Mutual Information Optimal Control Problem

Updated 6 July 2026

MIOCP is a framework that integrates mutual information into control design, sensing, and decision making to optimize system performance.
It encompasses methodologies that maximize, constrain, or regularize mutual information across dynamic, static, and hybrid systems using tools from reinforcement learning, active sensing, and optimal experimental design.
The framework has significant implications in fields such as sensor scheduling, mixing protocols, detector design, and intrinsic control with actionable optimization strategies.

Searching arXiv for recent and foundational papers on mutual-information optimal control and closely related formulations. Querying the arXiv API for papers matching “mutual information optimal control” and adjacent terms. $I[\mathbf r_T;\mathbf r_0]$ 6

The Mutual Information Optimal Control Problem (MIOCP) denotes a family of optimal control, design, sensing, scheduling, and decision problems in which mutual information enters the formulation as an objective, a constraint, a regularizer, a stopping cost, or an operational performance metric. In the surveyed literature, MIOCP includes formulations that maximize information transfer from latent state to measurements, penalize information flow from state to control, minimize retained information after mixing, or enforce explicit information budgets on randomized policies. Accordingly, the “control” variable may be a feedback policy, a sensor gain, an attention location, a sampling rule, a pulse-sequence parameter, a shear protocol, a radar stopping decision, a wireless scheduling action, or even a static detector geometry (Enami et al., 7 Jul 2025, Shafieepoorfard et al., 2015, Zhao et al., 2021, Wozniak et al., 18 Mar 2025).

1. Conceptual scope and defining formulations

In the surveyed work, no single canonical equation exhausts MIOCP. Rather, the common structure is that the optimization variable changes the joint law of two random objects whose mutual information is operationally meaningful. In discrete-time linear control, the relevant quantity is the mutual information between state and action, obtained after optimizing a learnable prior in a KL-regularized stochastic control problem (Enami et al., 7 Jul 2025). In intrinsic-control reinforcement learning, the quantity is the mutual information between agent state and surrounding state, written as $I(S^a;S^s)$ , with the interpretation that the agent should have “maximum control on its surrounding” (Zhao et al., 2021). In active sensing with hard attention, the objective is $\max_{l_t} I(s_{t+1}; l_t)$ , where the control is a hard attention location (Sahni et al., 2021). In detector design, the optimization is $\max_\theta I(x;y)$ , where $\theta$ parameterizes detector geometry and $x$ and $y$ denote truth-level particle variables and detector response (Wozniak et al., 18 Mar 2025). In low-Reynolds-number mixing, the control problem is to minimize or erase the mutual information between initial and final tracer positions, so that efficient mixing corresponds to low $I[\mathbf r_T;\mathbf r_0]$ (Cocconi et al., 5 Feb 2025).

A second major class imposes mutual-information constraints rather than maximizing or minimizing mutual information directly. Rationally inattentive control constrains the mutual information between state and action under a randomized stationary policy, $I(\mu,\Phi)\le R$ , and then minimizes average or discounted cost (Shafieepoorfard et al., 2015). Closely related formulations appear in sequential detection, where a stopping cost is built from mutual information or stochastic observability of Gaussian targets, and in information-freshness problems, where sampling is controlled to maximize time-average expected mutual information between the current source state and the delivered samples (Krishnamurthy et al., 2011, Sun et al., 2018).

A third class uses mutual information as a regularizer. In discrete-time linear systems, the objective

$\min_{(\pi,\rho)} \mathbb{E}\!\left[ \sum_{k=0}^{T-1} \left\{ \frac{1}{2}\|u_k\|_{R_k}^{2} +\varepsilon\,\mathcal{D}_{\mathrm{KL}\!\left[\pi_k(\cdot\mid x_k)\,\|\,\rho_k(\cdot)\right] \right\} +\frac{1}{2}\|x_T-\mu_{x_{\mathrm{fin}}}\|_{F}^{2} \right]$

becomes a mutual-information-regularized control problem because optimizing the prior $\rho$ turns the KL term into $\max_{l_t} I(s_{t+1}; l_t)$ 0 (Enami et al., 7 Jul 2025). An analogous mechanism underlies mutual-information optimal density control and its connection to generalized Schrödinger bridges (Enami et al., 10 May 2026).

2. Canonical problem classes

Problem class	Representative formulation	Representative sources
MI maximization	$\max_{l_t} I(s_{t+1}; l_t)$ 1 or $\max_{l_t} I(s_{t+1}; l_t)$ 2	(Zhao et al., 2021, Wozniak et al., 18 Mar 2025, Jha et al., 2022)
MI minimization / information erasure	$\max_{l_t} I(s_{t+1}; l_t)$ 3	(Cocconi et al., 5 Feb 2025)
MI-constrained control	minimize control cost subject to $\max_{l_t} I(s_{t+1}; l_t)$ 4	(Shafieepoorfard et al., 2015)
MI-regularized control	cost $\max_{l_t} I(s_{t+1}; l_t)$ 5, with optimized $\max_{l_t} I(s_{t+1}; l_t)$ 6 yielding $\max_{l_t} I(s_{t+1}; l_t)$ 7	(Enami et al., 7 Jul 2025, Enami et al., 10 May 2026)
MI-based stopping, reward, or freshness	stopping cost, intrinsic reward, or sampling rule derived from MI	(Krishnamurthy et al., 2011, Sun et al., 2018, Zhao et al., 2021)

Across these classes, mutual information is almost always interpreted through KL divergence. Representative identities used in the literature include

$\max_{l_t} I(s_{t+1}; l_t)$ 8

and entropy decompositions such as $\max_{l_t} I(s_{t+1}; l_t)$ 9 or $\max_\theta I(x;y)$ 0 in optimal experimental design (Wozniak et al., 18 Mar 2025, Jha et al., 2022). In neural settings, mutual information is commonly estimated rather than computed analytically. Both detector-design optimization and intrinsic-control RL employ the Donsker–Varadhan representation and the Mutual Information Neural Estimator (MINE) (Wozniak et al., 18 Mar 2025, Zhao et al., 2021).

The surveyed papers also differ on whether mutual information measures useful preservation or useful destruction. In detector design, MRI pulse-sequence design, and intrinsic control, larger mutual information is desirable because it means the controlled system preserves or reveals more information about a latent variable or parameter (Wozniak et al., 18 Mar 2025, Jha et al., 2022, Zhao et al., 2021). In mixing, smaller mutual information is desirable because the aim is to erase information about initial tracer position (Cocconi et al., 5 Feb 2025). In rational inattention, the controller is penalized or constrained for using too much state information, so lower mutual information corresponds to limited attention (Shafieepoorfard et al., 2015).

3. Mathematical structure and variable choices

The meaning of MIOCP depends critically on what the random variables are. In state–action regularization for linear systems, the mutual information is $\max_\theta I(x;y)$ 1, and penalizing it discourages control inputs that are too informative about the state (Enami et al., 7 Jul 2025). In rationally inattentive control, the same state–action dependence is constrained by $\max_\theta I(x;y)$ 2, where $\max_\theta I(x;y)$ 3 is a state distribution and $\max_\theta I(x;y)$ 4 is a randomized stationary policy (Shafieepoorfard et al., 2015). In intrinsic control, the variables are the “agent state” $\max_\theta I(x;y)$ 5 and the “surrounding state” $\max_\theta I(x;y)$ 6, and the formal objective is $\max_\theta I(x;y)$ 7 (Zhao et al., 2021). In hard-attention control, the coupling is between attention location and next state, $\max_\theta I(x;y)$ 8, although the practical objective is a hybrid of entropy regularization and reconstruction-error-based reward (Sahni et al., 2021).

A separate axis of variation is whether the problem is dynamic, static, or hybrid. Classical control examples optimize a policy over time, as in discrete-time linear systems, rationally inattentive Markov control, and continuous-time sensor gain scheduling (Enami et al., 7 Jul 2025, Shafieepoorfard et al., 2015, Zinage et al., 2021). By contrast, detector design treats the static detector configuration $\max_\theta I(x;y)$ 9 as the control variable, so the “control” is a geometry rather than a time-indexed action; the paper explicitly notes that unlike classic MIOCP formulations that optimize a control policy over time, here the control is the static detector configuration (Wozniak et al., 18 Mar 2025). Hyperpolarized $\theta$ 0C-pyruvate MRI occupies an intermediate position: the control is a finite sequence of flip angles $\theta$ 1, and the state evolves according to low-fidelity ODE or high-fidelity PDE kinetics (Jha et al., 2022).

Several papers exploit special structure to convert mutual information into explicit control objectives. In low-Reynolds-number mixing, linear Gaussian dynamics imply a closed-form covariance $\theta$ 2, so minimizing $\theta$ 3 reduces to maximizing $\theta$ 4, or equivalently maximizing a nonnegative action functional $\theta$ 5 of the shear protocol $\theta$ 6 (Cocconi et al., 5 Feb 2025). In continuous-time Gauss-Markov estimation, Duncan’s theorem gives

$\theta$ 7

turning the MIOCP into a deterministic optimal control problem over Riccati dynamics for the estimation-error covariance $\theta$ 8 (Zinage et al., 2021). In rationally inattentive control, the dynamic problem is decomposed into a rate-distortion problem with a Bellman-type distortion $\theta$ 9 or $x$ 0, where the Bellman error carries future-cost information (Shafieepoorfard et al., 2015).

4. Solution methods and structural results

The linear-Gaussian literature emphasizes closed-form structure. For discrete-time linear systems with Gaussian policy and Gaussian prior classes, the fixed-prior subproblem yields a unique Gaussian optimal policy through a Riccati recursion, and the fixed-policy subproblem yields the unique optimal prior as the marginal distribution of the control under that policy (Enami et al., 7 Jul 2025). These results support an alternating minimization algorithm that updates policy and prior in turn. In mutual-information optimal density control, a closely related alternating optimization is derived in closed form and is shown to coincide with the alternating optimization of a generalized Schrödinger bridge with reference refinement (Enami et al., 10 May 2026). A further refinement establishes existence of an optimal policy and derives sufficient conditions on the temperature parameter $x$ 1 under which the optimal policy becomes stochastic or deterministic; notably, in this MIOCP, sufficiently small $x$ 2 can make the policy stochastic, while sufficiently large $x$ 3 can make it deterministic (Enami et al., 29 Jul 2025).

Other MIOCPs admit strong qualitative structure. Sequential detection with mutual-information stopping cost proves that the optimal stopping policy is monotone on the partially ordered set of positive definite covariance matrices, which in turn motivates monotone parametrized policies learned by SPSA (Krishnamurthy et al., 2011). Information freshness through queues shows that the optimal sampling policy is a threshold policy: a new sample is taken once a conditional mutual information term reduces to a threshold, and the threshold equals the optimum value of the time-average expected mutual information being maximized (Sun et al., 2018). Continuous-time sensor-gain control for minimum-information estimation derives Pontryagin minimum principle conditions; in the scalar case, the optimal gain is bang-bang except for the possibility of an intermediate value on the switching surface, and the number of switches is at most two (Zinage et al., 2021).

A distinct group of papers uses variational or neural surrogates. MUSIC alternates between collecting trajectories, estimating $x$ 4 with a MINE critic, and updating the policy with DDPG, SAC, or PPO using a transition-level intrinsic reward (Zhao et al., 2021). Generative Intrinsic Optimization builds a policy-iteration scheme in which the intrinsic reward is $x$ 5, and introduces a variational posterior $x$ 6 together with a generative dynamics model $x$ 7 so that mutual information appears through a VAE-style lower bound (Ma, 2023). Detector optimization uses a local deep-learning surrogate loop: nearby detector designs are simulated in Geant4, an MI estimate is produced by MINE, an MLP surrogate $x$ 8 is trained, and the detector parameters are updated by gradient descent on the surrogate (Wozniak et al., 18 Mar 2025).

Several non-Gaussian applications admit exact or greedy solutions. Information-optimal mixing derives the exact maximizers of the protocol functional under both fixed total shear and fixed total dissipation per unit volume: the first is a universal delta pulse at mid-time, and the second is a universal time-reversal-symmetric Jacobi-elliptic protocol solving a Duffing equation (Cocconi et al., 5 Feb 2025). In routing with mutual information accumulation, structural theorems imply that in each stage at most one node transmits, enabling optimal greedy algorithms for minimum delay routing, minimum energy routing under a delay constraint, and minimum delay broadcast, albeit with exponential worst-case complexity after reduction (Urgaonkar et al., 2010). In delay-optimal power-aware scheduling with mutual information accumulation, a frame-based Lyapunov drift formulation yields an online algorithm whose per-frame subproblem reduces to a dynamic program over remaining mutual information, with an $x$ 9-type delay optimality gap under an average power constraint (Wei et al., 2015).

5. Representative domains and operational interpretations

In reinforcement learning, MIOCP frequently appears as intrinsic motivation or active sensing. MUSIC formalizes intrinsic control as maximizing $y$ 0 and reports that unsupervised MUSIC learns meaningful manipulation behaviors, including reaching, pushing, sliding, and picking up an object in FetchPickAndPlace without any task reward (Zhao et al., 2021). Generative Intrinsic Optimization instead couples external reward with a mutual-information term between action and a future sequence $y$ 1, yielding a control objective that seeks “maximally informative consequences” while supporting a convergent policy-iteration scheme (Ma, 2023). Hard-attention control treats attention location as the control input and uses a learned Dynamic Memory Map; the attention policy is trained to look where the world model is most wrong, and the resulting representation supports downstream control in partially observable tasks (Sahni et al., 2021).

In sensing, estimation, and sequential decision, mutual information operates as a resource metric. Sequential radar detection uses mutual-information-based stochastic observability as a stopping cost and proves a monotone policy structure on covariance matrices (Krishnamurthy et al., 2011). Continuous-time sensor gain scheduling minimizes a weighted sum of mutual information and mean-square estimation error for a Kalman-Bucy filter over a Gaussian channel (Zinage et al., 2021). Information freshness through queues reinterprets age as decay of $y$ 2 and obtains an exact threshold sampling rule (Sun et al., 2018). These formulations show that MIOCP is not restricted to choosing physical actuation; it also includes choosing when to observe, how strongly to observe, and when to stop observing.

In scientific design, mutual information is used as a task-agnostic scalar objective. Hyperpolarized $y$ 3C-pyruvate MRI formulates optimal experimental design over pulse-sequence flip angles by maximizing the mutual information between noisy measurements $y$ 4 and uncertain parameters $y$ 5. A time-varying flip-angle scheme yields a higher mutual information value than a constant scheme, but the constant scheme gives the best accuracy and precision under noise-corrupted inference; for the MRI data examined, pyruvate and lactate flip angles of $y$ 6 and $y$ 7 are the best choice in terms of accuracy and precision of parameter recovery (Jha et al., 2022). End-to-end detector design in high-energy physics uses $y$ 8 to optimize calorimeter layer thicknesses and finds that MI-optimized designs are closely aligned with reconstruction-based and physics-informed designs (Wozniak et al., 18 Mar 2025).

In physics and communication, mutual information can represent irreversibility or progress. Low-Reynolds-number mixing measures mixing quality by the mutual information between initial and final tracer positions, deriving exact universal optimal protocols and interpreting the result as a minimum energetic cost of erasing information (Cocconi et al., 5 Feb 2025). Multi-hop wireless routing with rateless codes treats the state as decoded nodes plus residual mutual information at undecoded nodes and optimizes routing or broadcast under mutual-information accumulation (Urgaonkar et al., 2010). Single-link opportunistic scheduling likewise uses accumulated mutual information toward packet completion as the controlled state variable and optimizes average delay subject to average power (Wei et al., 2015).

6. Distinctions, misconceptions, and unresolved issues

A recurring misconception is that MIOCP always means “maximize mutual information.” The surveyed literature shows a broader picture. Some problems maximize mutual information, as in intrinsic control, detector design, and optimal experimental design (Zhao et al., 2021, Wozniak et al., 18 Mar 2025, Jha et al., 2022). Some minimize it, as in information-optimal mixing (Cocconi et al., 5 Feb 2025). Some constrain it, as in rationally inattentive control (Shafieepoorfard et al., 2015). Some obtain it indirectly by optimizing a prior inside a KL regularizer, thereby turning policy regularization into $y$ 9 (Enami et al., 7 Jul 2025). Some use it as a stopping or threshold variable rather than as an objective in the conventional sense (Krishnamurthy et al., 2011, Sun et al., 2018).

A second misconception is that any mutual-information objective is automatically sufficient for control. This is explicitly contradicted by the representation-learning analysis of mutual-information objectives. For state-based rewards, maximizing forward information $I[\mathbf r_T;\mathbf r_0]$ 0 is sufficient for control, but state-only transition information $I[\mathbf r_T;\mathbf r_0]$ 1 and inverse information $I[\mathbf r_T;\mathbf r_0]$ 2 are not sufficient in general; adding $I[\mathbf r_T;\mathbf r_0]$ 3 to the inverse objective does not repair the insufficiency (Rakelly et al., 2021). A closely related practical caveat appears in MRI design: a time-varying flip-angle design can achieve higher mutual information while giving worse accuracy and precision under noise-corrupted inverse inference (Jha et al., 2022). These results indicate that the random variables entering the mutual-information term matter at least as much as the presence of mutual information itself.

A third distinction concerns stochasticity. In maximum-entropy control, increasing temperature typically increases policy randomness. In mutual-information-optimal control of linear systems, that intuition does not carry over directly because the prior is optimized jointly with the policy. The linear-systems analysis shows that sufficiently small $I[\mathbf r_T;\mathbf r_0]$ 4 can yield a stochastic optimal policy, whereas sufficiently large $I[\mathbf r_T;\mathbf r_0]$ 5 can force a deterministic one; the same phenomenon appears in the alternating optimization algorithm (Enami et al., 29 Jul 2025). This is directly relevant to safety-critical applications, which motivates formulations that impose Gaussian boundary or density constraints to control state uncertainty while retaining MI regularization (Enami et al., 10 May 2026).

Finally, several papers identify limitations that delimit the present scope of MIOCP. MUSIC requires a meaningful agent–surrounding split, assumes full observability, and depends on the quality of the learned MI estimator (Zhao et al., 2021). Hard-attention control notes one-glimpse-per-state constraints, noisy-TV failure modes, and poor performance of one variational mutual-information approximation (Sahni et al., 2021). Detector optimization relies on surrogate learning because Geant4 is stochastic and non-differentiable in detector parameters (Wozniak et al., 18 Mar 2025). Alternating minimization for linear-system MIOCP decreases the objective monotonically, but objective convergence can occur much earlier than convergence of the decision variables, so stopping criteria based solely on the objective can be misleading (Enami et al., 7 Jul 2025).

Taken together, these formulations present MIOCP not as a single theorem or algorithm but as a unifying information-theoretic viewpoint on control: choose actions, schedules, sensing policies, or designs so as to regulate the statistical dependence that matters for performance, whether that means preserving information, erasing it, rationing it, or trading it against other costs.