrid dynamical systems

The most explicit control-theoretic use appears in the Goebel–Sanfelice–Teel style hybrid setting, where state evolution occurs either through flows or through jumps. The plant is written as

$\mathcal H=\left\{ \begin{array}{lll} \dot x = f(x,u_c) & &(x,u_c)\in C,\[2mm] x^+ = g(x,u_d) & &(x,u_d)\in D, \end{array} \right.$

with $C$ the flow set, $$\rho_d(x)=\arg\min\{|u_d|:\ u_d\in \mathcal T_d(x)\}$$0 the jump set, $$\rho_d(x)=\arg\min\{|u_d|:\ u_d\in \mathcal T_d(x)\}$$1 the flow map, and $$\rho_d(x)=\arg\min\{|u_d|:\ u_d\in \mathcal T_d(x)\}$$2 the jump map. The framework uses hybrid arcs on hybrid time domains and assumes the hybrid basic conditions: $$\rho_d(x)=\arg\min\{|u_d|:\ u_d\in \mathcal T_d(x)\}$$3 and $$\rho_d(x)=\arg\min\{|u_d|:\ u_d\in \mathcal T_d(x)\}$$4 closed, $$\rho_d(x)=\arg\min\{|u_d|:\ u_d\in \mathcal T_d(x)\}$$5 continuous, and $$\rho_d(x)=\arg\min\{|u_d|:\ u_d\in \mathcal T_d(x)\}$$6 continuous (Sanfelice, 2020).

For a compact target set $$\rho_d(x)=\arg\min\{|u_d|:\ u_d\in \mathcal T_d(x)\}$$7, the control Lyapunov function $$\rho_d(x)=\arg\min\{|u_d|:\ u_d\in \mathcal T_d(x)\}$$8 is required to satisfy class-$$\rho_d(x)=\arg\min\{|u_d|:\ u_d\in \mathcal T_d(x)\}$$9 bounds and decrease conditions both along flows and across jumps. The jump-side dissipation inequality is

$\mathbf 1_{\theta \le \lambda_s}$0

The controlled-jump rule is then obtained by defining

$\mathbf 1_{\theta \le \lambda_s}$1

setting $\mathbf 1_{\theta \le \lambda_s}$2, and forming the admissible jump-control set

$\mathbf 1_{\theta \le \lambda_s}$3

Equivalently, for $\mathbf 1_{\theta \le \lambda_s}$4, $\mathbf 1_{\theta \le \lambda_s}$5 is the set of all jump inputs satisfying

$\mathbf 1_{\theta \le \lambda_s}$6

The jump law itself is the pointwise minimum-norm selection

$\mathbf 1_{\theta \le \lambda_s}$7

This construction has a direct flow analogue,

$\mathbf 1_{\theta \le \lambda_s}$8

with

$\mathbf 1_{\theta \le \lambda_s}$9

The paper distinguishes individual inputs for flows and jumps, a common input $z$0, and the cases in which input enters only through flows or only through jumps. For the common-input case it requires

$z$1

and defines

$z$2

The main formal results validate the construction under lower semicontinuity and convexity hypotheses. Proposition 1 states practical stabilization of the sublevel set $z$3 for the restricted hybrid system $z$4. Theorem 1 extends stabilization of $z$5 to the full hybrid system $z$6. Theorem 2 gives global asymptotic stability of $z$7 under additional local invariance and lower-semicontinuity assumptions near $z$8. Under convex admissible sets and convexity of $z$9 in the controls, $$\Delta X_t(z) = E_{\alpha_t,t}(z)X_{t-} + F_{\alpha_t,t}(z)u_{2,t}(z)$$0 and $$\Delta X_t(z) = E_{\alpha_t,t}(z)X_{t-} + F_{\alpha_t,t}(z)u_{2,t}(z)$$1 have nonempty closed convex values, so each has a unique minimum-norm element; if $$\Delta X_t(z) = E_{\alpha_t,t}(z)X_{t-} + F_{\alpha_t,t}(z)u_{2,t}(z)$$2 have closed graph, the selections $$\Delta X_t(z) = E_{\alpha_t,t}(z)X_{t-} + F_{\alpha_t,t}(z)u_{2,t}(z)$$3 are continuous (Sanfelice, 2020).

The examples make the rule concrete. In the “rotate and dissipate” example, the jump law becomes

$$\Delta X_t(z) = E_{\alpha_t,t}(z)X_{t-} + F_{\alpha_t,t}(z)u_{2,t}(z)$$4

while the flow law is a bang-bang selector $$\Delta X_t(z) = E_{\alpha_t,t}(z)X_{t-} + F_{\alpha_t,t}(z)u_{2,t}(z)$$5 based on the sign of $$\Delta X_t(z) = E_{\alpha_t,t}(z)X_{t-} + F_{\alpha_t,t}(z)u_{2,t}(z)$$6. In the impact-controlled pendulum example, the minimum-norm jump control is

$$\Delta X_t(z) = E_{\alpha_t,t}(z)X_{t-} + F_{\alpha_t,t}(z)u_{2,t}(z)$$7

and in the timer desynchronization example the minimum-norm choice is zero as well (Sanfelice, 2020).

3. Stochastic jump mechanisms, jump-size control, and nonlocal HJB structure

In stochastic models, “controlled-jump rule” often does not mean deterministic selection of a jump instant. In controlled Hawkes jump-diffusions, the state satisfies

$$\Delta X_t(z) = E_{\alpha_t,t}(z)X_{t-} + F_{\alpha_t,t}(z)u_{2,t}(z)$$8

so a point of the driving Poisson measure is accepted only if the auxiliary mark satisfies $$\Delta X_t(z) = E_{\alpha_t,t}(z)X_{t-} + F_{\alpha_t,t}(z)u_{2,t}(z)$$9. The control enters the drift, diffusion, jump size, and jump feedback, but does not directly choose jump times deterministically. Because a general Hawkes kernel makes the system non-Markovian, the paper approximates $$\sigma(t_k^+)=\arg\min_{i}\{\chi(t_k)^T P_i \chi(t_k)\}$$0 by a finite sum of exponentials, augments the state with variables $$\sigma(t_k^+)=\arg\min_{i}\{\chi(t_k)^T P_i \chi(t_k)\}$$1, and obtains a Markov jump-diffusion amenable to dynamic programming and HJB analysis (Khabou et al., 15 Jul 2025).

A more direct jump-size control appears in constrained stochastic linear-quadratic control under regime switching, where the state equation contains both a conventional control $$\sigma(t_k^+)=\arg\min_{i}\{\chi(t_k)^T P_i \chi(t_k)\}$$2 and a jump-size-dependent control $$\sigma(t_k^+)=\arg\min_{i}\{\chi(t_k)^T P_i \chi(t_k)\}$$3: $$\sigma(t_k^+)=\arg\min_{i}\{\chi(t_k)^T P_i \chi(t_k)\}$$4 Both controls are constrained within general closed cones. The optimal value and optimal feedback control depend on solutions to multidimensional fully coupled stochastic Riccati equations, formulated as BSDEJs, and the feedback law is given by pointwise minimizers of Hamiltonian-like quadratic forms (Shi et al., 2024).

In mean field games with controlled jump-diffusion dynamics, the jump component is modeled through a controlled increment: $$\sigma(t_k^+)=\arg\min_{i}\{\chi(t_k)^T P_i \chi(t_k)\}$$5 The abstract model allows the control to affect the drift, volatility, and jump size, while the time-dependent intensity $$\sigma(t_k^+)=\arg\min_{i}\{\chi(t_k)^T P_i \chi(t_k)\}$$6 is given. Existence is established in a relaxed formulation, and under suitable conditions relaxed optimal controls can be replaced by Markovian, and under an additional convexity assumption by strict Markovian, controls (Benazzoli et al., 2017).

Several related literatures place the controlled jump inside a nonlocal HJB or stochastic target framework. In singular control for multidimensional jump-diffusions with infinite activity and finite variation, the controlled state evolves as

$$\sigma(t_k^+)=\arg\min_{i}\{\chi(t_k)^T P_i \chi(t_k)\}$$7

and the value function solves, in the almost everywhere sense, an HJB equation with gradient constraint,

$$\sigma(t_k^+)=\arg\min_{i}\{\chi(t_k)^T P_i \chi(t_k)\}$$8

(Kelbert et al., 2017). In controller-stopper games with controlled jumps, the controller chooses $$\sigma(t_k^+)=\arg\min_{i}\{\chi(t_k)^T P_i \chi(t_k)\}$$9, where $CU$0 affects jump amplitudes, and the analysis proceeds through stochastic target problems with stopping and the Stochastic Perron method; the resulting HJB operators are nonlocal because jumps enter through integral terms (Bayraktar et al., 2016). In slow–fast controlled jump diffusions with $CU$1-stable noise, the original singularly perturbed nonlocal HJB equation converges to an effective averaged HJB equation after averaging over the ergodic measure of the fast component (Zhang et al., 2024).

A distributional formulation appears in one-dimensional jump processes with deterministic drift, uncontrolled jump forcing, and controlled jump forcing. There the controlled current can be written in terms of antecedent and posterior densities,

$CU$2

so the control is described directly by pre-jump and post-jump distributions rather than by an underlying control amplitude (Rondoni, 2019). A distinct optimal-control formulation governs continuous-time Markov jump processes through controlled transition rates $CU$3; when the reduced traffic is unconstrained the optimum is singular, whereas bounded or fixed reduced traffic yields a discrete Hamilton–Bellman–Jacobi-type system (Muratore-Ginanneschi et al., 2012).

4. Event-driven switching, co-state jumps, and sampled-data min-jumping rules

In state-dependent switched optimal control, the central jump law may apply not to the state but to the adjoint variable. For a two-mode system with switching surface $CU$4, fixed initial state, fixed final time, and continuous state across the interface, variational analysis yields adjoint equations on each side of the switching instant $CU$5, continuity of the Hamiltonian,

$CU$6

and the explicit co-state jump law

$CU$7

For a time-varying interface $CU$8, the denominator is modified to include the interface velocity term $CU$9. The paper uses this jump law in the GEL algorithm, which solves the resulting hybrid optimal control problem by combining a boundary-value solve with a scalar Newton update in the switching time (Zhou et al., 2022).

A different event-driven use appears in aperiodic sampled-data control of impulsive and switched impulsive systems. Measurements arrive at times $$CU\in \mathcal{C}_{m+2}^{(n+1)}\setminus \mathcal{C}_{m+1}^{(n+1)}$$0 with

$$CU\in \mathcal{C}_{m+2}^{(n+1)}\setminus \mathcal{C}_{m+1}^{(n+1)}$$1

and the controller chooses at each sampling instant which jump map or mode to apply next and what sampled-data control input to hold until the next sample. For the augmented state $$CU\in \mathcal{C}_{m+2}^{(n+1)}\setminus \mathcal{C}_{m+1}^{(n+1)}$$2, the min-jumping rule is

$$CU\in \mathcal{C}_{m+2}^{(n+1)}\setminus \mathcal{C}_{m+1}^{(n+1)}$$3

with an analogous formula for switched impulsive systems involving $$CU\in \mathcal{C}_{m+2}^{(n+1)}\setminus \mathcal{C}_{m+1}^{(n+1)}$$4. The synthesis condition is first expressed as a semi-infinite discrete-time Lyapunov-Metzler inequality and then rewritten as equivalent clock-dependent matrix inequalities involving differentiable matrix functions $$CU\in \mathcal{C}_{m+2}^{(n+1)}\setminus \mathcal{C}_{m+1}^{(n+1)}$$5. A convexified design form follows from inverse variables, and the infinite-dimensional constraints are approximated by sum-of-squares relaxations. The paper proves that the SOS relaxation is non conservative provided that the degree of the polynomials is sufficiently large, and notes that acceptable results are obtained for low polynomial degree. It explicitly identifies the rule as the sampled-data analogue of the classical continuous-time min-switching rule of Geromel–Colaneri (Briat, 2017).

5. Jumping control in legged robotics

In legged robotics, the relevant usage shifts from discontinuous stochastic or hybrid state updates to controlled jumping motions across stance, flight, and landing. CAJun is a hierarchical framework with a high-level centroidal policy trained with reinforcement learning at $$CU\in \mathcal{C}_{m+2}^{(n+1)}\setminus \mathcal{C}_{m+1}^{(n+1)}$$6 Hz and a low-level leg controller running at $$CU\in \mathcal{C}_{m+2}^{(n+1)}\setminus \mathcal{C}_{m+1}^{(n+1)}$$7 Hz. The policy outputs stepping frequency $$CU\in \mathcal{C}_{m+2}^{(n+1)}\setminus \mathcal{C}_{m+1}^{(n+1)}$$8, desired base velocity $$CU\in \mathcal{C}_{m+2}^{(n+1)}\setminus \mathcal{C}_{m+1}^{(n+1)}$$9, and swing-foot residuals, while the low-level controller handles swing-leg tracking and stance-leg ground-reaction-force optimization. The stance optimizer is reformulated by ignoring the inequality constraints initially, solving the unconstrained least-squares problem in closed form, and then clipping or projecting the result back into the friction cone. The paper reports that this makes the whole stack roughly $\mathcal H=\left\{ \begin{array}{lll} \dot x = f(x,u_c) & &(x,u_c)\in C,\[2mm] x^+ = g(x,u_d) & &(x,u_d)\in D, \end{array} \right.$0 faster, enabling training in about $\mathcal H=\left\{ \begin{array}{lll} \dot x = f(x,u_c) & &(x,u_c)\in C,\[2mm] x^+ = g(x,u_d) & &(x,u_d)\in D, \end{array} \right.$1 minutes on a single GPU. It also reports zero-shot transfer to a Unitree Go1, continuous adaptive jumping, and gap crossing with a maximum width of $\mathcal H=\left\{ \begin{array}{lll} \dot x = f(x,u_c) & &(x,u_c)\in C,\[2mm] x^+ = g(x,u_d) & &(x,u_d)\in D, \end{array} \right.$2 cm, which is over $\mathcal H=\left\{ \begin{array}{lll} \dot x = f(x,u_c) & &(x,u_c)\in C,\[2mm] x^+ = g(x,u_d) & &(x,u_d)\in D, \end{array} \right.$3 wider than existing methods (Yang et al., 2023).

A bipedal counterpart is the jumping control framework for BRUCE, which combines an offline kino-dynamic motion planner based on centroidal momentum dynamics, a real-time heuristic landing planner, and a low-level whole-body or joint controller. The planner enforces

$\mathcal H=\left\{ \begin{array}{lll} \dot x = f(x,u_c) & &(x,u_c)\in C,\[2mm] x^+ = g(x,u_d) & &(x,u_d)\in D, \end{array} \right.$4

together with kinematic consistency, discrete-time integration, and friction-cone constraints. During flight, the heuristic landing planner updates nominal foot placement using linear and angular momentum feedback,

$\mathcal H=\left\{ \begin{array}{lll} \dot x = f(x,u_c) & &(x,u_c)\in C,\[2mm] x^+ = g(x,u_d) & &(x,u_d)\in D, \end{array} \right.$5

plus a lateral clearance term $\mathcal H=\left\{ \begin{array}{lll} \dot x = f(x,u_c) & &(x,u_c)\in C,\[2mm] x^+ = g(x,u_d) & &(x,u_d)\in D, \end{array} \right.$6. The low-level controller uses a QP-based whole-body controller during contact and joint-level PD control during flight. Hardware and simulation experiments on BRUCE include in-situ jump, directional jump, twisting jump, step jump, push recovery in the air, somersaults, and jump down; the reported performance includes maximum COM jump height about $\mathcal H=\left\{ \begin{array}{lll} \dot x = f(x,u_c) & &(x,u_c)\in C,\[2mm] x^+ = g(x,u_d) & &(x,u_d)\in D, \end{array} \right.$7 cm, maximum jump distance about $\mathcal H=\left\{ \begin{array}{lll} \dot x = f(x,u_c) & &(x,u_c)\in C,\[2mm] x^+ = g(x,u_d) & &(x,u_d)\in D, \end{array} \right.$8 cm, and twisting angle about $\mathcal H=\left\{ \begin{array}{lll} \dot x = f(x,u_c) & &(x,u_c)\in C,\[2mm] x^+ = g(x,u_d) & &(x,u_d)\in D, \end{array} \right.$9 degrees (Zhang et al., 2023).

These robotics papers do not treat jump events as random arrivals or state resets in the stochastic-control sense. Instead, the controlled-jump mechanism is embedded in phase logic, contact creation or loss, and flight-phase landing adaptation. This suggests that the robotics usage is operational rather than variational: it specifies how to generate, track, and adapt a jumping maneuver rather than how to choose among admissible discrete state updates.

6. Controlled jump in the Clifford hierarchy

In quantum information, “controlled jump” refers to a level jump in the qubit Clifford hierarchy induced by coherent control of a Clifford unitary. The hierarchy is defined recursively by

$C$0

For a Clifford unitary $C$1, the paper introduces Pauli periodicity: $C$2 The central theorem states that if $C$3 has Pauli periodicity $C$4, then

$C$5

Equivalently,

$C$6

The proof uses controlled-unitary identities such as $C$7, a control-$C$8 conjugation identity that makes repeated squaring appear, and a block-diagonal lemma. The result is sharp: membership in $C$9 and strict exclusion from $$\rho_d(x)=\arg\min\{|u_d|:\ u_d\in \mathcal T_d(x)\}$$00 are both proved. The paper also establishes the periodicity bound

$$\rho_d(x)=\arg\min\{|u_d|:\ u_d\in \mathcal T_d(x)\}$$01

which implies that if $$\rho_d(x)=\arg\min\{|u_d|:\ u_d\in \mathcal T_d(x)\}$$02 is strictly in level $$\rho_d(x)=\arg\min\{|u_d|:\ u_d\in \mathcal T_d(x)\}$$03, then the target Clifford must act on at least

$$\rho_d(x)=\arg\min\{|u_d|:\ u_d\in \mathcal T_d(x)\}$$04

qubits. Explicit infinite families saturate this scaling, including Clifford permutations and the family

$$\rho_d(x)=\arg\min\{|u_d|:\ u_d\in \mathcal T_d(x)\}$$05

for which the Pauli periodicity is

$$\rho_d(x)=\arg\min\{|u_d|:\ u_d\in \mathcal T_d(x)\}$$06

As an application, the paper proposes logical catalyst states enabling logical $$\rho_d(x)=\arg\min\{|u_d|:\ u_d\in \mathcal T_d(x)\}$$07 phase gates via phase kickback from a single jumped Clifford (Xu et al., 25 Feb 2026).

7. Unifying themes and recurrent distinctions

A plausible unifying view is that a controlled-jump rule specifies a map from a pre-jump configuration to an admissible controlled update, together with a criterion selecting one update among several possibilities. In the hybrid CLF setting the criterion is Lyapunov decrease with minimum Euclidean norm; in regime-switching LQ control it is quadratic optimality under cone constraints; in sampled-data impulsive systems it is minimization of a quadratic Lyapunov certificate at sampling instants; and in the Clifford hierarchy it is the smallest number of repeated squarings needed for a Clifford to become Pauli (Sanfelice, 2020, Shi et al., 2024, Briat, 2017, Xu et al., 25 Feb 2026).

Several distinctions prevent the term from having a single universal meaning. First, a controlled-jump rule does not necessarily mean direct control of jump times. In Hawkes jump-diffusions, jumps are accepted through $$\rho_d(x)=\arg\min\{|u_d|:\ u_d\in \mathcal T_d(x)\}$$08, so control acts through intensity modulation and jump amplitude rather than deterministic timing (Khabou et al., 15 Jul 2025). Second, the jumping quantity need not be the physical state. In state-dependent switched optimal control, the state is continuous at the interface while the co-state satisfies an explicit jump law normal to the switching surface (Zhou et al., 2022). Third, “jump” can denote a structural change rather than a dynamical discontinuity: in the Clifford hierarchy, it refers to the strict rise of $$\rho_d(x)=\arg\min\{|u_d|:\ u_d\in \mathcal T_d(x)\}$$09 to level $$\rho_d(x)=\arg\min\{|u_d|:\ u_d\in \mathcal T_d(x)\}$$10 (Xu et al., 25 Feb 2026).

The literature also shows that controlled-jump rules are usually accompanied by a certification mechanism. Hybrid CLF rules rely on asymptotic-stability theorems; stochastic formulations lead to HJB equations, BSDEJs, martingale problems, or stochastic target problems; sampled-data jumping rules rely on Lyapunov-Metzler inequalities and SOS relaxations; robotics implementations rely on centroidal dynamics, whole-body QP control, and flight-phase landing heuristics; quantum constructions rely on exact hierarchy membership theorems and symplectic periodicity bounds (Kelbert et al., 2017, Bayraktar et al., 2016, Zhang et al., 2023, Yang et al., 2023).

Taken together, these works show that the controlled-jump rule is best understood as an umbrella expression for control laws governing discrete updates under rigorous admissibility, stability, optimality, or hierarchy-membership conditions. The exact mathematical content depends entirely on the surrounding formalism.