Discrete-Time Control Lyapunov Functions

• DCLFs are stability certificates for discrete systems that enforce a decay condition, e.g., V(xₖ₊₁) ≤ (1-κ)V(xₖ), ensuring exponential stability.
• They are applied in various control scenarios such as MPC, event-triggered control, and reinforcement learning to secure system performance.
• Optimization-based synthesis embeds DCLF conditions into QCQPs and MPC constraints, reducing conservativeness and enhancing control robustness.

Searching arXiv for recent and foundational papers on discrete-time control Lyapunov functions to ground the article in published work. Discrete-time control Lyapunov functions (DCLFs) are Lyapunov-type certificates for controlled systems evolving on discrete time indices, typically written as $x_{k+1}=f(x_k,u_k)$. In the cited literature, a DCLF is used to encode positive definiteness together with a one-step or multi-step decrease condition under an admissible control input, yielding stability guarantees for nonlinear systems, sampled-data implementations, distributed model predictive control (MPC), event-triggered controllers, reinforcement learning, and neural verification pipelines (Olkin et al., 3 May 2026, Taylor et al., 2021, Wang et al., 27 Aug 2025, Wu et al., 2023).

1. Formal definitions and canonical inequalities

A standard discrete-time formulation considers the system

$$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$

with $f$ continuous, $f(0,0)=0$, and locally Lipschitz in $x$. In this setting, a continuously differentiable function $V:\mathbb R^n\to\mathbb R_{\ge 0}$ is a discrete-time CLF on a domain $\mathcal D\ni 0$ if there exist class-$\mathcal K_\infty$ functions $\underline\alpha,\overline\alpha$ and a constant $\kappa\in(0,1)$ such that

$$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$0

and, for every $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$1, there exists $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$2 satisfying

$$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$3

This yields the equivalent one-step contraction

$$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$4

which is the discrete-time exponential-decay form used explicitly in stability analysis and CLF-guided reinforcement learning (Olkin et al., 3 May 2026).

A closely related classical formulation uses a continuous $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$5 together with class-$$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$6 functions $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$7 and a scalar $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$8 such that

$$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$9

Equivalent treatments replace the right-hand side by a positive-definite decrement $f$0, writing $f$1. Under these conditions, the closed loop is asymptotically stable (Lazar, 2010).

In distributed multi-agent formation control, the same idea is specialized to pairwise deviation coordinates. With

$f$2

a continuously differentiable $f$3 is called a DCLF if there exist constants $f$4, $f$5, and $f$6 such that

$f$7

and

$f$8

Equivalently,

$f$9

Defining $f(0,0)=0$0, this becomes $f(0,0)=0$1 with $f(0,0)=0$2 (Wang et al., 27 Aug 2025).

2. One-step, finite-step, and average-descent variants

A recurrent distinction in the literature is between strict one-step decrease and relaxed multi-step decrease. The one-step case gives the most direct exponential estimate: if a feedback policy always selects a control satisfying

$f(0,0)=0$3

then

$f(0,0)=0$4

and the origin is exponentially stable (Olkin et al., 3 May 2026).

A common misconception is that a DCLF must decrease at every sampling instant. Several cited works explicitly relax that requirement. Finite-step control Lyapunov functions (fs-CLFs) require contraction only after a fixed number $f(0,0)=0$5 of steps. For the system

$f(0,0)=0$6

an fs-CLF $f(0,0)=0$7 satisfies

$f(0,0)=0$8

for an admissible finite-step feedback $f(0,0)=0$9 and a contraction $x$0 with $x$1 for $x$2. Repeated application of $x$3 renders the closed-loop trajectories asymptotically $x$4-stable in the target set $x$5 (Noroozi et al., 2019).

Generalized DCLFs extend the relaxation further by enforcing average descent over a horizon $x$6. For

$x$7

a generalized DCLF $x$8 of horizon $x$9 satisfies

$V:\mathbb R^n\to\mathbb R_{\ge 0}$0

This average-descent constraint permits temporary increases in $V:\mathbb R^n\to\mathbb R_{\ge 0}$1 so long as there is net decay over the window. Under the stated small-control property, the existence of such a generalized DCLF implies convergence and Lyapunov-type asymptotic stability (Fürnsinn et al., 2022).

The dissipativity-based control dissipation function (CDF) framework yields an equivalent multi-step perspective. Here one imposes the dissipation inequality

$V:\mathbb R^n\to\mathbb R_{\ge 0}$2

together with a cyclically negative supply: $V:\mathbb R^n\to\mathbb R_{\ge 0}$3 Positive-definite storage and cyclically negative supply yield asymptotic Lyapunov stability, and the cited paper states that a CDF with cyclically negative supply is equivalent to a finite-step CLF (Lazar, 2021).

Variant	Decrease requirement	Representative source
One-step DCLF	$V:\mathbb R^n\to\mathbb R_{\ge 0}$4	(Olkin et al., 3 May 2026)
fs-CLF	$V:\mathbb R^n\to\mathbb R_{\ge 0}$5	(Noroozi et al., 2019)
Generalized / dissipative form	Average descent or cyclically negative supply over $V:\mathbb R^n\to\mathbb R_{\ge 0}$6 or $V:\mathbb R^n\to\mathbb R_{\ge 0}$7 steps	(Fürnsinn et al., 2022, Lazar, 2021)

This family of relaxations is used to reduce conservativeness. The flexible-CLF overview by M. Lazar states that classical CLFs can be overconservative, and that making the contraction rate a decision variable allows non-monotonicity to be explicitly linked with a decision variable optimized on-line (Lazar, 2010).

3. Optimization-based synthesis and MPC embeddings

DCLF conditions are frequently embedded directly into finite-horizon optimization. In sampled-data stabilization via quadratically constrained quadratic programs (QCQPs), the zero-order-hold implementation of a continuous-time control-affine system is represented by a discrete-time map $V:\mathbb R^n\to\mathbb R_{\ge 0}$8, and a DCLF is required to satisfy

$V:\mathbb R^n\to\mathbb R_{\ge 0}$9

For a quadratic candidate $\mathcal D\ni 0$0, the Euler-expanded decrease inequality becomes a quadratic constraint in $\mathcal D\ni 0$1,

$\mathcal D\ni 0$2

and the controller is synthesized by the QCQP

$\mathcal D\ni 0$3

Under feedback linearizability, local exponential stability of the zero-dynamics, and consistency of the Euler approximation with the exact map, the resulting closed loop is uniformly ultimately bounded, with the ultimate bound made arbitrarily small by choosing the sampling period $\mathcal D\ni 0$4 small (Taylor et al., 2021).

In distributed safety-critical MPC for multi-agent systems, DCLFs appear as terminal constraints. Using estimated neighbor states, the predicted deviations are

$\mathcal D\ni 0$5

The terminal constraint is imposed as

$\mathcal D\ni 0$6

and the paper states that the terminal inequality

$\mathcal D\ni 0$7

drives all deviations to zero asymptotically. Under Assumption 1, feasibility at $\mathcal D\ni 0$8 implies recursive feasibility for all future $\mathcal D\ni 0$9, and the receding-horizon DSMPC law ensures $\mathcal K_\infty$0 for any prescribed tolerance $\mathcal K_\infty$1, while safety is maintained by the barrier constraints (Wang et al., 27 Aug 2025).

The fs-CLF and generalized-DCLF formulations also enter MPC directly. One approach uses the fs-CLF as the running cost in contractive multi-step MPC, contractive updated multi-step MPC, and classic MPC. Another embeds average-descent constraints into a flexible-step MPC OCP and then implements a variable number of open-loop controls before the next optimization. Under the stated terminal-set and small-control assumptions, these schemes guarantee recursive feasibility and asymptotic stability (Noroozi et al., 2019, Fürnsinn et al., 2022).

4. Sampled-data, stochastic, and event-triggered DCLFs

In stochastic sample-and-hold control, the DCLF viewpoint is applied to a continuous-time system sampled every $\mathcal K_\infty$2 with the control held constant between sampling instants. On a ball $\mathcal K_\infty$3, a function $\mathcal K_\infty$4 is called a DCLF if there exist class-$\mathcal K_\infty$5 functions $\mathcal K_\infty$6 and numbers $\mathcal K_\infty$7 such that

$\mathcal K_\infty$8

and the sample-to-sample increment satisfies an almost-sure bound

$\mathcal K_\infty$9

together with a corresponding bound in expectation. The central result states, roughly, that if there is a generally non-smooth control Lyapunov function, then the stochastic system can be practically stabilized in sample-and-hold mode. The construction uses Moreau–Yosida regularization, inf-convolution, proximal subgradients, and lower Dini derivatives, and yields almost-sure practical stability and practical stability in mean under sufficiently small $\underline\alpha,\overline\alpha$0, noise bound $\underline\alpha,\overline\alpha$1, and optimization error $\underline\alpha,\overline\alpha$2 (Osinenko et al., 2022).

Event-triggered parameterized control for discrete-time linear systems uses the quadratic DCLF

$\underline\alpha,\overline\alpha$3

and imposes the contractive prediction funnel

$\underline\alpha,\overline\alpha$4

At each event time $\underline\alpha,\overline\alpha$5, the controller solves a convex QCQP for the parameter vector $\underline\alpha,\overline\alpha$6, with quadratic constraints

$\underline\alpha,\overline\alpha$7

Between events it applies the time-parameterized input $\underline\alpha,\overline\alpha$8, monitors the Lyapunov-based predictor $\underline\alpha,\overline\alpha$9, and triggers replanning when the predictor violates the funnel or a small-ball threshold. The resulting closed loop is globally uniformly ultimately bounded, and the inter-event times are non-trivial, with $\kappa\in(0,1)$0 (Rajan et al., 2024).

These sampled-data and event-triggered formulations address a point emphasized in the QCQP-based sampled-data paper: continuous-time CLF-based quadratic programs do not address the gap between continuous-time design and discrete-time sampled implementation, which can lead to poor performance on hardware platforms (Taylor et al., 2021).

5. Reinforcement learning and neural verification

A DCLF can be used as an explicit stability prior in reinforcement learning. In the CLF-guided RL formulation, the discrete-time decay condition

$\kappa\in(0,1)$1

is enforced indirectly through reward shaping. The per-step CLF-shaped reward is

$\kappa\in(0,1)$2

where the first term penalizes large CLF values and the second penalizes decay violations. The cited work states that, by standard arguments in potential-based reward shaping, adding these terms does not alter the optimal policy but strongly encourages actions satisfying the discrete-time Lyapunov decrease. Its discrete-time theorem then gives exponential stability for any feedback policy that always selects a control satisfying the DCLF inequality (Olkin et al., 3 May 2026).

The same paper verifies the theory numerically on the double integrator and cart-pole and implements CLF-guided rewards on a walking humanoid robot. For the double integrator, $\kappa\in(0,1)$3 is derived from the discrete Lyapunov equation, and the paper reports a numerical decay rate $\kappa\in(0,1)$4 for typical $\kappa\in(0,1)$5. For the cart-pole, the nonlinear decay condition is shown in the ball $\kappa\in(0,1)$6 with $\kappa\in(0,1)$7, and the learned policy respects the discrete-time Lyapunov condition (Olkin et al., 3 May 2026).

Neural Lyapunov control for discrete-time systems parameterizes both $\kappa\in(0,1)$8 and $\kappa\in(0,1)$9 as feed-forward ReLU networks, then verifies positivity and decrease conditions by mixed-integer linear programming. The method combines three ingredients: a mixed-integer linear programming approach for verifying the discrete-time Lyapunov stability conditions, a method for computing verified sublevel sets, and a heuristic gradient-based method for finding counterexamples. The experiments on four standard benchmarks are reported to significantly outperform state-of-the-art baselines; on path tracking the method outperforms recent neural Lyapunov control baselines by an order of magnitude in both running time and the size of the region of attraction, and on cartpole and PVTOL it is stated to be the first automated approach to return a provably stable controller (Wu et al., 2023).

6. Conservativeness, practical design choices, and representative applications

The literature repeatedly treats conservativeness as a central design issue. The flexible-CLF overview by M. Lazar states that a classical CLF enforces a cone with a fixed, predefined shape and that such a requirement often proves to be overconservative. Flexible CLFs therefore introduce on-line decision variables such as $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$00 and $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$01, and solve a small nonlinear program of the form

$$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$02

subject to

$$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$03

The stated stability theorem guarantees $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$04 if the on-line program remains feasible and feasibility implies $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$05 whenever $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$06, while allowing the sequence $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$07 to be non-monotone (Lazar, 2010).

The same overview sketches fast-sampling applications to a Buck–Boost DC–DC converter with sampling interval $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$08 and an electromagnetic actuator controlled at $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$09. It states that the required on-line program solves in well under the sampling time on a standard 32-bit microcontroller, that the flexible-CLF region of attraction is up to $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$10–$$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$11 larger than that guaranteed by the best quadratic CLFs for the same constrained linearizations, and that transient performance measured in hardware experiments improves by $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$12–$$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$13 versus classical CLF–MPC schemes that enforce strict monotonicity (Lazar, 2010).

Concrete parameter choices also appear in distributed MPC. In the cited multi-agent simulation, each vehicle uses

$$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$14

with horizon $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$15 or $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$16 and weighting matrices $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$17. Under these settings, all inter-agent deviations converge to within the desired $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$18, while the CBF-based safety constraints prevent collisions with obstacles or other agents (Wang et al., 27 Aug 2025).

The CDF framework shows the same tradeoff through horizon parameters. For interconnected synchronous generators with nonlinear coupling and state/input constraints, the paper reports that with kernel $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$19 and horizon $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$20, one needs $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$21 to enforce cyclically negative supply and achieve convergence; with $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$22, $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$23 suffices because the $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$24-step CDF is already a one-step CLF (Lazar, 2021).

Several limitations are stated explicitly. For fs-CLFs, the existence of an fs-CLF and the minimal step size $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$25 must be established a priori; contractive MPC introduces a potentially nonconvex terminal constraint; and classic MPC without terminal ingredients typically needs longer horizons $$x_{k+1}=f(x_k,u_k), \qquad x_0\in\mathbb R^n,\;u_k\in\mathcal U,$$26 (Noroozi et al., 2019). A plausible implication is that DCLF design is often less about a single canonical inequality than about choosing an admissible decrease notion—one-step, finite-step, average-descent, or dissipativity-based—that matches the sampling, communication, and computational structure of the control architecture.