Minimum Attention Control (MAC)

• Minimum Attention Control (MAC) is a control-design paradigm that minimizes controller complexity by penalizing the sensitivity of the control law to state and time variations.
• MAC is implemented through various methods such as continuous variational formulations, linear Gauss–Markov covariance steering, and LP-based receding-horizon schemes.
• MAC techniques offer practical benefits including reduced update frequencies, extended inter-execution times, and enhanced robustness in both traditional and reinforcement learning contexts.

Searching arXiv for recent and foundational papers on Minimum Attention Control to ground the article in published work. Minimum Attention Control (MAC) is a control-design paradigm in which the controller is required to accomplish a specified task while using as little “attention” as possible. In Brockett’s formulation, attention is the dependence of the control law on state and time: a constant input is the easiest law to implement, and any dependence on $x$ or $t$ requires attention. Subsequent work developed this idea along several lines: continuous variational formulations for nonlinear systems, linear sampled-data laws that maximize inter-execution times, receding-horizon schemes that minimize the number of input changes, and, more recently, linear Gauss–Markov covariance steering in which attention is priced directly on the drift gain and its time variation (Sabbagh et al., 7 Dec 2025).

1. Historical origin and technical meaning

The modern MAC literature traces the term “attention” to Brockett’s observation that “the easiest control law to implement is that of a constant input,” with Wiener’s cybernetics providing an earlier conceptual backdrop in which control and communication are inseparable and the controller acts on the basis of “messages” from sensing and computation (Sabbagh et al., 7 Dec 2025). In this usage, attention is not an informal synonym for saliency or focus; it is a technical quantity that measures how much the control law changes with respect to state and time.

For a feedback law $u(x,t)$, Brockett’s minimum attention criterion is

$$J(u)=\frac{1}{2}\int_0^T\int_{\mathcal X}\left\|\frac{\partial u}{\partial x}\right\|^2+\left\|\frac{\partial u}{\partial t}\right\|^2\,dx\,dt.$$

The term $\|\partial u/\partial x\|^2$ measures spatial attention, and $\|\partial u/\partial t\|^2$ measures temporal attention (Lee et al., 2019). In this sense, MAC penalizes the implementation complexity of the law itself rather than the energy of a particular realized input trajectory.

A central distinction within the literature is that “attention” has been instantiated in more than one mathematically precise way. In Brockett-style continuous formulations it is a smooth quadratic penalty on derivatives of the law. In sampled-data linear systems it can mean the inverse of the time elapsed between two consecutive executions of the control task, so that larger inter-execution times correspond to less attention (Donkers et al., 2011). In receding-horizon formulations it can mean the number of nonzero increments $\Delta u_k=u_k-u_{k-1}$, so that attention is the frequency of input updates rather than the size of the input itself (Teja et al., 28 Jul 2025). These are not interchangeable definitions, but they are all resource-aware formulations of the same general question: how little controller activity is sufficient to meet the objective?

The notion of vanishing attention is correspondingly precise. A law requires vanishing attention if it is independent of both time and state while still meeting the control specification. In Brockett’s language the ideal “no attention” law is a constant input; in the linear Gauss–Markov setting this corresponds in particular to $\dot A_t=0$, and, when $\alpha=0$, any constant $A_t\equiv A$ meeting the covariance endpoint conditions has zero temporal attention (Sabbagh et al., 7 Dec 2025).

2. Continuous variational formulations for nonlinear systems

A major step toward a practical MAC methodology for nonlinear systems was to restrict the feedback to the affine-in-state form

$t$0

with $t$1 and $t$2, both $t$3 (Lee et al., 2019). Under this structure,

$t$4

so Brockett’s attention functional becomes quadratic in $t$5, $t$6, $t$7, and the state distribution.

The nonlinear system is represented in Liouville form. With density $t$8 governed by

$t$9

the optimization penalizes both terminal density mismatch and attention: $u(x,t)$0 Within this class, existence is proved by a direct compactness argument: under bounded state space, smooth dynamics, and equicontinuous uniformly bounded sequences $u(x,t)$1, there exists a uniformly convergent subsequence yielding a minimizing law $u(x,t)$2 (Lee et al., 2019).

The corresponding first-order optimality system couples the Liouville PDE, a backward adjoint PDE, and an Euler–Lagrange stationarity equation. With Lagrange multiplier $u(x,t)$3, variation with respect to $u(x,t)$4 gives

$u(x,t)$5

with terminal condition $u(x,t)$6, while variation with respect to $u(x,t)$7 gives

$u(x,t)$8

These equations do not yield a Riccati reduction; instead they define a PDE-constrained optimal control problem.

Computation is carried out by a one-shot method that updates $u(x,t)$9 and $$J(u)=\frac{1}{2}\int_0^T\int_{\mathcal X}\left\|\frac{\partial u}{\partial x}\right\|^2+\left\|\frac{\partial u}{\partial t}\right\|^2\,dx\,dt.$$0 directly. The procedure uses an iLQR-generated nominal trajectory for initialization, a Monte Carlo Liouville solver for $$J(u)=\frac{1}{2}\int_0^T\int_{\mathcal X}\left\|\frac{\partial u}{\partial x}\right\|^2+\left\|\frac{\partial u}{\partial t}\right\|^2\,dx\,dt.$$1, adjoint evaluation along characteristics, and line-searched gradient steps in the functional variables. Under the paper’s degeneracy condition on the second variation of $$J(u)=\frac{1}{2}\int_0^T\int_{\mathcal X}\left\|\frac{\partial u}{\partial x}\right\|^2+\left\|\frac{\partial u}{\partial t}\right\|^2\,dx\,dt.$$2 with respect to $$J(u)=\frac{1}{2}\int_0^T\int_{\mathcal X}\left\|\frac{\partial u}{\partial x}\right\|^2+\left\|\frac{\partial u}{\partial t}\right\|^2\,dx\,dt.$$3, the resulting algorithm converges to a local minimizer (Lee et al., 2019). In the two degree-of-freedom robot-arm example, the computed MAC laws are characterized by small feedback gain magnitude early in the movement and increased feedback near the terminal pose, with feedforward dominant early and diminishing toward zero near the end.

3. Linear Gauss–Markov covariance steering

The 2025 linear Gauss–Markov treatment revisits MAC in a setting where the state process is Gaussian and zero mean, and the design variable is the drift matrix itself rather than an additive control channel (Sabbagh et al., 7 Dec 2025). The dynamics are

$$J(u)=\frac{1}{2}\int_0^T\int_{\mathcal X}\left\|\frac{\partial u}{\partial x}\right\|^2+\left\|\frac{\partial u}{\partial t}\right\|^2\,dx\,dt.$$4

with covariance evolution

$$J(u)=\frac{1}{2}\int_0^T\int_{\mathcal X}\left\|\frac{\partial u}{\partial x}\right\|^2+\left\|\frac{\partial u}{\partial t}\right\|^2\,dx\,dt.$$5

The task is to steer the system from $$J(u)=\frac{1}{2}\int_0^T\int_{\mathcal X}\left\|\frac{\partial u}{\partial x}\right\|^2+\left\|\frac{\partial u}{\partial t}\right\|^2\,dx\,dt.$$6 to $$J(u)=\frac{1}{2}\int_0^T\int_{\mathcal X}\left\|\frac{\partial u}{\partial x}\right\|^2+\left\|\frac{\partial u}{\partial t}\right\|^2\,dx\,dt.$$7 over a fixed horizon $$J(u)=\frac{1}{2}\int_0^T\int_{\mathcal X}\left\|\frac{\partial u}{\partial x}\right\|^2+\left\|\frac{\partial u}{\partial t}\right\|^2\,dx\,dt.$$8.

For linear feedback $$J(u)=\frac{1}{2}\int_0^T\int_{\mathcal X}\left\|\frac{\partial u}{\partial x}\right\|^2+\left\|\frac{\partial u}{\partial t}\right\|^2\,dx\,dt.$$9, the state-gradient term is $\|\partial u/\partial x\|^2$0, and the time-derivative term is $\|\partial u/\partial x\|^2$1. Replacing Brockett’s spatial integral by expectation with respect to the Gaussian state law yields the attention functional

$\|\partial u/\partial x\|^2$2

with $\|\partial u/\partial x\|^2$3 (Sabbagh et al., 7 Dec 2025). The first term is spatial attention and the second temporal attention. No LQR-type input or state penalty is added: the only cost is attention.

For $\|\partial u/\partial x\|^2$4, existence of a minimizer $\|\partial u/\partial x\|^2$5 is established by the direct method in the calculus of variations. The optimal covariance satisfies uniform interior SPD bounds $\|\partial u/\partial x\|^2$6, and the KKT system introduces a symmetric adjoint $\|\partial u/\partial x\|^2$7 satisfying

$\|\partial u/\partial x\|^2$8

$\|\partial u/\partial x\|^2$9

$\|\partial u/\partial t\|^2$0

with $\|\partial u/\partial t\|^2$1 (Sabbagh et al., 7 Dec 2025). The stationarity condition is third-order-like in the sense that it couples $\|\partial u/\partial t\|^2$2, $\|\partial u/\partial t\|^2$3, and the derivative of $\|\partial u/\partial t\|^2$4. A Sobolev bootstrap then yields $\|\partial u/\partial t\|^2$5. The resulting controller is linear in state but generally smoothly time varying, and the optimality system is a nonlinear two-point boundary-value problem rather than a Riccati equation.

The two endpoint cases have distinct structures. For pure spatial attention, $\|\partial u/\partial t\|^2$6, the cost reduces to $\|\partial u/\partial t\|^2$7, and stationarity becomes algebraic: $\|\partial u/\partial t\|^2$8 The coupled $\|\partial u/\partial t\|^2$9 system is reminiscent of a Hamiltonian system, and again all optimal variables are $\Delta u_k=u_k-u_{k-1}$0 (Sabbagh et al., 7 Dec 2025). In the zero-noise case $\Delta u_k=u_k-u_{k-1}$1, Proposition 4 shows that if $\Delta u_k=u_k-u_{k-1}$2 with symmetric $\Delta u_k=u_k-u_{k-1}$3 and skew-symmetric $\Delta u_k=u_k-u_{k-1}$4, then $\Delta u_k=u_k-u_{k-1}$5 is constant and the optimal gain is conjugate-constant, with explicit formulas for both $\Delta u_k=u_k-u_{k-1}$6 and $\Delta u_k=u_k-u_{k-1}$7.

For pure temporal attention, $\Delta u_k=u_k-u_{k-1}$8, the cost

$\Delta u_k=u_k-u_{k-1}$9

is degenerate, because any constant $\dot A_t=0$0 has zero cost whenever it satisfies the covariance boundary conditions. The paper resolves this by a perturbation-and-selection principle: solve the mixed problem for $\dot A_t=0$1 with small $\dot A_t=0$2, then let $\dot A_t=0$3. If the feasible set of constant gains is nonempty, the limit is the smallest-norm constant gain that achieves the steering task (Sabbagh et al., 7 Dec 2025). In the zero-noise case, feasibility requires

$\dot A_t=0$4

which yields a logarithmic Procrustes problem on $\dot A_t=0$5.

The same paper also places MAC within covariance steering and information geometry. Unlike standard linear Schrödinger bridge formulations, there is no separate control input $\dot A_t=0$6; the drift matrix $\dot A_t=0$7 itself is controlled, and the cost measures attention rather than input energy. In the zero-noise limit the paper identifies a Fisher–Rao geodesic

$\dot A_t=0$8

and defines a Fisher-inducing cost $\dot A_t=0$9 whose unique minimizer is the Fisher pair $\alpha=0$0. Under uniform SPD bounds, $\alpha=0$1, so small attention implies closeness to Fisher–Rao geodesics in an information-geometric sense (Sabbagh et al., 7 Dec 2025).

4. Sampled-data, LP-based, and receding-horizon MAC

For linear sampled-data systems, an influential formulation interprets attention as the inverse of the time elapsed between two consecutive executions of the control task (Donkers et al., 2011). The plant is

$\alpha=0$2

with zero-order hold $\alpha=0$3 on $\alpha=0$4, and only a finite set of admissible inter-execution intervals

$\alpha=0$5

is allowed. The MAC problem is then to choose both $\alpha=0$6 and $\alpha=0$7 so that the closed loop is globally exponentially stable with prescribed rate $\alpha=0$8 and gain $\alpha=0$9, while making $A_t\equiv A$0 as large as possible.

The key tool is the extended control Lyapunov function (eCLF), typically of the form

$A_t\equiv A$1

For each candidate interval $A_t\equiv A$2, one enforces

$A_t\equiv A$3

together with a control magnitude bound $A_t\equiv A$4 (Donkers et al., 2011). The “extended” aspect is that these inequalities are imposed at multiple points in the interval, which improves transient guarantees by making the exponential gain depend on the maximal spacing $A_t\equiv A$5 between consecutive points in $A_t\equiv A$6, rather than only on the largest interval. Because $A_t\equiv A$7 inequalities can be written as linear inequalities, the per-step MAC law reduces to online linear programs. The controller incrementally tests feasibility for increasingly large $A_t\equiv A$8 and selects the largest feasible interval. The same paper also formulates an anytime attention control problem, in which the scheduler chooses the interval and the controller maximizes performance under that externally imposed attention budget.

A different discrete notion appears in the receding-horizon framework of Minimum Attention Model Predictive Control (MAMPC), where attention is quantified by the number of nonzero input changes (Teja et al., 28 Jul 2025). For the discrete-time linear model

$A_t\equiv A$9

the standard MPC objective

$t$00

is augmented by a zero-norm constraint on input increments over an extended window that includes past optimal inputs: $t$01 This is the paper’s core encoding of minimum attention: the controller is allowed to move the actuators only a limited number of times over the combined past-and-future horizon (Teja et al., 28 Jul 2025).

Because the resulting problem $t$02 is nonconvex, the paper introduces an auxiliary variable $t$03 and solves a relaxed problem $t$04 with objective

$t$05

subject to $t$06 (Teja et al., 28 Jul 2025). The solution method is alternating minimization: the outer step is a quadratic program in $t$07, and the inner step is the best $t$08-sparse approximation of $t$09, obtained analytically by hard thresholding the $t$10 largest absolute components.

The reported case studies show the expected trade-off. In the quadruple tank system, after discarding initial transients, standard MPC yields sparse densities $t$11, $t$12, and tracking error $t$13, whereas MAMPC with $t$14 yields $t$15 and $t$16 (Teja et al., 28 Jul 2025). In the solid oxide fuel cell stack, MAMPC again reduces sparse density, but with a more visible increase in tracking error, illustrating the paper’s conclusion that MAC-style receding-horizon sparsity is especially favorable for systems with slow dynamics.

5. Reinforcement learning and meta-learning variants

Minimum attention has also been incorporated directly into reinforcement learning as a differentiable regularizer on the policy (Lee et al., 22 May 2025). In that setting, the control objective is not derived analytically from a known model; instead, the policy $t$17 is trained to maximize task reward while minimizing the Brockett-style penalty

$t$18

The per-step regularized reward is

$t$19

so minimum attention appears as a penalty on policy Jacobian norm and time variation rather than on actuator increments or execution times (Lee et al., 22 May 2025).

The paper develops this idea in a model-based meta-RL setting with an ensemble of learned dynamics models $t$20. For each task/model $t$21, one performs an inner-loop gradient ascent step

$t$22

where $t$23 already contains the MAC-regularized reward. The outer loop then maximizes the average post-adaptation return across the ensemble by a TRPO update, again with the attention penalty embedded in the objective (Lee et al., 22 May 2025). The paper notes that PPO, TRPO, and SAC were explored, with SAC working best in practice.

The stated interpretation is that penalizing $t$24 discourages high-gain, brittle policies, while penalizing $t$25 discourages jerk and fast retuning. The empirical results are reported on MuJoCo locomotion tasks including HalfCheetah, Hopper, and Walker2D, under perturbations such as crippled legs, mass changes, and slopes. In HalfCheetah meta-training, MB-MPO without MAC achieves reward $t$26, feedback norm $t$27, feedforward norm $t$28, and energy $t$29, whereas MB-MPO plus MAC with $t$30 achieves reward $t$31, feedback norm $t$32, feedforward norm $t$33, and energy $t$34 (Lee et al., 22 May 2025). In crippled-leg meta-testing at 200K steps, MB-MPO gives reward $t$35 and energy $t$36, while the $t$37 MAC variant gives reward $t$38 and energy $t$39. The same work explicitly states that it does not provide Lyapunov or convergence theorems; its claims about stability and robustness are empirical and are supported by reduced variance in learning curves and OOD performance.

Within MAC proper, this RL formulation is significant because it translates Brockett’s smooth derivative penalty into a form usable by gradient-based policy optimization. This suggests a bridge between analytical MAC design and learned controllers, but the paper itself identifies full theoretical stability analysis and broader applicability to purely model-free RL as future work (Lee et al., 22 May 2025).

6. Related paradigms, information-geometric links, and terminological ambiguities

MAC is closely connected to several neighboring research areas, but it is not identical to any one of them. In continuous stochastic control, the linear Gauss–Markov formulation is explicitly a covariance steering problem between Gaussian endpoint distributions, with a cost on the drift law rather than on control energy (Sabbagh et al., 7 Dec 2025). This places it near linear Schrödinger bridge methods, but the variables and objective differ: there is no separate $t$40, and the cost is the spatial/temporal sensitivity of the law $t$41. In resource-aware control, MAC intersects with event-triggered and self-triggered control, sparse feedback design, minimum data-rate control, and hands-off control, yet its defining penalty may be a smooth derivative norm, an inverse inter-execution time, or a zero norm on input increments, depending on the formulation (Donkers et al., 2011).

A common misconception is to treat all of these formulations as merely different numerical implementations of the same optimization problem. The literature does not support that interpretation. Brockett-style MAC is continuous and differentiable in the control law, sampled-data MAC is a scheduling-and-control co-design over a finite set of inter-execution times, and MAMPC is an $t$42-constrained reference-tracking problem in which sparsity is imposed on successive input differences rather than on derivatives of the feedback map. A plausible implication is that comparisons across MAC papers must attend carefully to the operative notion of attention; otherwise, empirical conclusions about sparsity, robustness, or smoothness may not be commensurate.

The acronym itself is also overloaded outside control theory. “MAC” in “Compositional Attention Networks for Machine Reasoning” denotes Memory, Attention, and Composition, a recurrent architecture with separate control and memory states for visual question answering, not minimum attention control (Hudson et al., 2018). “MAC” in “Mean Activation Approximated Curvature” denotes a second-order optimizer that approximates the Fisher information matrix using mean activations and, for transformers, attention scores in the preconditioner, again unrelated to control-theoretic MAC (Seung et al., 10 Jun 2025). In technical writing, disambiguation is therefore necessary whenever the acronym appears without expansion.

Across its control-theoretic variants, MAC can be summarized as a family of formulations in which the performance objective is coupled to an explicit price on controller activity. In Brockett’s original sense the activity is how strongly the control law depends on $t$43 and $t$44; in sampled-data and receding-horizon senses it is how often the controller recomputes or changes commands; in the linear Gauss–Markov paradigm it is the magnitude and time variation of the drift gain that steers Gaussian uncertainty profiles. The unifying theme is not a single optimization template, but the insistence that implementation effort itself is a first-class control cost (Sabbagh et al., 7 Dec 2025).