There’s a Hard Limit to How Efficiently We Can Control Noisy Systems
Even with infinite time, some systems can’t be steered precisely without a minimum cost—noise creates an energy floor that can’t be overcome.
The Science
Control theory is the science of steering dynamical systems—robots, power grids, spacecraft, even biological networks—from one state to another. For decades, engineers have relied on a classic result: the minimum-energy control (MEC) problem, which asks: what’s the least amount of energy needed to move a system from point A to point B in a fixed time? For deterministic systems—those without randomness—the answer is elegant and well-known. It depends on a matrix called the controllability Gramian, which captures how easily inputs can influence the system’s state. If the system is controllable, you can always find a finite-energy input to reach your target.
But the real world is noisy. Sensors drift, actuators jitter, environments fluctuate. When randomness enters the picture, the problem changes fundamentally. In their 2026 paper, Vaishnavi Sharma and Vaibhav Katewa tackle the stochastic version of minimum-energy control—what they call the Stochastic MEC (SMEC) problem—for linear time-invariant systems driven by Gaussian noise (Sharma & Katewa, 2026). Their model is a discrete-time system:
where $\mathbf{x}(t)$ is the state, $\mathbf{u}(t)$ the control input, and $\mathbf{w}(t) \sim \mathcal{N}(0, \Sigma)$ i.i.d. Gaussian noise. The initial state is also random: $\mathbf{x}(0) \sim \mathcal{N}(0, \Sigma_0)$. The goal is to minimize the control energy $\sum_{t=0}^{T-1} \mathbf{u}(t)^\top Q_0 \mathbf{u}(t)$ while ensuring the terminal state $\mathbf{x}(T)$ is close to a desired $\mathbf{x}_f$.
The twist? Exact reachability is impossible. Because of noise, $\mathbf{x}(T)$ is a random variable. So instead of demanding $\mathbf{x}(T) = \mathbf{x}_f$, the authors define stochastic terminal constraints that bound how far the state can deviate, in a probabilistic sense. They consider two types: constraints on the normalized error $\mathbf{e}_T = \mathbf{x}(T) - \mathbf{x}f$, weighted by its covariance $\Sigma{e_T}$, and constraints on the absolute error. Within these, they explore chance constraints (e.g., “the error stays within bounds with 95% probability”), second-moment bounds, and Mahalanobis distance limits.
Crucially, they focus on open-loop control: the input sequence $\mathbf{u}_T = [\mathbf{u}(T-1)^\top, \dots, \mathbf{u}(0)^\top]^\top$ is fixed in advance, not adapted based on observations. This makes the control deterministic, so it can only shift the mean of the terminal state, not reduce its variance. That’s a key distinction from closed-loop (feedback) control, where adaptive inputs can reshape the entire distribution.
What They Found
The authors show that the SMEC problem can be cast as a convex Quadratically-Constrained Quadratic Program (QCQP), which is computationally tractable. For the normalized error constraint:
where $C_T$ is the standard controllability matrix, $Q_T = I_T \otimes Q_0$, and $\gamma$ is a tolerance parameter tied to the desired confidence level. The covariance $\Sigma_{e_T}$ depends on the system dynamics and noise, but not on the control input—highlighting that open-loop control cannot reduce uncertainty.
They derive the optimal solution using KKT conditions, introducing a novel stochastic controllability matrix $C_{T,s}$ and stochastic Gramian $W_{T,s} = C_{T,s} Q_T^{-1} C_{T,s}^\top$. The optimal control takes the form:
which mirrors the deterministic MEC solution but with stochastic counterparts. When the tolerance $\gamma$ is loose, no control is needed ($\mathbf{u}_T^* = 0$); when tight, the full corrective input is applied.
But the most striking result comes when they analyze the optimal cost $J_T^* = \mathbf{x}f^\top W{T,s}^\dagger \mathbf{x}_f$ as the time horizon $T$ increases. In deterministic systems, longer horizons allow gentler, lower-energy inputs—energy can be arbitrarily reduced by spreading it out. Not so in the stochastic world.
They prove a fundamental limit: under mild conditions (specifically, if $A \Sigma_0 A^\top + \Sigma \succeq \Sigma_0$, meaning the noise doesn’t shrink over time), the optimal cost $J_T^*$ is non-increasing in $T$ and converges to a positive lower bound:
This means: no matter how much time you have, you cannot steer the system with less than $J_\infty^*$ energy. Noise imposes an energy floor.
Optimal Control Energy vs. Time Horizon
As the time horizon T increases, the optimal energy J_T^* decreases but converges to a positive lower bound J_∞^* > 0, demonstrating the fundamental energy floor.
| Label | Value |
|---|---|
| T=1 | 4.2 |
| T=2 | 3.1 |
| T=3 | 2.5 |
| T=4 | 2.2 |
| T=5 | 2.05 |
| T=6 | 2.01 |
| T=7 | 2.003 |
| T=8 | 2.001 |
This isn’t just theoretical. Consider a satellite needing to reach a precise orbit. In a noiseless world, you could use tiny thruster pulses over weeks to get there with near-zero energy. But with sensor noise, atmospheric drag fluctuations, or solar pressure, there’s a minimum fuel cost—no amount of patience reduces it further. The same applies to quantum systems, where measurement and decoherence noise limit control precision, or to insulin delivery in artificial pancreases, where physiological variability sets a floor on dosing accuracy.
The paper also shows that the deterministic MEC is a special case of SMEC. When the noise covariance $\Sigma_{e_T} \to 0$, or when the tolerance $\gamma \to \underline{\gamma}_T$ (the minimum feasible error), the stochastic solution converges to the deterministic one. This unification is elegant—it shows that classical control theory sits within a broader stochastic framework.
Figure 1, reproduced from the paper, shows the function $f(\nu_T)$ that determines the Lagrange multiplier. As $\nu_T \to \infty$, $f(\nu_T) \to \underline{\gamma}_T$, the minimum achievable normalized error. This asymptotic behavior underpins the existence of the energy limit.
Why This Changes Things
This work shifts how we think about control in uncertain environments. Engineers have long assumed that “given enough time, you can do it gently.” That intuition fails in stochastic settings. The existence of a fundamental energy lower bound has profound implications:
Design trade-offs are unavoidable. You can’t just buy your way out of noise with time. System designers must confront the energy-precision trade-off head-on. Want higher precision? You’ll pay in energy, and there’s a point of diminishing returns.
Feedback isn’t just helpful—it’s essential for breaking the floor. The paper focuses on open-loop control, where the energy floor exists. But in closed-loop control, where measurements are used to adapt inputs, it may be possible to reduce uncertainty and thus lower the energy cost. This highlights a deep principle: information is a resource that can circumvent physical limits. The energy floor applies only when you’re flying blind.
It redefines “controllability.” In deterministic systems, a system is controllable if you can reach any state. In stochastic systems, even if the mean can be steered, the variance may prevent high-confidence reachability. The authors’ stochastic Gramian $W_{T,s}$ captures this new notion: it blends control authority with noise propagation. A system might be deterministically controllable but stochastically weak if noise quickly corrupts the state.
It impacts energy-constrained systems. For drones, satellites, or medical implants, energy is scarce. Knowing the minimum energy required to achieve a task prevents over-engineering and informs mission planning. If the energy floor exceeds available power, the task is simply infeasible—no amount of clever control can fix that.
Consider autonomous vehicles. At high speeds, small steering errors can lead to large deviations due to noise in tire grip or road conditions. The SMEC framework suggests there’s a minimum correction energy for lane-keeping, regardless of how far ahead the system plans. This could inform the design of fail-safe systems or energy budgets for long-haul autonomy.
What’s Next
The paper opens several avenues. First, the analysis assumes linear dynamics and Gaussian noise. Real systems are often nonlinear or have non-Gaussian disturbances (e.g., sudden wind gusts, sensor dropouts). Extending SMEC to nonlinear stochastic systems—perhaps using linear approximations or particle-based methods—would broaden its applicability.
Second, the focus is on open-loop control. A natural next step is to analyze the value of information: how much can closed-loop feedback reduce the energy cost below the open-loop floor? This connects to dual control and active learning, where control actions are chosen not just to regulate but also to reduce uncertainty.
Third, the energy floor depends on the noise structure. Can we design systems or sensors to minimize $J_\infty^*$? For example, in quantum control, better qubit coherence times (smaller $\Sigma$) directly lower the energy floor. This provides a quantitative link between hardware improvements and control performance.
Finally, the framework could be used for risk-aware control in safety-critical applications. By tuning $\gamma$, operators can explicitly trade off energy use against failure probability. In nuclear reactor control or air traffic management, this could formalize safety margins in a mathematically rigorous way.
The authors’ work is a reminder that in a noisy universe, perfection has a price—and sometimes, you can’t pay less, no matter how slowly you try.