Neural Event-Triggered Control with Optimal Scheduling
Authors: Luan Yang
Jingdong Zhang
Qunxi Zhu
Wei Lin
Venue: ICML 2025 poster (2025)
Keywords: Event-triggered control
Stability guarantee
Limit resources
Optimal scheduling
Primary area: optimization->everything_else
TL;DR: We design a controller that only sends updates when absolutely necessary, without sacrificing safety or stabilization performance.

Abstract: Learning-enabled controllers with stability certificate functions have demonstrated impressive empirical performance in addressing control problems in recent years. Nevertheless, directly deploying the neural controllers onto actual digital platforms requires impractically excessive communication resources due to a continuously updating demand from the closed-loop feedback controller. We introduce a framework aimed at learning the event-triggered controller (ETC) with optimal scheduling, i.e., minimal triggering times, to address this challenge in resource-constrained scenarios. Our proposed framework, denoted by Neural ETC, includes two practical algorithms: the path integral algorithm based on directly simulating the event-triggered dynamics, and the Monte Carlo algorithm derived from new theoretical results regarding lower bound of inter-event time. Furthermore, we propose a projection operation with an analytical expression that ensures theoretical stability and schedule optimality for Neural ETC. Compared to the conventional neural controllers, our empirical results show that the Neural ETC significantly reduces the required communication resources while enhancing the control performance in constrained communication resources scenarios.

Neural Event-Triggent Control with Optimal Scheduling
Luan Yang" Jingdong Zhang 12
Qunxi Zhu 134
Wei Lin 1234
Abstract
Triggering Times
Control
Trattoriiess
0.4
Learning-enablee controllers with stability cer- tificate functions have demonstrated impressive empirical performance in addressing control prob- lems in recent years. Nevertheless, directly de- ploying the neural controllers onto actual digital platforms requires impractically excessive com- munication resources due to continuously updat- ing demand from the closed-loop feedback con- troller. We introduce a framework aimed at learn- ing the event-triggered controller (ETC) with opti- mal scheduling, i... minimal triggering times, to address this challenge in resource-constraing sce- narios. Our proposed framework, denoted by Neu- ral ETC, includes two practical algorithms: the path integral algorithm based on directly simulat- ing the event-triggens triggeedd dynamics, and the Monte Carlo algorithm derived from new theoretical re- sults regarding lower bound of inter-event time. Furthermore, we propose a projection operation with an analytical expression that ensures theoret- ieal stability and schedule optimality for Neural ETC. Compared the conventional neural con- trollers, our empirical results show that the Neural ETC significantly reduces the required commu- nication resources while enhancing the control performance in constrained communication re- sources scenarios.
Inter-eennt Thme
Neural ETC: (a)
(b)
Neural ETC
16
NLC: 1999
NLC
Anm
Time
Time
Time
Figure Comparison of Neural ETC (yellow) and neural Lya- punov control (NLC, purple) in stabilizing the Lorenz system under the event-triggered control setting. (a): The inter-event time of consecutive triggering events. The dashed lines represent the minimal inter-event time of each control. (b): The control values acting variable æ in the control process. (c): The controlled trajectories of the variable x.
engineering. Previous research in the field of cyberneties has applied the Lyapunov stability theory to formulate stabi- lizing policies for linear or polynomial dynamical systems, including the linear quadratic regulator (LQQR) (Khalil, 2002) and the sum-of-squares (SOS) polynomials, using the semi- definite planning (SDP) (Parrilo, 2000). Stabilizing more intricate dynamical systems with high dimension and nonlin- earity, as encountered in real applications, has prompted the integration of machine learning techniques into the cyber- netics Community(Tsukamenn
al., 2021). Recent advance- ments in learning neural networks based controllers with cer- tificate functions, such as Lyapunov function (Chang et al., 2019; Zhang et al., 2022a), LaSalle function (Zhang et al., 2022b), barrier functions (Qin et al., 2020) and contraction metrics (Sun al., 2021), have demonstrated outstanding performance in controlling diverse dynamics (Dawson et al., 2022). Nevertheless, it is noteworthy that all these con- trollers require updating the control signal continuously over time, leading to considerable communication cost between controller and platform.
1. Introduction
Stabilizing the complex nonlinear systems represents a formidable focal task within the realms of mathematics and
"Equal contribution 'Reccarch Institute of Intelligent Complex Systems, Fudan University, China. "School of Mathematical Sci- ences, LMNS, and SCMS, Fudan University, China. 'State Key Laboratory of Medical Neurobiology and MOE Frontiers Center for Brain Science, Institutes of Brain Science, Fudan University, China. "Shanghai Artificial Intelligence Laboratory, China. Cor- respondence to: Qunxi Zhu <qxzhul66 fudan.edu.cn.>, Wei Lin <wlin@ edw.a..ddd.c......
The periodic control is mostly advocated for implement- ing feedback control laws on digital platforms (Franklin et al., 2002). However, such implementations often incur significant peviisoovsi of the communication network, especially in the recently developed large-scale resource- constrained wireless embedded control systems (Lemmon, 2010). To mitigate this issue, event-triggerin mechanism is introduced to generate sporadic transmissions across the feedback channels of the system. Compared to periodic
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Neural Event-Triggeree Triggered Control with Optimal Scheduling
control which updates the control signal at a series of pre- defined explicit times, event-triggered control updates the control signal at the instants when the current measurement violates predefined triggering condition, thereby triggering a state-dependent event (Heemels et al., 2012). Given that these instants are implicitly determined by the state trajec- tories, the scheduling of computation and communication resources for event-triggere control becomes a very chal- lenging problem, involving the minimization of the number of events and the increase of inter-event time. While signifi- cant strides have been made in stabilizing specific dynamics with veenttriggered control in recent years, the task of de- signing event-triggered control for general nonlinear and large-scale dynamics with optimal scheduling remains an open problem (Aarzén, 1999; Tabuada, 2007: Heemels et al., 2008; Henningsson al., 2008).
system under our Neural ETC, we propose the projec- tion operation that rigorously endows our Neural ETC with stability and schedule optimality, Instead of solv- ing an optimization problem to obtain the projection, we provide analytical expression for our projection operation, leading to a fast implementation for our framework.
Finally, we evaluate Neural ETCs on a variety of representative physical and engineering systems. Compared to existing stabilizing controllers, we find that Neural ETCs exhibit significant superi- ority in decreasing the triggering times and max- imizing the minimal inter-event time. The code for reproducing all the numerical experiments is re- leased at Event-riigered-Contoll (hyperlink of Neural ETC).
Our goal is to design event-triggered control for general complex dynamics, ensuring both stability guarantee and optimal scheduling, i.e., to implement event-triggered con- trol with the minimal triggering times and the maximal inter-event time. Fig. depicts the comparison of con- trol performance of the Neural ETC and the NLC in the event-triggered realization stabilize Lorenz dynamic. In Fig. 1(a)-1(b) (b)), it is evident that the triggering times of Neural ETC are significantly fewer than those of NLC, and the minimal inter-event time of consecutive triggering times of Neural ETC considerably exceeds that of NLC. These disparities lead to the different behaviors of the controlled trajectories, as depicted in Fig. I(c). Under Neural ETC, the trajectory rapidly converges to the target state, while the NLC exhibits violent oscillation around the target.
2. Background
Notations. Denote by || ||. the L²-norm for any given vector in R Denote by |ICDD) the maximum norm on continuous function space C(D). For A (axy): matrix of dimension d x r. denote by || ||A = Σi 2i=1 -j=1 d Frobenius norm. Denote max(a, 0) by (a) Denote x y as the inner product of two vectors.
of the
2.1. Neural Lyapunov Control
To begin with, we consider the feedback controlled dynami- cal system of the following general form:
Contribution. The principal contributions of this paper can be summarized as follows:
d = f(r, u(x)) - fu(()), a E RR, u RT (1)
where fu(z) R is the Lipschitz- continuous vector field acting on some prescribed open set D C Rd. The solution initiated at time to from æo under controller is denoted by æ (t; to, 2o). For brevity, we let the unstable equilibrium x" E D be origin, i.c., f(0, 0) = O. One major problem in cybernetics field is to design stabilizing con- troller u(x) (Wiener, 2019) such that lim wn (1; to xo) =
We propose Neural ETC, a framework for learning event-triggered controllers ensuring both stability guar- antee and optimal scheduling, where the exponential stability comes from the devised event function.
Specifically, we firstly propose a path integral approach to realize the implementation of the machine learning framework based on the root solver and neural event ODE solver that calculate the trainable event triggering times. Secondly, we theoretically address the estima- tion of the minimal inter-event time of the event trig- gered controlled system, which leading to the Monte Carlo approach of our framework that circumvents the expensive computation cost of back-propagation through ODE solvers. The two approaches trade off in terms of stabilization performance and training effi- ciency, which is convenient for users to flexibly choose the specific approach according the task in hand.
10000
O, for any initial value 30 E D.
Theorem 2.1. (Mao, 2007) Suppose there exists continu- ously differentiable function V 'D - R that satisfies the following conditions: (i) ( = O, (ii) V(x) 2 c||=||? for some constants c,p > O, (iii) and L, V < -VV, for some 8 > 0. Then, the system is exponentially stable at the ori- gin, that is, lim sup log || (t: to æo)il - Here V called a Lyapunov function.
Previous works parameterize the controller and the Lya- punov function as U Vo. and integrate the sufficient con-
"Lea V represent the Lie derivative of V along the direction Sur fu..c.. i.c., L.s, V VV fu-
To theoretically guarantee the stability of the controlled
2

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Neural Event-Triggeree Triggered Control with Optimal Scheduling
ditions (i)- in Theorem 2.1 for Lyapunov stability into the loss function (Chang al., 2019; Zhang al., 2022a; Dawson et al., 2023). The learned Lyapunov Ve plays role of stability certificate function.
based on the known model f and time limit T.
min - ) + (illu(())l((()
min
Remark 2.2. Unlike model-free reinforcement learning (RL) approaches that search for an online control policy guided by a reward function along the trajectories of the dynamical systems. (Kaelbling et al., 1996), the neural Lyapunov con- trol searches for an offline policy and a certificate function V that proves the soundness of the learned policy (Daw- son et al., 2022). Nevertheless, updating the feedback pol- icy continuously in the implementation process incurs pro- hibitive high communication cost.
S.L. ++)) ((x()), u(<((t)), t E (x+1)
(0) xo D, lim x(t) 0,
(2)
The major difficulty of this problem comes from that the implicitly defined triggering times are not equidistant, and are only known when the events are triggered (Miskowicz. 2018). The majority of existing works focus on the stabi- lization performance of event-triggered control and often omit the communication cost of updating the control value at triggering moments. In what follows, we propose neural event- seett-riiggeed control (Neural ETC) framework to address both the stabilization and the communication cost issues of event-triggered control.
2.2. Event-triggered Control
Although the feedback controller works well in numerical simulations, updating and implementing the controller con- tinuously is impractical in most real-world digital platforms under communication constraints (Aström & Bernhardsson, 1999). To conquer this weakness, weent-triggerd stabilizing control introduced as follows (Heemels et al., 2012),
3. Method
Closed-loop controlled dynamics. The dynamics under event-triggered control is generally an open-loop system with controller varying from different triggering time in- tervals. In order to simplify the theoretical analysis and to utilize the existing numerical tools for ODE solvers, we transform the event-triggered controlled system to the closed- loop version via augmenting the dynamics with er- ror state e(t) x (txx) xxt), t [t,, tx+1) and update operation e(f++1) = O. Then we obtain the closed-loop controlled dynamics as
Definition 2.3. (Event (Event-triggered Control) Consider the con- trolled system (I), the event-triggered controller is defined as u(t) u(x(tt)), t S < tg+1, where the triggering time is decided by (x+1 (t >> l4 h(x((()= = 0} for some predefined event function h. The largest lower bound T² of [4x+1 } is called as minimal inter-event time. For example, if there exists a Lyapunov function V for the feedback controlled system (1), then the event function is set to guarantee the Lyapunov condition on each event trig- gering time interval, i.e., VV f(r((,), u(xr.) < 0, E [tu t2+1).
a = f(x u(x + e)), ffz, u(s + e)), t E [tx, tx+1).
In the next sections, we construct the event function with exponential stability guarantee and deduce the theoretical estimation of minimal inter-event time based the aug- mented dynamics of (æ, e).
Problem Statement. We assume that the zero solution of the uncontrolled system in Eq. (1) is unstable, i.e. lim, Tu=o to: xo) * O. We aim at stabilizing the zero solution using event-triggered control based on neu- ral networks (NNs) with optimal scheduling, i.e., the least triggering times, which is urgently required by the digital platforms wherein the communication resources of updat- ing the control value are limited. Notice that in an average sense, the triggering times are inversely proportional to the inter-event time, our goal is equivalently to leverage the NNs to design an appropriate controller u with u())= = 0 such that the controlled system under wein-triggered imple- mentation is steered to the zero solution with the maximal inter-event time. We summarize the problem formulation as the following optimization problem, where the triggering time {tx t < T}} depends on the controller u and the triggering mechanism, and T < oo is the prefixed time limit according to the specific tasks. We aim at devising controller u and triggering mechanism to solve this problem
Event function for exponential stability. We consider the exponential Lyapunov stability for controlled system (1) such that the corresponding Lyapunov function defined in Theorem 2.1 satisfies the stability condition L,, V < -8V and V (xx) 2 <<[[==]), where a class-K function For brevity, we fix 8 = in this paper such that the decay expo- nent of the Lyapunov function is 1. Since the evint-triggered controller is a discrete time realization of the original feed- back controller u, the corresponding exponential decay rate of the Lyapunov function is less than 1. Therefore, we design the event function h = h(x,, e) as
h VV(x) (f(z, u(x + e)) f(x, u(x)))
(3)
A continuous function a : (0, co) - (0, oo) is said to helong to class- K if it is strictly increasing and a(0) =
3

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Neural Event-Triggeen Triggered Control with Optimal Scheduling
VV
-Xxx(xxxxxxxx + + +()x fx,
xx
Initial
Target
trigger
untriggered
NLC
Neural (TC()
Communication Cost
Low
High
Figure 2. Illustration of the Neural ETC with optimal scheduling.
with 0 < 1, such that the event veetttriggered controlled
parametrization in (Zhang al., 2022a) as follows,
system satisfies
Ve ICNNo(x) ICNNo + =||==|? u diag (2: NN or NN (x) (
VV(z) f(x, f(x, e)) s VV(x) -fr, u(z)) + +v)
(5)
(1)))))(z))
(4)
where diag(m) transforms vector to diagonal matrix with (diag()));; = 6, ICNN, and NN represent the input convex neural network and the feedforward neural networks, respectively, the detailed formulation is provided in Appendix 1.3.1. We minimize the continuous function norm |uoloco) by regularizing the Lipschitz constant of the neural network, ization method in (Yoshida & Miyato, 2017) to minimize the spectral norm of the weight matrices (W Vi in us with the regularization term Lup E (WW,)) +.,)? To solve the substantially non-convex optimization prob- lem, we relax the original hard constraint L Vo(x)) + Vo(x) S a soft constraint in the loss function as Lytab EN (Esn, Vo(x.) + Vo(z.))
Hence, the exponential stability of the event-triggered con- trolled system is assured with exponential decay rate a.
As illustrated in Fig. 2, the NLC is used as an example method be compared with the Neural ETC. The control value is updated when an event is triggered, i.e., the event function & equals to zero. Our method achieves the expo- nential stability and has the least triggered events, leading to the lowest communication cost of updating the control value.
apply the spectral norm regular-
3.1. Path Integral Approach
Parameterization. In order to design the feedback con- troller such that its event-triggered implementation stabilize the unstable equilibrium efficiently and has the largest mini- mal inter-event time, we consider the following parameter- ized optimization problem.
Calculate gradients of t.. To proceed, we handle the objective function related to the triggering times. Instead of directly training the parameters Φb based on the direct samples of Ve. Us and f(z, u (x)) as done in neural certificate-bast controllers, we have to numerically solve the controlled ODEs to identify the triggering times. To proceed, we need to calculate the gradients of t4 for opti- mizing --- optimization. We employ the neural event ODE method as: tk+1, (t4-11) = ODESol versooteeventtccc (tx.).. f, us tk), where ODESolveEvent is proposed by (Chen et al., 2020), which introduces root solver and adjoint method (Pontryagin, 2018) to the numerical solver and deduce the gradient dha from the implicit function Be theorem (Krantz. & Parks, 2002).
min 0.6
min
+ (Illu(()))oc())
(1uST) 1x+1 (k)
s.t. i == f(x, us (x + e)), t [tx, tx+1),
f(x. (x a + +e), E [t,, tx+1);
- term loss function during gradient- based
=0) wo e(tx) 0, Vo(0) 0, (0) = 0,
tx41 inf (t het (x()) = 0}
t>th
a([[=]) Vo(x)) 0, Li., Vo(x) + Vo(x) $ O.
Here, d is a predefined weight factor, T the temporal length of the controlled trajectory. a is class-K function, and he,
L, Vo
(f(x ux( + c)) - f(x, uz(x))) - (Vo(x(t)) is the pa- rameterized event function. To ensure the neural functions
Reduce computation complexity, We denote by ty (z) the km triggering time from initial value to = 0, x(0) = x. The computation cost of SeeSoveEEennt is O K KTd)),
Vo, us satisfy some constraints naturally, we adopt the

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where qa, (.,(.) 2 0 is the output of the NNs (Wehenkel & Louppe, 2019). We regularize the inverse of integrand to minimize the Lipschitz constant of o.. We apply the spectral norm regularization Lnp defined above to minimize Neural Weent-Triggered Control with Optimal Scheduling
where M is the batch size of the initial value (0)
j=
orem 3.2 into the training process is unrealistic and cum- bersome, because the error state e in condition (iii) should depend on æ and we cannot determine the sampling distri- bution of e before training. To solve this problem, we split the inequality in condition (iii) into a sufficient inequality group as
K = 1. M Em (0)), (0)( = (z) :
l S T}
is the number of triggering times before T, and L
the iteration times in the root solver. In this case, the com-
putation cost pivots the sampled batch and its variance is hard to decrease. In addition, the numerical error in ODE solver accumulates over the triggering time sequence (4+)- To mitigate these issues, according to the time invariance property of ODEs, i.e., fx+1 - -ty (xx)) = t (xx())),
VV. (f(e, u( + e)) - ((x))(())))) s ((le])
(7) (8)
G.. V(x) - -((1=))
we recast the problem of solving M batch trajectories (tx(tz)) & S (0) ~ (o(x)) J of controlled ODE as solving MK trajectories (t)( 4 I (0) qo (x) up to t1. Here, K represent the expectation of K. In practice, we directly treat MK together as a single hyperparameter M. Then the triggering times contribute into the loss function as Levent = M Σ [[]] Finally, we train the overall parameterized model with the total loss function as follows,
Lr. V(2, u(x e)) - -a([=)] + +[lee))
The dependence on x of right term y in Eq. (7) can be omitted when the state space 'D is bounded, which occurs in most real-world scenarios. The Eq. (7) implies that the Lipschitz constant of y is related to the Lipschitz constant of u. Furthermore, we notice that if we replace the event function in Theorem 3.2 by the following event function with stability guarantee,
L[ 0) Luab + X1 Ltip + Levvent
(6)
The whole training procedure is summarized in Algorithm
h (V-(f(x ut + e)) f(z, (u(z)))-a((le)) (9)
Remark 3.1. A more reasonable augmented distribution should takes the form as (x()) = K-1 > 4s (xr). where Ftg ~ qss (z)) is deduced from the initial distribu- tion to and the ODE integration from to 4-- Since la
then the inter-event time of these two event functions hold the relation +*+1, fk.b S * - + due to Eq. (7). Therefore, the inter-event time of event function h is also lower bounded by Tu in Theorem 3.2. We summarize the results in the following theorem.
varies for different initial value and cannot be determined in advance, fix qo qu for simplicity.
3.2. Monte Carlo Approach
Theorem 3.3. For the event-triggered controlled dynamics in Eq. (1) with event function h defined in Eq. (9), the state space D is bounded, the Eqs. (7), and the conditions (i). (ii) in Theorem 3.2 hold, then the minimal inter-event time et d1,- here la-
Although the proposed algorithm works efficiently in low dimensional ODEs, the high computation cost and accumu- late error caused by the ODE solver affect its performance in higher dimensional tasks. To circumvent this drawback, we propose a Monte Carlo approach for training the Neu- ral ETC. Inspired by the event- triggered scheduling theory in (Tabuada, 2007), we provide the following estimation of minimal inter- time.
is lower bounded by Ti. = Iy log al.- Tit is the Lipschitz constant of a1, is constant depending on V, f, DD
The proof is provided in Appendix A. With this theorem, we come to a Monte Carlo approach for training the Neural ETC framework by directly learning the parameterized func- tions Vo, oop,, and control function us simultaneously, as well as regularizing the Lipschitz constants of us and ap.- For constructing neural class-K functions, we adopt the monotonic NNs to construct the candidate class-KC function as
Theorem 3.2. Consider the event-triggered controlled dy- namics in Eq. (1), the following assumptions are satisfied: (i) f u²) - f(x, u) < l (I|=' xI| + ||u uIl); (ii) [u(x') u(x)) I| < Lu |x - ==:l: (iii) f V(x, u(x e)) S [[([ + (([ee]) for some class-K K functions a, Y with a ()((ee)) < P|le||. Then, the minimal inter-event time implicitly defined by event function h = (((eell) is lower bounded by Th = 1y log P+1 14
ae., (z) = = 1. g0, (s)
(10)
The detailed proof is provided in Appendix A.1.2. Accord- ing to the theorem, the lower bound of minimal inter-event time increases as Lipschitz constants of a o and u de- crease. Therefore, we can maximize the minimal inter-event time by regularizing these Lipschitz constants. Nonethe- less, directly integrating the conditions and results of The-
the Lipschitz constant of controller u Finally, we train
5

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Neural Event-Triggered Triggered Control with Optimal Scheduling
the overall model with the loss function follows,
potential function Vg. obtain the theoretical stability guarantee for our approach. Based the Theorem 4.1 and Theorem 3.2, we could provide necessary condition for the optimal event-triggered control with the largest minimal inter-event time by utilizing the lower bound of the inter- event time and the projection operation.
N
Lata
(Ctu Vo(zz)) + a.0 (0, (axi)
N
1=1
Ma
(11)
La
Ma i=1 -0 (zi))
Theorem 422. (Optimality guarantee) Denote the Lipschitz constant of the controller u on state space as Lu then the optimal control with the largest minimal inter-event time satisfies.
L(Ø, 0, 0x) = Tutab + X1 Ltip + -
The specific training procedure of this algorithm, dubbed Neural ETC-MC, is shown in Algorithm 2.
u E arg min Lu- L((V)
(13)
Remark 3.4. To obtain stronger exponential decay rate of V, multiply the term a[[==]] in Eq. (9) by (0, 1) in realization. Similarly to Eq. (4), the controlled vector under event h satisfied
Furthermore, for any candidate controllers the optimal condition can be simplified as
T m(u,u(V)) E arg min ((
(14)
VV. (f(x, u(x S - (11-0)a([12]))
(12)
This theorem is direct result from the Theorem 4.1 and Theorem 3.2, and the projection operation simplifies the constrained necessary condition in Eq. (13) to the uncon- strained condition Eq. (14). We can easily provide optimal- ity guarantee for the neural network controller us and the Lyapunov function Vo by regularizing the Lipschitz constant of π (u,,,
Then the exponential decay rate of V is 1 a. The lower bound of inter-event time can be obtained by replacing 4,-1 with o - 1 in Theorem 3.3.
4. Theoretical Guarantee for Stability and Optimality
In this section, we provide several theoretical results for rigorously guaranteeing the stability and optimality of our neural controllers. Firstly, we note that the NNs trained on finite samples cannot guarantee the Lyapunov stability condition in the loss function is satisfied in the whole state space with infinite data points. To circumvent this weak- ness, we introduce the projection operation in the following theorem.
5. Experiments and Analysis
In this section, we demonstrate the superiority of the Neural ETCs over existing methods using series of experiments from low dimensional tasks to high dimensional tasks, then we unravel the key factors of Neural ETCs. More details of the experiments can be found in Appendix A.3.
5.1. Benchmark Experiments
Theorem 4.1. (Stability guarantee) For a candidate con- troller u and the stable controller space U(V) {u f V V < 0}, we define the projection operator as,
Benchmark dynamical systems.
(I) Gene Regulatory Network (GRN) plays a central role in describing the gene expression levels of mRNA and pro- teins in cell (Davidson & Levin, 2005), here we consider a two-node GRN, 11 01 +07
TI ((,u(V)) A u max (0, f. V - V) VV.
V
a²
b1 3"
e"
kx1, 2o2
If the controller has affine actuator, then we have T (u,u(VV) E UVV), the projected controller is Lipschitz continuous over the state space 'D if and only if 'D is bounded. Furthermore, under the triggering mechanism
02 +a b2 4" - kzz, where the tunable parameters 11 a2 b, and b2 represent the strengths of auto or mutual regulations. We aim at stabilizing the system from one at- tractor to another attractor via only tuning aj in time interval [0, 20).
VV(2) [f(z, u(x +e)) f(x, ((a))]][oV[l = 0,, E 1), =1(4))=((1)) (++1)
(2) Lorenz system a fundamental model in atmospheric science (Lorenz, 1963): å a x). y pz y
the controlled system under (f) is assured exponential stable with decay rate 1 o, and the inter-event time has positive lower bound.
æ2, ; xy 3z. For this chaotic system, we stabilize its unstable zero solution by an fully actuated controller
u (uz U, Uz) in time interval [0, 20.
We provide the proof in Appendix A.1.4. By applying the projection operation to the learned controller us and
(3) Michaelis-MMeri Menten model for subcellular dynamics (Cell) captures the collective behavior of the coupled cells (San-
6

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Neural Event- Event-Triggeree Triggered Control with Optimal Scheduling
Table 1. Comparison studies of benchmark models and dynamical systems. Best results bolded. Averaged over 5 runs. The dimension of tasks are: GRN (2-D), Lorenz (3-D), Cell (100-D).
Method
Nomber of triggors
Mininal inter-ovent time"
MSB under litile triggers..
GRN
Lorenz
Cell
GRN
Loreaz
Cell
GRN
Lorenz
Cell
BALSA (Fan et 2020)
12(=-4)
273(+24)
18C=4)
0.29)+0.07)
60-4c260-4)
3e-3(::1e3)
005
7.20)+2.255
29.75(=12.72)
LQR (Heemels 2012)
1816(214)
2000(:0)
1)
6e-3(13e-))
0.02(10.02)
5.19(10.59)
53.02(17.03)
2o-3(22e-3)
Quad-NLC (Jin al., 2020)
1914(=107)
433(1337))
551(1-220)
6e-6(11e-6)
4e-6(=10e-7)
5.2
7.00/41.28)
18.40)=1.44)
NLCC (Chang al., 2019)
23(=2)
1602(4795)
15(1:3)
Se-8(18e-9)
1o-8(211-7)
6c-6(111:55
0.20(10.05)
10.56(121.14)
27.76(=5.52)
IRL ETC (Xue al., 2022)
131(428)
2000C
170(=14)
Re-3(4)
0.00010.00)
3e-3(48e-5)
4.94(10.77)
9.76/42.13)
38.12(=0.11)
Cirtic-Actor ETC (Chong et al.. 2023)
605()293)
69(18)
330(25)
5e-K(3e-55
2o-3(228-1)
1c:3(18--5)
8.09(10.81)
7.22(42.05)
38.13/=0.01)
ETS & Liu, 2019)
81(487)
(120(45.62)
0.05(=2e-16)
9c-4(155-4)
0.02
0.19:42.277
2.42(41.02)
4.96(77533)
PGDNLC (Yang et al., 2024)
23(22)
340(2216)
60t::15)
0.37
7e-1(2Bo-1)
0.39(
11.35(17.69)
41.0%
Neural ETC-PI (ours)
20(-33)
20(141
[1(20.000
0.26110.17)
0.02/00000"
0.95(200003)
0.05(144.33
0.110200112
3e 1e-9)
Neural ETC-MC (ours)
4(:44)
13(21)
2(10000)
15.52(28.54)
0.01(#8e-3)
07.18(20.60)
0.07(10.03)
0.29(10.31)
1.66(0011))
This model has two important equilibrium phase, inactive phase indicating the malignant state and active state indicating the benign state. We consider n = 100 and regulate the high dimensional model from the inactive phase to the active phase by only tuning the topology structure {= in time interval (0, 30).
hedrai et al., 2022): 4; = - Bz, + Σi=1 Aij 1447
methods.
For the communication cost, our Neural ETCs need the least number of triggers in the same time interval while have the largest minimal inter-event time compared to other methods, leading to the most optimal scheduling in actual implementation. In addition, the MSE results illustrate our Neural ETCs have the ability in stabilizing the systems at various scales with limited communication resources. We also find the Neural ETC-PI and Neural ETC-MC form the trade off in scheduling and the stabilization performance, we further compare them in the next section.
)
All these systems have application scenarios and urgently call for event- -ringerrggeeed control with minimal communica- tion burden, we summarize the motivation for selecting the them in Appendix A.3.5.
The results underpin the practicability of the Neural ETCs. Take GRN model for an example, the auto regulation strength ai can be adjusted externaly through the appli- cation of repressive or inductive drugs in a typical exper- imental setting (Wang et al., 2016). In reality, the drugs can only be administered a few times and it takes time for the drug to take effect, requiring the controller should only be updated at several times with large interval. Therefore, while all the benchmark methods successfully regulate the GRN to the target gene expression level in simulation, only the Neural ETC-MC is acceptable.
We benchmark against the exten- sively used neural Lyapunov control (NLC) method (Chang et al., 2019), an improvement version of neural Lyapunov control via constructing quadratic I Lyapunov function pro- posed in (Jin et al., 2020), dubbed as Quad-NLC here, a integral reinforcement learning (IRL) based ETC (Xue et al., 2022), a critic-actor neural network based ETC method (Cheng al., 2023), and two latest SOTA methods ETS (Li & Liu, 2019) and PGDNLC (Yang al., 2024). We also compare with the classic linear quadratic regulator (LQR) method, BALSA (Fan et al., 2020), an online control policy based on the quadratic programming (QP) solver, and our Neural ETC variants: Neural ETC-PI and Neural ETC-MC. We implement all the control methods with the similar kinds of event functions proposed in Eqs. (3). For a fair com- parison, we set the number of hidden units per layer such that all learning models have nearly the same number of total parameters. We provide further details of model selection, hyperparameter selection and experimental configuration in Appendix A.3.
Benchmark methods.
Combining online and offline policy. In the context of event-triggered control, Table demonstrates that the online control method outperforms other offline policies. However, the online policy's computational cost is high due to solving the quadratic programming (QP) problem at each realiza- tion time. In contrast, our Neural ETCs achieve superior performance compared to online methods while maintaining the same computation cost as the offline policy during the control process. The event-triggeres control employs an event function that continuously assesses whether an event is triggered, effectively acting as an online solver to deter- mine real-time control values. Consequently, we can view event-triggered control as an online realization of the offline policy, inheriting the advantages of both online and offline approaches
Results. Table 1 summarizes the control performance re- sults in terms of the triggering times in the same temporal length, the minimal inter-event time and the mean square error (MSE) between the target state and the controlled trajectories with no larger than 10 triggering events, repre- senting the control performance in limited communication resources. We see that our Neural ETC variants achieve su- perior performance compared to the other online and offline
7

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Neural Event- Tringerrgeeeed Control with Optimal Scheduling
5.2. Comparison Between Neural ETCs
terized controllers are already stabilization controllers in the future work. For reference, in Table 1 Neural ETC-PI is using 0.5, X2 0.05 and Neural ETC-MC is using σ = 0.5. X2 = 0.1.
We further evaluate the strengths and weaknesses of the Neural ETC-PI and Neural ETC-MC. As shown in Table 2, the Neural ETC-MC is more efficient in training process, especially in the high dimensional tasks. Nevertheless, the temporal variance of the controlled trajectories of Neural ETC-PI is far below that of Neural ETC-MC, implying Neural ETC-PI is more robust in the control process. These two algorithms thus are complementary in applications. In addition, the training time of Neural ETC-PI in 2-D GRN and 3-D Lorenz has significant difference, the reason is that the minimal inter-event time of the former is larger than the latter (see Table 1), requiring more time to solve tj.
Table 3. Performance under various event loss weight Au.
Method
Neural ETC-PI
Neural ETC-MC
X2
0.005
0.05
0.5
0.01
0.1
1.0
Triggering times
114
29
34
37
10
10
Min Inter-event time
0,010
0.008
0.025
0.02
0.07
0.06
(MSE)
8e-8
7e-4
0.32
3.92
0.25
0.53
5.4. Essential Factor of Neural ETC
Table 2. Comparison of Neural ETCs (denoted by NETC) in terms of training time and variance of stabilized trajectories.
We investigate the essential factor in the Neural ETC frame- work that determine the optimization of scheduling. We plot the convexity of V function (Tr( (T) and the strength of the variation of controller Vll in the training process, and compare their evolution with triggering times of the cor- responding trained controller. Fig. 4 shows that the || Vull plays a leading role in minimizing the triggering times as it has strong negative correlation to the triggering times while the convexity of V function does not. We also observe an early convergence phenomenon of the triggering times and Vull simultaneously in Neural ETC-PI from Fig. (c))
Training time |
Temporal variance |
Model
NETC-PI
NETC-MC
NETC-PI
NETC-MC
GRN
1230
32
5e-4
7c-3
Lorenz
503
29
4e-3
0.09
FHN
4634
62
3e-15
2.78
5.3. Ablation Study
Neural BTC-PI
Neural ETC-MC
Nearal RTC-PI
Neural FTOME
20
Tringoring Times
(a)
Inter Time
(b)
[
Orooddy
Tiggering
10
10
Epocla
Epodla
Epodas
10
0.1
0.9
0.1
0.9
Figure 4. (a) Convexity calculated as the trace of the V2V on 1000 points in [-2.5, 2.5) (b) Triggering times and (c) norm of Vu the training process.
Figure 3. The solid lines are obtained through averaging the sampled trajectories, while the shaded areas stand for the variance regions.
6. Related Work
The parameter corresponds to the exponential decay rate of Lyapumov function along controlled trajectory in Eqs. (12). We investigate the influence of a in apply- ing Neural ETC variants to Lorenz dynamic. The results in Fig. 3 suggests the best choice is a = 0.8. Then we investigate the influence of weight factor X2 of event loss in Eqs. (11, and summarize the results in Table 3. We find the small A leads to poor triggering scheduling because the event loss does not play leading role in training, the large X2 will break the stabilization performance because the optimization function of event loss is not guaranteed to satisfy the stabilization loss. This phenomenon inspires us to extend the framework to the setting where the parame-
Neural control with certificate functions. Previous works in neural control establish the performance guaran- tee via using the certificate functions, including Lyapunov function for stability (Giesl & Hafstein, 2015; Chang et al., 2019), barrier function for safety (Zhang et al., 2022b; Ames et al., 2016; Taylor al., 2020; 2019; Taylor & Ames, 2020; Peruffo et al., 2021), and contraction metrics for stability in trajectory tracking (Singh al., 2021; Tsukamoto et al., 2021). However, all these feedback controllers require im- practically high communication cost for updating the con- troller continuously when deployed on the digital platforms. We solve this challenge in limited communication resources and improve the performance guarantee at the same time.
8

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Neural Svent-Triggered Control with Optimal Scheduling
Event-triggered control. The pioneering works (Aström & Bernhardsson, 1999; Åarzén, 1999) highlighted the ad- vantages of event-based control against the periodic imple- mentation in reducing the communication cost. Since then, (Tabuada, 2007) investigates the sufficient conditions for avoiding the Zeno behavior in event-triggered implemen- tations of stabilizing feedback control laws, (Henningsson al., 2008) extends the event-triggered control to the linear stochastic system and (Heemels et al., 2008) gives the sys- tem theory of event-triggered control scheme for perturbed linear systems. Machine learning methods have also been introduced to the ETC settings, (Xue et al., 2022; Cheng et al., 2023) employ the critic-actor RL. structure to solve the dynamic Hamilton-Jacobi-Bele Bellmmnn equation under the ETC, (Funk al., 2021) cultivates a model-free hierarchical RL method to optimize both the control and communication policies for discrete dynamics, and (Baumann et al., 2018) applies deep RI. to ETC in the nonlinear systems. All the previous works focus on the stabilization analysis of the controlled systems, the existence of the minimal inter-event time (and hence avoids the Zeno behavior), and directly in- troducing machine learning methods to ETC. To our knowl- edge, we are the first to study the optimization scheduling problem of ETC in the continuous dynamics.
8. Conclusion
This work focuses on a new connection of machine learning and control field in the context of learning event-triggered stabilization control with optimal scheduling. In contrast to the existing learning control methods, the learned event- triggered control, named Neural ETC, only updates the con- trol value in very few times when an event is triggered. As consequence, our Neural ETC can be deployed on the actual platform where the communication cost for updating the control value is limited (e-g. tuning the protein regulation strength in cell via drugs). The superiority of the Neural ETC over the existing methods is demonstrated through series of representative dynamical systems.
Acknowledgements
L. Yang is supported by the China Scholarship Coun- cil (No. 202406100251). Q. Zhu supported by the China Postdoctoral Science Foundation (Grant No. 2022M7208177 17), by the Shanghai Postdoctoral Excellence Program (Grant No. 2021091), and by the STCSM (Grants No. 2151 11511100200, No. 22ZR 1407300, and No. 23YF1402500). W. Lin is supported NSFC (Grant No. (1925103), the IPSMEC (Grant No. 2023ZKZD04), and the STCSM (Grants No. 22JC1401402, No. 22IC1402500, and No. 2021SHZDZX0103).
7. Scope and Limitations
ODE solver. The use of the fixed step ODE solvers in finding the triggering times in the training process is less optimal than the adaptive ODE solver. One can still improve the performance of the framework by applying the adaptive solvers with higher accuracy tolerance with stronger com- puting platform. However, in practice the performance of the Neural ETC did not decrease substantially when using adaptive solvers. In addition, the employ of ODE solvers in the Neural ETC-PI may not always work, especially for sys- tems described by stiff equations, stiff-based ODE solvers can be introduced mitigate this issue (Kim et al., 2021).
Neural ETC for SDEs. Although the current Neural ETC framework works efficiently in ODEs, many real-world scenarios affected by the noise are described by stochas- tic differential equations (SDEs). The major challenge for establishing the Neural ETC framework for SDEs ensues from the stochasticity of the triggering time. Specifically, the triggering time in SDEs, i infrzo(1 (t: h((=(())) = 0} initiated from any fixed (0) with (0)) < 0, is stop- ping time. Therefore, f is random variable and can take different values in different sample paths. In this case, none of the existing methods can find t for SDEs as a counterpart of ODESolveEvent for ODEs.
9

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Neural Event-Triggerss Triggered Control with Optimal Scheduling
Impact Statement
Dawson, C.,, Qin, Z., Gao, S., and Fan, C. Safe nonlinear
control using robust neural lyapunov-harrin functions. In Conference Robot Learning, pp. 1724-1735. PMLR, 2022.
This paper presents work whose goal to advance the field of Machine Learning. There are many potential societal consequences of our work, none which feel must be specifically highlighted here.
Dawson, C., Gao, S., and Fan, C. Safe control with learned certificates: A survey of neural Iyapunov, barrier, and contraction methods for robotics and control. IEEE Trans- actions on Robotics, 2023.
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Neural Event- Trent-Triggered Control with Optimal Scheduling
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Neural Event- Triggered Control with Optimal Scheduling
A. Appendix
A.I. Proofs and Derivations
In this section, we introduce some basic notations and then provide the proof's of the theoretical results.
1.1.1. NOTATIONS
Throughout the paper, we employ the following notation. Let (zz, y) be the inner product of vectors æ, y R For second continuous function f(x) Fd IR, let V denote the gradient of f(x), that is, V denote the Hessian matrix of f. For the two sets A, B, let A C B denote that A is covered in B. Denote by log the base logarithmic function. Denote by || 1-1 ll the L²-norm for any given vector in R. Denote by | the absolute value of scalar number or the modulus length of complex number. For A (asz), matrix of dimension d x r, denote by ||A| Σ Σj= dly the Frobenius norm.
Notations.
A.1.2. PROOF OF THEOREM 3.2
Theorem A.1. Consider the event-triggered controlled dynamics in Eq. (1), if the following assumptions are satisfied: (i) [a'',,, u²) ((x,u)), S l (I|æ' æ|| + ||ul - wll): (ii) [|((x') - u(x)) S Lu bul| ælI; (iii) C V(x, u(x + e)) S -a([)=]) + (([ee])) for some class- functions a, 7 with a (>((eel)) S P|le||- Then, the minimal inter-event time implicitly defined by event function h a(1==])) ((lee]) lower bounded by Th 1 log
p+
From the condition (iii) and the definition of the event function, we have the triggering time happens after Plle|| |a||- Therefore, the inter-event time lower bounded by the minimal inter-event time defined by the event function h [[] - || Now come deduce the estimation of the inter-event time of h, i.e., the time from ||e|| 0 to
||l| L |=||_ Consider the dynamic of |e||
we have
e
(e 2)
dt
dt (xx Te xx)
(e)e)) 1//2/ T é(x
T x)
(2 (a) 11/2 #tee ( e)
x
e TTè
x TT
||e|
||||||
||=0
|||]||]
ac
||e| [e]][[=]]
|e||
3c
|e||
|e||
|az]]
If (xx,
u(x
+ e)ll
|||||
1 +
[x:
+ 1yyyuu (1|20||
+ ||el])
|e|
1 +
||0|||
||=]|
||l||
|||||
1 (.) 1y
+
||=
|a=||
|ell xc]
By denoting z
we have the triggering time of happens after the variable = increases from 0 to p. The dynamic of
is
I (1 + (w) + 1ybuz) (1
20 O,
2T =
P
13

---

Neural Event- Triggered Control with Optimal Scheduling
We have
dz bdt, (1 az y) where a - 1yhn [ b= = (1 + (w). Then we have
de
1
d=
(1 + (1 (1 + 2)
a
1 az
a(1 + 2)
(dlog(1 + az) d log(1 2))
bdt
By integrating the above equation, we have
1 p) bT
log(1 + p) log((1 +
a
1+ p
T b(a - 1)
log
1 +
1
1+ p
log
6(1 a)
1 1 a
p
1
P+1
log
I
1ytu P + 1y(1+
which completes the proof.
A.1.3. PROOF THEOREM 3.3
Theorem A.2. For the event-triggered controlled dynamics in Eq. (1) with event function h defined in Eq. (9), if the state space D is bounded, the Eqs. (7),() and the conditions (i), (ii) in Theorem 3.2 hold, then the minimal inter-event time is lower bounded by Ti, = Ty log 1111
here La is the Lipschitz constant of a g/u
d 7
From the Eqs. (7), we know that the triggering time defined by h in Eq. 9 is larger than that defined by in Theorem 3.2. Notice in Theorem 3.2 P is a tight upper bound Lipschitz constant of a -1 o Since the state space D is bounded, from Eq. 7, we set y as the tight estimation of VV (f(x, u(x + e)) f(x, u(x))), the Lipschitz constant ? can be bounded by
max mED
[V
Then we get
Lip(a - < max [V(
=ED
By denoting e = max (*)]ll and replace with da- by in Theorem 3.2, we obtain the final estimation Tj.
14

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Neural Event- Event-Triggeree Triggered Control with Optimal Scheduling
A. 1.4. PROOF OF THEOREM 4.1
Theorem A.3. (Stability guarantee) For candidate controller and the stable controller space UVV {u: L
0}, we define the projection operator
T (u)) é u max Lg. V..V -V). VV.
|V
If the controller has affine actuator, then we have π Tu, (V)) E UV),, the projected controller is Lipschitz continuous over the state space D if and only if is bounded. Furthermore. under the triggering mechanism
VV(x) -f(x u(x + e)) - f(x,,((()))) u(x))) - oV =0, a (0, x(tx) - x(t), t [tx, tx+1)
the controlled system under uu, is assured exponential stable with decay rate o, and the inter-event time has positive lower bound.
Proof. begin with, we check the inequality constraint in U(V satisfied by the projection element, that is
f. V Mu=s(uuVV) S -V.
Since the controller has affine actuator, from the definition of the Lie derivative operator, we have
Cr V u= VV (f + max(0, C. V + +V)
VV)
VV 2
max 0, C V +V)
=V. (f u)) VV
VV
VV
C.V + -VV.
Next, we show the equivalent condition of the Lipschitz continuity of projection element. Notice that Lip(D), then have
म (u, E Lip(D) ? max(0, L.V VV) VV E Lip(D).
VVI2
VV
Since ||
is continuous unit vector, and naturally is Lipschitz continuous, we only need to consider the remaining
[V]
max (0, L. V +V)
According to the definition, all the functions occured in this term are continuous, so we only need to bound this term to obtain the global Lipschitz continuity, that is
term
/VII
max (0, C. +V)
ma C, V + V) < +o0.
E Lip(D) + sup
|VV||
|VV||
xED
When L.VV V < -V, obviously we have max(0, L., + V) 0 < too, otherwise, since V 2 lP we have
C.V +V LvV = =||=||" 0(1 s0(12> + 00).
Thus, have
max(0, (0, Lu + +V)
sup mED
< to0
||||| < +oc #ED
sup
[V
The positive lower bound of the inter-event time comes from the Theorem 3.2. We complete the proof.
15

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Neural Event- Event-Triggerrs Triggered Control with Optimal Scheduling
A.2. Algorithms
In this section, we provide the algorithms of Neural ETC-PI (1) and Neural ETC-MC (2), Firstly, supplement the warm up stage for path integral algorithm to accelerate the convergence of training process.
Warm up. At the beginning of the training process, the stability constraint is not satisfied, which leads to the solution of the event function h does not exist. To ensure the training process can proceed smoothly, we pre-train the parameterized model with
L 0, (ci)) Luaab + A Liip
(15)
Algorithm Neural ETC-PI: Path Integral Algorithm
1: hyperparameters:
M,
D Sample size and batch size D. Learning rate and max iterations Data distrituion, weight factors From Eq. (5)
8.m
p(D),>1 \1, A2
2: initialize
(ob 0)
3: generate dataset DN {x (D)
4: for r = 1: m do
5: w + w V (ww)
Warm up in Eq. (15)
6: end for
7: for do
8: (x,( ~ DN
Sample batch data
9: 4,,1 (44,1)
www.ooeeseeee...oo..o. (ensolveRvent((((()) u 0)
10:
(w)
From Eq. (6)
11: end for
12: return ug. Ve
Algorithm 2 Neural ETC-MC: Monte Carlo Algorithm
hyperparameters:
N. Ma \1, X2
Sample sizes and weight factors b Learning rate and max iterations Distributions of state and erro > Eqs. (5),(10)
8.m
2: initialize w = (Φ, 0,) NN
3: generate dataset { x (x.) Ma ((D) x p(x)
4: for m do 3V (w))
5:
From Eq. (11)
6: end for
7: return ug. Vo. ae.
16

---

Neural Event- Event-Triggeree Triggered Control with Optimal Scheduling
A.3. Experimental Configurations
In this section, we provide the detailed descriptions for the experimental configurations of the benchmark dynamical systems and control methods in the main text. We implement the code on single 17-10870 CPU with 16GB memory, and we train all the parameters with Adam optimizer.
A. NEURAL NETWORK STRUCTURES
For constructing the potential function V, we utilize the ICNN as (Amos et al., 2017):
oW6 + bb).
214 = o(U,z; + W, + b.). i= 1,
p(x) = 2k:
((x) (p(x)) p()) + s||=||?
where is the smoothed ReLU function defined in the main text, W; Rhixd Uj (R+, (0))hx>>>> Rd. and, for simplicity, this ICNN function is denoted by ICNN(ho: hi," ,hg-1). We set 1e-3 as default value for all the experiments;
The class-C function is constructed as:
q1 ReLU(W6ss + bo);
4++ ReLU(W,qq b;), 1, 2.
9 FLU((W--1g)) 19 bx-1),
a a(x) !,"
(x(s)ds
where W; [R and this class-N. function denoted by KC(ho, 11, (k);
The neural control function (nonlinear version) is constructed as:
æ1 [[ppectralNoxxm(We bo)),
2;+1 F[spectzalNorr(Miia: + b;)), i i= 1,
NN(x) rpectralNoomm//
(xz) diag(æ x NN))))))) or NN(x) NN(x"),
where F(-) is the activation function, SpectralNoz is the spectral norn function from (Yoshida & Miyato, 2017), W; E Rh and this control function is denoted by Control hi, h4+1). 1). Since deploy the ApectralNorm package in our algorithm, the weight factor X for Lipschitz constant of u is automatically set the default value in this package and we do not tune it in our experiments due to its good performance.
The standard neural network is constructed as:
21 F(W + bo):
1;+1 = F(W + bi), 1= 1, k- NN(x) Wg
where F(.) is the activation function, and this standard function is denoted by MLP(ho, h1, hh,4),
A.3.2. GENE REGULATORY NETWORK
Here we model the controlled gene regulatory network (GRN) as
n
x7 +
01
kx
g
+
8
x
s"
42
02
by
kxz
27
g
17

---

Neural Event- Event-Triggered Control with Optimal Scheduling
where the under-actuated control u only the protein regulation strength d1- We specify d1 02 1, bj b2 0.2, 2, k 1.1, 0.5. The two attractors of the original model is
P1: 25))))) (0.62562059, 0.62562059). P2 : (xf.2)) = (0.0582738, 0.85801853).
We aims at stabilize the attractor P3 with low protein concentration to P1 with high protein expression level. We slightly modify the neural networks s.t. (PP)) = 0, u P1) = 0. e.g. V = V()) V(P1), v = u(x)) - u P1). Since these two attractors are close the Euclidean space, it hard for algorithms to identify them from states with numerical error. To address this issue, we rescale the original system as म 10 22 10zg to enlarge the attractors. For training controller u, we uniformly sample 1000 data from the state region [-10, 10]. We test the performance under different learning rate Ir {0.01, 0.03, 0.05) and pick the best one, the considered control methods set as following,
Neural ETC-PI. We parameterize (()) as ICNN((2 10, 10, 1), u(x) as Control 20, 20, 1) with J ReLU. We set the iterations for warm up as 500, the iterations and batch size for calculating the triggering times as 50 and 10, the learning rate Ir 0.01, the weight factor for event loss as X2 10
1000
Neural ETC-MC. We parameterize (x) as ICNN((2, 20, 1), u(x) as (2ntrol 20, 1). We set the iterations as 500 + 50, the learning rate as Ir 0.05, the weight factor for event loss as X2 0.1.
NLC. We parameterize V (a)) as MLP(2, 20, 20, 1), u(x) as MLP(2, 20, 20, 1). We set the iterations as 500 + 50, the learning rate as Ir = 0.05, the loss function is
(Cmn)Vo(as)) + (Vo(z.))) + Vo(P1)²
L
N
P Quad-NLC. We parameterize ((x) as (a - PI) MLP(2, 20, 2) MLP(2, 20, 2)(x - Pi), u(x) as MLP(2, 20, 20, 1). We set the iterations as 500 + 50, the learning rate as = 0.05, the loss function is
Voll)) + Vo(P))
Volx.) Volxx)
BALSA. For this QP based method, we the object function
1 |112 + p1d}. u,di, 2
min
s.t.C V - V S d1,
where is the relaxation number. We choose V = = Pi 11122 P1 = 50. We solve this problem with the QP solver in cvxopt in Python package.
LQR. We linearize the controlled dynamic near the target P1 as
t = A(x P1) + Bu,
n(r])n-1
n( m-
@1 (8" + (xt)
bi (8" (x:)
A
n( n-1
n(x5) n
02 (s" + (2
k by S + (2
(xi)=
(8" + (xf)n)z A
B
18

---

Neural Event- Triggered Control with Optimal Scheduling
We set the cost matrix in LQR
0
Q
10
R (0.1)
and solve the problem via lgr method in Matlab. (ax - P (a Pi). the controller is v (ax Pi) where K E R
The obtained Riccati solution S forms the Lyapunov function V
RIx2 is returned by the lqr solver. The Lie
derivative of the Lyapunov function is -xx P)) Q (a - Pi) with Q Q + K RK.
According to the implementation setting in (Cheng al., 2023), we consider the following event- triggered controller parametrized by the critic neural network We and the actor neural network Wa.
Critic-Actor ETC.
f(x) + (()))(x),
(*(x) = min f (x su u Ru) dt,
("(x) x Wa
u (xx)) = Wa
e, = Wa a + 29 (v)
Ka e Ea:
T
Wa oK.
OW.
E. VV* [f(x) (()) Tx +x++u u Ru,
K. = _e ec:
2
W. 0K.
OW.
here f is the original dynamics described above, g is the actuator taking the form,
x
(s" )2
The cost matrix Q, R are the same as that in LQR. In the event-triggered mode, the weights of critic and actor NN, W. and Wa, obeying the evolution dynamics as follows,
W. = 0, t E [t,, tx+1);
0Ka
W = W. aa aw.
, la+1,
W. = 0, 1 [lk (1+1;;
Oke
W We ae ow.""
tk+1,
where a, and a. are the learning rates of the critic and actor NNs, respectively. For the initial value of W. and W.. we employ the solutions from the above LQR solver W. S, W. K. We set the learning rate ag Ga = le - 2, the event function is set as h = |e| éthres, Ethres = 0.2 according to (Cheng et al., 2023), here = (ez, (ez];6ry Gzg) are the variables of error dynamics. The event function here is different to our proposed stability guaranteed function because of the lack of Lyapunov function in this method, we note that V* is only an auxiliary function used to find the dynamics of W.. W, and cannot be verified as a Lyapunov function. We have tuned the hyperparameters ae, Qg E {5e 4, le 3, le 2, 1e - 1}, ethres E (0.1,0.2. 0.3, 0.4, 0.5) and fix the parameters with the best performance.
19

---

Neural Event-Triggeres Control with Optimal Scheduling
Similarly to the Critic-Actor ETC, (Xue al., 2022) transformed the optimization control problem RL problem via abstracting the Hamilton-Jacchi-BR Bellman equation as the value function and approximating the optimal value function based preset basis activation function. Specifically, we consider the control problem parametrized by the critic neural network W as follows,
IRL ETC.
f(x) + g())u(x).
(x) = min /' (a Qr + u Ru) dr, (m'Qx=
0
(((x)= x T Wæ,
1 R R'g (2
"' u'(x) no
2m
1+|
x T Qx + E 1/"
E c((-r))
2no "n s)rdds dr, diag(R) (r1,"* ,mm);
1
K LB²
2
W 0K
OW
here/ f is the original dynamics described above, I the actuator taking the form,
21
g (s"h x: )
0
The cost matrix Q, R are the same as that in LQR. In the event-triggeree mode, the weight W of critic NN is updated as,
W = 0, t E [tk, tx+1),
W W 3 OK
aw 1 Lk+1
with 33 being the learning rates of the weight. We initialize the weight as W = (Wi, = 4)2x2. We set the learning rate as le - 2, the event function set as |e| (1 ( | X = 0.6 according to (Cheng et al., 2023). In the original work (Xue al., 2022), historical data is considered as multiple integral on time interval [t> t + [] C [t tx+1) like E Ej To simplify the caleulation, here merge the multiple historical data single integral on time interval [tx, t + [] with | min(l", +++1 -t), here [* 1.2 is predefined length of historical data. We tuned the hyperparameters in the same way with Critic-Actor ETC, and the final results are 0.1, 1.0, Xa = 0.1, Id(-).
Test configurations. For implementing the controller in the event-triggered mode, we set the event function for Neural ETC-PI, Quad-NLC, BALSA as
VV (f(x, e)) f(x, u(x)) -V(x))
event function for Neural ETC-MC as
VV (fffz,u(=+e f(z, u(x))) -
the event function for NLC
VV. (f(æ, u(x e)) (f u(x)))
the event function for LQR as (Heemels et al., 2012)
(a - 1)(x
Q T Pi) + 2(x - P1) SBKe,
where the a is set as 0.5 for all models. For the initial value, we set Bo = Ps + E., &, ~ U[-1,1],i 1, 1.... ,,,,, the random
seed is (2, 4, 5, 6, 7).
20

---

Neural Event-Triggenn Triggered Control with Optimal Scheduling
A.3.3. LORENZ SYSTEM
Here model the state of the Lorenz system under fully actuated control (W1, U2 U3) (2,
= oy 2) + u1
pz y .2 + U22
wy 3z u3.
We aim to stabilize the zero solution of this chaotic system. We consider a = 10, p 28, 3 8/3. For training controller u, we uniformly sample 5000 data from the state region [-10, 10). We construct the controllers follows.
Neural ETC-PI. We parameterize ((x) as ICNN(3, 64, 1), uxx:) as Control(3, 64, 64, 3) with F= ReLU. Since the Ode solver in the training process require high computational resources, we down-sample 2000 data from the original dataset for training. We set the iterations for warm up as 500, the iterations and batch size for calculating the triggering 100
times 100 and 10, the learning rate Ir 0.05, the weight factor for event loss as A2 2000
Neural ETC-MC. We parameterize (x) as ICNN((3, 64, 1), u(x) as Control (3, 64, 64, 3). We set the iterations as 500 100, the learning rate Ir 0.05, the weight factor for event loss X2 = 0.1.
NLC. We parameterize V(x) as MLP(3, 64, 64, 1), u(x:) as MLP(3, 64, 64, 3). We set the iterations as 500 + 100, the learning rate as Ir = 0.05, the loss function
Σ (CtngVo(az)) + (Vo(z;))*) + (Vo(0))+ L == N
notice that we select the last term the right hand side as (Vo (0)+ instead of (0) since the former performs better than the latter. We also resample 5000 data from [-5,5] 5] since the NLC performs poorly in the original dataset, the similar case holds for Quad-NLC.
Quad-NLC. We parameterize V(x) as x MLPP33, 64, MLPP(3, 3)æ,) u(z) as MLP(3, 64, 64, 3). We the iterations 500 100, the learning rate as Ir 0.05, the loss function is
N
(Sog Volxx)+ + VV(xz) + Vo)
BALSA. For this QP based method, we set the object function as
min _ 11 ///222222 2 + pi pidf.
1
u.di
s.t.C V dh.
We choose = 12 20.
LQR. We linearize the controlled dynamic near the zero solution
Ax + Bu,
0
4
P
-1
0
0
0
-3
1
0
B
0
1 0
0
0
1
21

---

Neural Event-Trisee Triggered Control with Optimal Scheduling
We set the cost matrix in LQR
5
0
Q
0
10
0
0
5
0.1
0
0
R
0.1
0
0.1
The obtained Riccati solution S forms the Lyapunov function V= = 1 Sx, the controller u = Kx where K E R333 returned by the lqr solver. The Lie derivative of the Lyapunov function is - Q x with Q = Q + RK.
Critic-Actor ETC. The updating procedure is the same that in Appendix A.3.2, the hyperparameters as a. au 5e 4, ethres 0.3, we note that for the chaotic system, the event-triggered dynamics is easy to explode when ac,a are slightly larger than le- 3. The actuator in this example is the identity matrix 13x3-
IRL ETC. The updating procedure is the same as that in Appendix A.3.2, we set the hyperparameters as 8 = le 2, a = 0.1, (.) = t-nh(-), 10, d = 1.0, X = 0.6. The actuator in this example is the identity matrix as g = 13x3-
Test configurations. We select the same event functions as those for GRN to implement the ovent-triggered control, except for setting 0.99 for LQR since it fails in the case 0.5. For the initial value, we randomly select points in the original dataset using numpy random. choice ce method in Python, and the random seeds are set as {3, 53,5, 8,8,9)
A.3.4. SERHAELIS-MENTER MODEL
Consider the coupled subcellular model under topology control as
100
x
4; Br,
A
E +
6A 1+ x
1
This dynamic has two attractor, inactive state P represents the cell apoptosis and the active P2 represents the reviving cell state. We aim at regulating the cell state to the reviving state through only tuning the diagonal topology structure, which can be achieved experimentally via drugs or electrical stimulation. Therefore, an ideal control should be updated as little as possible since the frequent stimulation may do harm to the cells. For training controller u (dA11, , 6A100.100). we uniformly sample 1000 data from the state region [-10, 10). Similarly to that in GRN, we modify the parameterized V and functions s.t. ((P) = O, u(P)) = O. We construct the controllers as follows.
Neural ETC-PI. We parameterize (ax) ICNN(110,, 64, 1), u( as Control (100, 64, 64, 100) with F = ReLU. Since the dimension of the task is very high, the ODE solver has very high computational cost in solving the triggering times. We set the the iterations and batch size for calculating the triggering times as 10 and 5. If the readers have more powerful computing device, larger iterations and batch size are recommended. We set the iterations for warm up as 500, the learning rate as Ir = 0.01, the weight factor event loss as A 1000
the case, we try combination of Neural ETC-PI and
Neural ETC-MC by setting the stabilzaition loss as
Lasb
1 N
Σ
(Crmg Vo(x:;) + ae, (11=:11))*
and we also penalize the Lipschitz constant of -1 by adding term -, the loss function with weight 0.1. The dataset {2x) for - is generated by equidistant sampling on [0. 10).
Neural ETC-MC. We parameterize V(x)) as ICNN(100, 200, 1), u(x) as Control(100, 200, 200, 100). We set the iterations as 500, the learning rate as Ir = 0.05, the weight factor for event loss as Xg = 0.1. The dataset {2,) for L is generated by equidistant sampling on [0, 5).
22

---

Neural Event- Triggered Control with Optimal Scheduling
NLC. We parameterize V as ML0( 200, 1), u(x) MLP((10,, 200, 200, 100). We set the iterations 500, the learning rate Ir 0.01, the loss function
Σ (o(o:))))))) + (Vo(z:))* Vo(0)²
Quad-NLC. We parameterize V(z) (x - Pa) MMMP 200, 100) ML0((000 200, 100) - P2), u(x) as MLP(100, 200, 200, 100). We set the iterations as 500, the learning rate as Ir = 0.01, the loss function is
L Σ (C Vo((m)) Vo(
BALSA. For this QP based method, we the object function
1 min _ 11/1 2 + pt pfdj.
with .d 2
S.t.C V V $ d1.
We choose ==
-00.
LQR. We linearize the controlled dynamic near the P3 solution as
A(x P2) Bu,
100
2x
(xt)²
- 8;
Asj (1 + (15)²)2
+ +
(1 + (x;))²²
We set the cost matrix in LQR
Q 10T 100
100 100;
R 0.011 1.01I100x100 100
The obtained Riceati solution S forms the Lyapunov function V = (ar- (( where K 100 is returned by the lqr solver. The Lie derivative the Lyapunov function is -(x (2))QQ((2 Po) with Q1
S(x
the controller is K(# P2)
Q + K RK.
Critic-Actor ETC. Similarly, we set the hyperparameters as a. 0g le ethirss 0.2. The actuator in this example is
2 00
g diag(
(16)
(1 + 2 2
(1
100
Since the dynamics of W. and W. are both -002-D, leading to a significantly high dimensional system, we reduce the dynamics as
V* (z: W (of, ... 2000) W. 100
u (x) diag W. (W)( ... " W, 100
to the 100-D systems, for the sake limited computational resources.
IRL ETC. We set the hyperparameters as 3 = le - 2, a = 0.1, = Id(-), n A, = 0.1, X = 0.6. The actuator in
this example
x7
x 100
= diag(
(17)
(1 + x
(1
2200
We reduce the dimension of the dynamics as the same with that in Critic-Actor ETC above.
23

---

Neural Event-Triggerss Triggered Control with Optimal Scheduling
Test configurations. We select the event functions those for GRN implement the event-triggered control. For the initial value, we æt Pi E.. & 0150.5, 0.5), = c..... 5, and the random seeds as {0,3, 4, 5, 6).
A.3.5. MOTIVATION OF SELECTING THE BENCHMARK SYSTEMS
In (Wang et al., 2016), a geometrical approach for switching the system from ROA of one equilibrium to another, through finite changes of the experimentally feasible parameters, wherein GRN system investigated their paper. Since our Neural ETC has similarity to the geometrical approach in terms of adding finite non-ine non-invasive control to the system, we also study GRN in our work. The Lorenz system classic chaotic systems possessing plentiful shapes of dynamical trajectories, hence, the control of Lorenz (or control of chaos in a more common sense) is of important position in control literature (Ott et al., 1990; Boccaletti et al., 2000), and the control of Lorenz system under event-triggered implementation also investigated in (Abdelrahim et al., 2015). In (Sanhedrai et al., 2022), topological reconstruction method to the structure of complex dynamics is proposed to revive the degenerate complex system via minimal interventions, i.e., reconstructing links or nodes as small as possible, and the Michaelis-Menten model describing the evolution dynamics of sub-cellular behavior is considered as an illustration. Since the cvent-triggered control aims at adding feasible control to the complex system intermittently. e.g., changing the network structure slowly time, we think it's meaningful to consider the Michaelis-Menna Menten model in our work if there are essentially same parts between our method with the topological reconstruction method.
24


Reviews:
[{"review_id": "hMr4YQKSS0", "reviewer": "Reviewer_6tBp", "summary": "This study proposes a neural-based learning method for optimal scheduling in event-triggered control problems. The proposed method formulates an optimization problem to optimize the triggering rule in control problems. Then, it demonstrates how to solve this problem using neural networks. Finally, theoretical guarantees for stability and optimality are discussed.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank the reviewer for the comments and valuable suggestions. We address the major concerns of the reviewer one by one.\n```\nQ1: I am not sure whether the values of the controller in Theorem 4.1 are continuous at the origin. This type of controller does not necessarily guarantee continuity at zero; if the values of $\\nabla V$ approach zero, the value of $\\pi(u, U(V))$ may diverge. Such a controller achieves the goal, but it may not be acceptable in real-world applications because it requires an excessively large control input.\n```\n**Response**: Many thanks for your comment. Actually, under the mild condition that the state space $x\\in X$ is bounded and the controller $u$ is Lipschitz continuous over the state space $X$, i.e., $u\\in Lip(X)$,  we can prove that the projected controller in Theorem 4.1 is continuous.  This is because that $\\pi(u,U(V))\\in Lip(X)\\iff \\frac{\\max(0,\\mathcal{L}\\_{f_u}V)}{\\Vert \\nabla V\\Vert^2}\\cdot\\nabla V\\in Lip(\\mathcal{X})$. Since $\\Vert\\frac{\\nabla V}{\\Vert\\nabla V\\Vert}\\Vert$ is a continuous unit vector, and naturally is Lipschitz continuous, we only need to consider the remaining term $\\frac{\\max(0,\\mathcal{L}\\_{f_u}V-cV)}{\\Vert \\nabla V\\Vert}$. According to the definition, all the functions occured in this term are continuous, so we only need to bound this term to obtain the Lipschitz continuity, that is, $\\frac{\\max(0,\\mathcal{L}\\_{f_u}V-cV)}{\\Vert \\nabla V\\Vert}\\in Lip(\\mathcal{X})\\iff\\sup_{x\\in\\mathcal{X}}\\frac{\\max(0,\\mathcal{L}\\_{f_u}V-cV)}{\\Vert \\nabla V\\Vert}<+\\infty.$ When $\\mathcal{L}\\_{f_u}V\\le cV$, obviously we have $\\max(0,\\mathcal{L}\\_{f_u}V-cV)=0< +\\infty$, otherwise, since $V\\ge\\varepsilon\\Vert x\\Vert^p$ and $c<0$, we have\n$\n\t\\mathcal{L}\\_{f-u}V-cV\\ge\\mathcal{L}\\_{f_u}V-c\\varepsilon\\Vert x\\Vert^p\\approx\\mathcal{O}(\\Vert x\\Vert^p)\\to\\infty(\\Vert x\\Vert\\to\\infty).\n$\nThus, we have \n$\n\t\\sup_{x\\in\\mathcal{X}}\\frac{\\max(0,\\mathcal{L}\\_{f_u}V-cV)}{\\Vert \\nabla V\\Vert}<+\\infty\\iff \\sup_{x\\in\\mathcal{X}}\\Vert x\\Vert<+\\infty,\n$\nwhich completes the proof. Following the response rules of ICML2025, we provide the revised Theorem 4.1 and the proof in the anonymous link https://anonymous.4open.science/r/Rebuttal_Neural-ETC-FF13/README.md.\n\n```\nQ2: In (2), I am wondering why this objective is designed in a way that also maximizes $\\Vert u(x)\\Vert_{C(\\mathcal{D})}$. This leads to excessively large control values when the obtained policy is applied.\n```\n**Response**: Thanks for your careful reading and constructive comment. It should be $\\min\\Vert u(x)\\Vert_{C(\\mathcal{D})}$ in the objective function instead of $\\max\\Vert u(x)\\Vert_{C(\\mathcal{D})}$, we have corrected this issue according to your comment. For a unified expression, we employ the objective function in Equation(2) as\n$\\min_{u}\\frac{1}{\\min_{t_k\\le T}(t_{k+1}-t_k)}+\\lambda_1\\Vert u(x)\\Vert_{C(\\mathcal{D})}$, which is consistent with the objective function in line 189.\n```\nQ3: It may be necessary to provide a more mathematical explanation of the computation of the derivative of $t_k$.\n```\n**Response**: Many thanks for your valuable comment. We take the computation of the derivative of $t_1$ as an example. Although in [R1] the authors provide a general mathematical expression for the derivative of $t_1$ (see Equation (9)-(11)), we can obtain a more concise expression thanks to the formulation of our event function $h(x(t),e(t))=h(x(t),x_0-x(t))$ and loss function $L=L(t_1)$ as\n$\\frac{\\partial t}{\\partial \\phi}=(\\frac{\\partial h}{\\partial x}\\frac{\\partial x}{\\partial t})^{-1}\\frac{\\partial h}{\\partial x}\\frac{\\partial x}{\\partial \\phi}$. Notice that $\\frac{\\partial x}{\\partial t}=f(x,u_{\\phi}(x))$, and the Adjoint Method [R2][R3] gives the computation method for $\\frac{\\partial x}{\\partial \\theta}=(\\frac{\\partial x_i}{\\partial \\phi})\\_{i=1,...,n}$ as\n$$\na(t)=\\frac{\\partial x_i(t)}{\\partial x(t)},\n\\frac{d a(t)}{dt}=-a(t)\\frac{\\partial  f(x,u_{\\phi})}{\\partial x},\n\\frac{d x_i}{d\\theta}=-a(t)\\frac{\\partial f(x,u_{\\phi})}{\\partial \\phi}.\n$$\nWith the abovementioned mathematical expressions, we could easily calculate the derivative of $t_1$. \n\n\nWe thank the reviewer again for your valuable comments. We do believe that the quality of the revised paper has been improved exceptionally from the theoretical aspect. Hopefully, the responses have sufficiently addressed the main concerns and then the reviewer will reconsider the assessment in support of the revised paper. We are looking forward to your feedback to make further improvements to the paper.\n\n**References**\n\n[R1]Chen, R. T., Amos, B., & Nickel, M. Learning Neural Event Functions for Ordinary Differential Equations. In International Conference on Learning Representations.\n\n[R2]Chen, R. T., Rubanova, Y., Bettencourt, J., & Duvenaud, D. K. (2018). Neural ordinary differential equations. Advances in neural information processing systems, 31.\n\n[R3]Pontryagin, L. S. (2018). Mathematical theory of optimal processes. Routledge.", "claims_and_evidence": "Some parts, especially the theoretical guarantee in Theorem 4.1, may need to be discussed more carefully (please see the question section below).", "methods_and_evaluation_criteria": "The proposed approach seems to be a feasible way to tackle the neural-based formulation of event-triggered control. The problem formulation in (2) appears to be natural, except for a few aspects of the objective function design (see the question section).", "theoretical_claims": "Yes, I checked the proofs.", "experimental_designs_or_analyses": "It may be necessary to provide more details about the gradients of $t_k$ in Section 3.1. While it explains how to compute the gradients, it is unclear to me how ODESolveEvent implements such computation.", "relation_to_broader_scientific_literature": "The contribution of this study is closely connected to control theory and control engineering. This study can be regarded as an approach to efficiently implementing event-triggered control in real complex systems.", "essential_references_not_discussed": "The references in the current manuscript are sufficient.", "other_strengths_and_weaknesses": "One of the strengths of this study is that it proposes a learning-based approach to event-triggered control problems. However, as pointed out in Experimental Designs or Analyses, it may be necessary to provide a more mathematical explanation of the computation of the derivative of $t_k$.", "other_comments_or_suggestions": "None.", "questions_for_authors": "1. I am not sure whether the values of the controller in Theorem 4.1 are continuous at the origin. This type of controller does not necessarily guarantee continuity at zero; that is, if the values of $\\nabla V$ approach zero, the value of $\\pi(\\boldsymbol{u}, \\mathcal{U}(V))$ may diverge. Such a controller achieves the goal, but it may not be acceptable in real-world applications because it requires an excessively large control input.\n2. In (2), I am wondering why this objective is designed in a way that also maximizes $\\|\\boldsymbol{u}(\\boldsymbol{x})\\|_{C(\\mathcal{D})}$. This leads to excessively large control values when the obtained policy is applied.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "wDJ46Awj0c", "reviewer": "Reviewer_QXef", "summary": "This paper presents a novel approach to learning event-triggered controllers with maximum inter-event times using neural networks. Compared to related works, the key innovation is that the entire framework is developed for continuous dynamics and continuous triggering times. The authors propose two approaches: Neural-ETC PI and Neural-ETC MC. Both methods are based on Neural Lyapunov Control, utilizing Lyapunov functions to design triggering conditions.\n\nThe first approach, Neural-ETC PI, directly maximizes the inter-event times by integrating over the system dynamics and triggering conditions, which can be computationally expensive. The paper introduces Neural-ETC MC to address this challenge, which avoids integrating the system dynamics. This method derives a lower bound on the inter-event times under certain conditions, and the neural networks are trained to satisfy these conditions. Finally, the authors propose a projection operation to guarantee stability after training.\n\nThrough simulation experiments, the paper demonstrates that both approaches significantly outperform existing event-triggered controllers, showcasing the effectiveness of their methods.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the overall positive feedback and the valuable comments. For the major comments, we are going to respond to them one by one.\n```\nQ1: Please explain and analyze the outstanding behaviour of Neural-ETC MC in detail.\n```\n**Response**: Many thanks for your valuable comment. The superiority of the Neural ETC MC over the Neural ETC PI lies in the regularization of the Lipschitz constants of function $\\alpha^{-1}$ in Theorem 3.2. According to the proof of Theorem 3.2 (see Appendix A.1.2), the triggering time $t_1$ is implicitly determined by the equation $\\frac{\\Vert e(t_1)\\Vert}{\\Vert x(t_1)\\Vert}=\\frac{1}{P}$ initiated from $\\frac{\\Vert e(0)\\Vert}{\\Vert x(0)\\Vert}=0$. Here $P$ is the tight upper bound of the Lipschitz constants of $\\alpha^{-1}\\circ\\gamma$ in Theorem 3.2. Therefore, by minimizing the Lipschitz constants of $\\alpha^{-1}$ and $\\gamma$, we equivalently maximise the inter-event time. In our paper, we only consider regularization of $\\alpha^{-1}$ because the Lipschitz constant of $\\gamma$ is positively correlated to the Lipschitz constant of controller $u$, which is regularized in the training process. \n```\nQ2: Where exactly does Neural-ETC MC minimize the inter-event times? Is it because the approach minimizes the Lipschitz constants?\n```\n**Response**: Many thanks for your comments. As explained in the Response to Q1, the Neural-ETC MC minimizes the inter-event time implicitly by regularizing the Lipschitz constants of $\\alpha^{-1}$ and controller $u$.\n```\nQ3: How have you adapted Schlüter et al. to the event-triggered control setting?\n```\n**Response**: Thanks for your careful reading. The original paper mistakenly cites the paper of Schlüter et al., we have corrected the citation as [R1][R2], here we just employ the classic LQR method with the proposed event-trigger mechanism as a baseline. The specific approach is to linearize the dynamics near the equilibrium and then apply the event-triggered LQR to stabilize the target state. We also supplemented two more methods as our baselines, one is the existing event-triggered control method for continuous dynamics and the other is the SOTA neural Lyapunov control method as suggested by Reviewer rjuM. Following the review response instruction of ICML2025, we provide the results in Table 1 at the anonymous link https://anonymous.4open.science/r/Rebuttal_Neural-ETC-FF13/README.md.\n```\nQ4: Why exactly are your methods named Path-Integral and Monte Carlo?\n```\n**Response**: Many thanks for your interesting comment. Our major aim is to maximize the inter-event time $t_{k+1}-t_k$ while stabilizing the dynamics. The first method calculates the inter-event time directly by integrating the temporal trajectories controlled dynamics, so we call it as a path-integral method. The second method implicitly maximizes the inter-event time by regularizing the Lipschitz constants of $\\alpha^{-1}(x)$ and $u(x)$ in the loss function. The Monte Carlo estimation using $\\frac{1}{N}\\sum_{i}^Nu(x_i)$ and $\\frac{1}{M}\\sum_{j}^M\\alpha^{-1}(y_j)$ over the finite training data are proportional to the Lipschitz constant of $\\Vert\\alpha^{-1}(x)\\Vert$ and $\\Vert u(x)\\Vert$, respectively. Therefore, we name it as the Monte Carlo method. The first method is an explicit method, while the second one is an implicit method. So we could also rename them as Neural ETC-EXPL and Neural ETC-IMPL.\n\n```\nCouldn’t one eliminate l_f in the calculation of \\tau_h in Theorem 3.2 and 3.3?\n```\n**Response**: Many thanks for your careful reading. Yes, we have eliminated  $l_f$ in the calculation of $\\tau_h$ in Theorem 3.2 and 3.3 accordingly.\n\n**Response to Other Strengths And Weaknesses**: Many thanks for your careful reading and helpful suggestions. We have checked the manuscript and revised the typos and grammar errors. We would like to put the discussion of more related works and the algorithms into the main text if the paper could be accepted and one extra page is permitted.  \n\nFinally, we would like to thank the reviewer again for his/her time and positive feedback on the paper. We may make further improvements according to your feedback.\n\n**References**\n\n[R1] Bellman, R., & Kalaba, R. E. (1965). Dynamic programming and modern control theory (Vol. 81). New York: Academic Press.\n\n[R2] Heemels, W. P., Johansson, K. H., & Tabuada, P. (2012, December). An introduction to event-triggered and self-triggered control. In 2012 ieee 51st ieee conference on decision and control (cdc) (pp. 3270-3285). IEEE.", "claims_and_evidence": "The properties of the controller and learning method are well supported by the proofs provided in the work. While the experimental results are promising, the comparisons to the state-of-the-art are based on only five runs. Increasing the number of runs could provide more statistically significant evidence to strengthen the conclusions.", "methods_and_evaluation_criteria": "The systems used make sense and are nicely motivated.", "theoretical_claims": "See above", "experimental_designs_or_analyses": "1. From the explanations of Neural-ETC PI and Neural-ETC MC, it follows that the set of controllers Neural-ETC MC can learn is a subset of those Neural-ETC PI can learn. Consequently, one might expect Neural-ETC PI to outperform Neural-ETC MC. However, according to Table 1, Neural-ETC MC significantly outperforms Neural-ETC PI and even surpasses all state-of-the-art methods by a factor of more than 100 in terms of the minimal inter-event time. This exceptional performance of Neural-ETC MC is noteworthy and could benefit from a more detailed discussion and analysis to explain the underlying reasons for these results.\n\n2. Table 1 also includes a comparison with the method of Schlüter et al., which is an event-triggered learning approach rather than an event-triggered control method. It would be helpful to understand how this method was adapted to suit the event-triggered control setting for this comparison.", "relation_to_broader_scientific_literature": "The work demonstrates, particularly in Table 1, that the proposed Neural-ETC methods outperform the state-of-the-art on three different benchmark systems. This highlights the suitability of the approach to directly learn neural networks that minimize inter-event times.", "essential_references_not_discussed": "Ok from my point of view", "other_strengths_and_weaknesses": "1. The paper's structure is commendable, and the explanations are clear and well-presented. However, there are occasional issues with misplaced articles and minor grammatical errors—for example: \"To mitigate this issue, **the** event-triggering mechanism is introduced to generate sporadic transmissions across the feedback channels of the system, compared to **the** periodic control which updates the control signal at a series of predefined explicit times.“ I recommend reviewing the manuscript for such grammatical inaccuracies, perhaps using a grammar-checking tool to enhance readability.\n2. While it is understandable that space constraints might necessitate referring readers to the appendix, this is frequently done throughout the paper. Although this is acceptable for detailed proofs and hyperparameters, it also includes essential content such as more detailed discussions of related work and Algorithms 1 and 2, which are crucial for understanding the proposed methods. Including these elements in the main text would improve the paper's clarity and accessibility.\n3. Furthermore, the naming of the proposed methods could be made clearer. It is somewhat unclear where the path-integral and Monte Carlo aspects come into play within the methods. Providing an explanation of these components and how they relate to the naming conventions would enhance the reader's understanding and improve the overall clarity of the paper.", "other_comments_or_suggestions": "1. Couldn’t one eliminate l_f in the calculation of \\tau_h in Theorem 3.2 and 3.3?", "questions_for_authors": "1. Please explain and analyze the outstanding behavior of Neural-ETC MC in detail.\n2. Where exactly does Neural-ETC MC minimize the inter-event times? Is it because the approach minimizes the lipschitz constants?\n3. How have you adapted Schlüter et al. to the event-triggered control setting?\n4. Why exactly are your methods named Path-Integral and Monte Carlo?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "vqhfGWSNwL", "reviewer": "Reviewer_rjuM", "summary": "This paper considers designing feedback controllers for continuous-time nonlinear systems, and the controller is updated only at certain chosen times, ensuring stability and using as few updates as possible. Experiments over three examples are provided to compare with several existing periodical control and event-trigger control techniques.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the overall valuable comments and respond to the reviewer's major concerns one by one. \n```\nQ1: The authors claimed in the Related Work that \"we are the first to study the optimization scheduling problem of ETC in the continuous dynamics\". Nonlinear continuous systems have been studied, for instance [Wang et al,2024].\n```\n**Response**: Thank you for your feedback. Our work addresses \"optimal scheduling\" in event-triggered control (ETC), focusing on minimizing trigger frequency or maximizing inter-event time. Previous research typically applies reinforcement learning to optimize communication costs for discrete systems, but rarely for continuous systems[R1]. We reviewed the suggested paper [Wang et al., 2024], which considers ETC for nonlinear continuous systems. However, its objective function, shown in Equation (5), involves uncertainty bound, system state and control values, differing from our focus on the number of triggerings or inter-event time, thus not conflicting with our claims.\n```\nQ2: The authors claimed that Theorem 4.2 is derived from Theorem 4.1 and Theorem 3.2. As Theorem 3.2 is in the section of the Monte Carlo approach, does it mean the result does not hold for the path integral approach? The authors claimed stability guarantee using the Monte Carlo approach, which is very counter-intuitive, as data-driven techniques cannot provide deterministic guarantees and only statistical results can commonly be expected with some specific sampling. Please clarify how deterministic stability is assured.\n```\n**Response**: Thanks for your comments. Theorem 3.2 demonstrates that maximizing inter-event time corresponds to minimizing the Lipschitz constants of $\\alpha^{-1}$ and $u$. We apply regularization to the Lipschitz constant of controller $u$ in both Path Integral and Monte Carlo approaches. For coherence, we positioned Theorem 3.2 in Section 3.2, although it could alternatively be placed at Section 3's start. Theorem 4.2 reveals that applying the projection operator from Theorem 4.1 to the learned controller in the post-training stage ensures optimality. This projection also works for the Path Integral approach.\n\nRegarding stability, Theorem 4.1’s projection operation $\\pi$ is applied to learned controllers in the post-training stage instead of the training stage. Initially, controllers are learned using finite data in Path Integral or Monte Carlo approaches, resulting in candidate controllers $u_{\\phi}$ that might not rigorously meet Lyapunov stability criteria. Then the projected controllers $\\pi(u_{\\phi})$ strictly fulfil the stability requirement. Therefore, this two-step process provides a rigorous stability guarantee for neural controllers.\n```\nQ3: The comparison with neural Lyapunov control is not fair. The authors should compare their approach with event-triggered control work, e.g., [Li et al, 2019], for exponential stability.\n```\n**Response**: Many thanks for your valuable comment. We supplement the numerical comparison with the event-triggered control method with exponential stabilization (named ETS) in the suggested paper, and we provide results in Table 1.pdf in the anonymous link https://anonymous.4open.science/r/Rebuttal_Neural-ETC-FF13/README.md (according to the Response rules of ICML2025). We note that [Li et al., 2019] assume a global Lipschitz condition (Assumption 1), which the Lorenz system does not satisfy, leading to poor performance in ETS. Additionally, their paper only designs state feedback controllers for linear systems, which is inadequate for our benchmark dynamics. Based on their theories, we developed a machine learning algorithm to derive the auxiliary function $V$ and controller $u$ that fulfil the exponential stabilization condition in Assumption 2 and Theorem 5. Our findings indicate that while ETS outperforms many benchmark methods, it remains inferior to our methods.\n```\nQ4: The authors did not use the SOTA neural Lyapunov control techniques as the baseline, e.g., [Yang et al, 2024], which makes the results less convincing.\n```\n**Response**: Thanks for your comment. We supplement the numerical comparison with this SOTA neural Lyapunov control technique, named as PGDNLC, in Table 1(see above link). The results show that the PGDNLC is still inferior to our methods. \n```\nQ5: Only three examples are given, and no clarification is given on how the variance is.\n```\n**Response**: Thanks for your careful reading and helpful comment. We have explained the motivation of selecting these three examples in Appendix A.3.5. We supplement the variance of each numerical experiment in Table 1, please refer to the above link.\n\nWe would like to thank the reviewer again for his/her time and positive feedback on the paper. We hope that the reviewer will be satisfied with the responses and the supplemented results as well, and then consider revising the assessment in support of the revised paper. We may make further improvements according to your feedback.", "claims_and_evidence": "The authors claimed in the Related Work that \"we are the first to study the optimization scheduling problem of ETC in the continuous dynamics\". In fact, nonlinear continuous systems have been studied, for instance [1].\n\n[1] Wang, Tengda, Guangdeng Zong, Xudong Zhao, and Ning Xu. \"Data-driven-based sliding-mode dynamic event-triggered control of unknown nonlinear systems via reinforcement learning.\" Neurocomputing 601 (2024): 128176.", "methods_and_evaluation_criteria": "The proposed method looks reasonable, though clarity should be improved.", "theoretical_claims": "In Sec 4.4, the authors claimed that Theorem 4.2 is derived from Theorem 4.1 and Theorem 3.2. As Theorem 3.2 is in the section of the Monte Carlo approach, does it mean the result does not hold for the path integral approach?\n\nIn addition, the authors claimed stability guarantee using the Monte Carlo approach, which is very counter-intuitive, as data-driven techniques cannot provide deterministic guarantees and only statistical results can commonly be expected with some specific sampling. Please clarify how deterministic stability is assured.", "experimental_designs_or_analyses": "First, the comparison with neural Lyapunov control is not fair. This paper uses the exponential stability condition to design the controller. The authors should compare their approach with event-triggered control work, e.g., [2], for exponential stability.\n\nSecond, the authors did not use the STOA neural Lyapunov control techniques as baseline, e.g., [3], which makes the results less convincing.\n\nFinally, only three examples are given, and no clarification is given on how the variance is. \n\n[2] Li, Fengzhong, and Yungang Liu. \"Event-triggered stabilization for continuous-time stochastic systems.\" IEEE Transactions on Automatic Control 65, no. 10 (2019): 4031-4046.\n[3] Yang, Lujie, Hongkai Dai, Zhouxing Shi, Cho-Jui Hsieh, Russ Tedrake, and Huan Zhang. \"Lyapunov-stable neural control for state and output feedback: A novel formulation.\" arXiv preprint arXiv:2404.07956 (2024).", "relation_to_broader_scientific_literature": "It is restricted to the control community.", "essential_references_not_discussed": "As mentioned previously, most recent works on neural Lyapunov control are not discussed, e.g., [3].", "other_strengths_and_weaknesses": "The paper is very dense without clear clarification of either the high-level idea, or the each key steps.\n\nThe authors aimed to maximize the inter-event time. However, in Equation (2), the inter-event time t_{k+1}-t_k is minimized. Please clarify Equation (2).\n\nIn Sec 3.1, why is the technique called path integral? What does the optimization above Eq. (5) mean? How is Eq. (5) constructed?\n\nAlso see Theoretical Claims for other clarity issues.", "other_comments_or_suggestions": "None.", "questions_for_authors": "Please see the comments above.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: We thank the authors for their submission.
The reviewers appreciated the novel formulation of neural event-triggered control as well as the learning-based design of the controller.
However, some parts of the paper, mentioned in the reviews, need clarification or rephrasing.
There are also some typos, and perhaps missing references that need to be addressed.
Decision: Accept (poster)