Decentralized Stochastic Control with Finite-Dimensional Memories: A Memory Limitation Approach
Open accessAbstract
Decentralized stochastic control (DSC) is a stochastic optimal control problem consisting of multiple controllers. DSC assumes that each controller is unable to accurately observe the target system and the other controllers. This setup results in two difficulties in DSC; one is that each controller has to memorize the infinite-dimensional observation history, which is not practical, because the memory of the actual controllers is limited. The other is that the reduction of infinite-dimensional sequential Bayesian estimation to finite-dimensional Kalman filter is impossible in general DSC, even for linear-quadratic-Gaussian (LQG) problems. In order to address these issues, we propose an alternative theoretical framework to DSC—memory-limited DSC (ML-DSC). ML-DSC explicitly formulates the finite-dimensional memories of the controllers. Each controller is jointly optimized to compress the infinite-dimensional observation history into the prescribed finite-dimensional memory and to determine the control based on it. Therefore, ML-DSC can be a practical formulation for actual memory-limited controllers. We demonstrate how ML-DSC works in the LQG problem. The conventional DSC cannot be solved except in the special LQG problems where the information the controllers have is independent or partially nested. We show that ML-DSC can be solved in more general LQG problems where the interaction among the controllers is not restricted.
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At higher energy availability (Q=1000), Π_zx and Π_zz take nonzero values, and memory-based estimation becomes optimal.
“In contrast, when more energy is available ( Q = 1000), Π zx and Π zz take nonzero values, and memory-based estimation becomes optimal [Fig. 1(e,f)].”
From Theoretical Analysis of Resource-Induced Phase Transitions in Estimation Strategies (2026) · cites this paper as [33]
Section 6.2 (page unavailable) says memory-related elements of Π are not always zero, but does not tie this to energy availability or Q=1000. Section 6.1 (page unavailable) says the Π-based control accounts for state estimation but is not optimal; the optimal control also estimates other controllers’ memories. (LLM, conf=0.84)
Experimental — accuracy not yet measuredResource limitations induce a phase transition between memoryless and memory-based estimation strategies in a minimal model of biological information processing.
“By applying this framework to a minimal model of biological information processing, we found that resource limitations induce a phase transition between memoryless and memory-based estimation strategies [Fig. 1] [33, 34].”
From Theoretical Analysis of Resource-Induced Phase Transitions in Estimation Strategies (2026) · cites this paper as [33]
The Introduction and Section 3.1 describe a framework that constrains controllers to finite-dimensional memories and optimizes how they use those memories. Section 6.1 compares estimation approaches, but our copy does not report a resource-induced phase transition between memoryless and memory-based strategies or apply the framework to a minimal model of biological information processing. (LLM, conf=0.83)
Experimental — accuracy not yet measuredAs Q increases, nonzero Π_zx and Π_zz emerge discontinuously while Π_zx = Π_zz = 0 remains optimal.
“As Q increases, Π zx , Π zz = 0 emerge discontinuously, while Π zx = Π zz = 0 remains optimal.”
From Theoretical Analysis of Resource-Induced Phase Transitions in Estimation Strategies (2026) · cites this paper as [33]
Section 6.2 reports that memory-related elements of Π are not always zero, which supports the limited point that nonzero elements occur. Neither the numerical experiments in Sections 6.1–6.2 nor the formulation in Section 5.1 reports a discontinuous emergence as Q increases or says that Π_zx = Π_zz = 0 remains optimal. Page numbers are unavailable in our copy. (LLM, conf=0.79)
Experimental — accuracy not yet measuredIncreasing M or F induces a transition from memory-based to memoryless estimation.
“Similar to the transition with respect to Q , those with respect to M and F also occur monotonically but in the opposite direction: increasing M or F induces a transitions from memory-based to memoryless estimation [33, 34].”
From Theoretical Analysis of Resource-Induced Phase Transitions in Estimation Strategies (2026) · cites this paper as [33]
Section 6.1 discusses how memory controls affect estimation, but does not describe a transition from memory-based to memoryless estimation. Searches for that transition and for increasing M or F found no passage establishing the claimed direction or monotonicity. (LLM, conf=0.73)
Experimental — accuracy not yet measuredThe state noise D also induces a similar nonmonotonic phase transition.
“Furthermore, the state noise D also induces a similar nonmonotonic phase transition [33, 34].”
From Theoretical Analysis of Resource-Induced Phase Transitions in Estimation Strategies (2026) · cites this paper as [33]
Section 6.2 (page unavailable) says state noise causes the actual state to diverge from its target, but does not describe a phase transition. Searches for “nonmonotonic phase transition” and “state noise D” found no passage linking a noise parameter D to such a transition. (LLM, conf=0.75)
Experimental — accuracy not yet measured
The resource-induced phase transition exhibits discontinuity, nonmonotonicity, and a scaling relation even in a linear-quadratic-Gaussian setting.
“This transition also exhibits several nontrivial features, including discontinuity, nonmonotonicity, and a scaling relation, even in a linear-quadratic-Gaussian (LQG) setting [Fig. 2] [33, 34].”
From Theoretical Analysis of Resource-Induced Phase Transitions in Estimation Strategies (2026) · cites this paper as [33]
Section 5 and Section 6 study memory-limited decentralized control in an LQG setting, but the passages do not report a resource-induced phase transition or its claimed features. Searches for phase transition, discontinuity, nonmonotonicity, and scaling relation found no passage addressing those claims. Page numbers were not available in our copy. (LLM, conf=0.74)
Experimental — accuracy not yet measuredThe phase transition is discontinuous even though the model is linear and Gaussian.
“This phase transition is discontinuous even though this model is linear and Gaussian [Fig. 2(a)].”
From Theoretical Analysis of Resource-Induced Phase Transitions in Estimation Strategies (2026) · cites this paper as [33]
Section 5 and the numerical experiments in Section 6 describe linear-quadratic-Gaussian control models, but no passage found reports a phase transition or says that one is discontinuous. Searches for “phase transition” and for a discontinuous jump or threshold did not locate the claimed result in our copy. (LLM, conf=0.75)
Experimental — accuracy not yet measuredAt low Q, Π_zx = Π_zz = 0 is the only solution and is optimal.
“At low Q , Π zx = Π zz = 0 is the only solution and clearly optimal.”
From Theoretical Analysis of Resource-Induced Phase Transitions in Estimation Strategies (2026) · cites this paper as [33]
Section 5.2 gives Riccati equations and an optimal-control function, while Section 6.1 compares controls involving Π. Neither passage says that at low Q, Π_zx = Π_zz = 0 is the only solution or that this solution is optimal. Searches for low Q, zero solutions, and the named Π components found no passage establishing the claim. (LLM, conf=0.72)
Experimental — accuracy not yet measuredAt low energy availability (Q=100), the memory control gains Π_zx and Π_zz vanish, indicating that no observational information is encoded into memory.
“At low energy availability ( Q = 100), the memory control gains, Π zx and Π zz , vanish, indicating that no observational information is encoded into the memory z t [Fig. 1(c,d)].”
From Theoretical Analysis of Resource-Induced Phase Transitions in Estimation Strategies (2026) · cites this paper as [33]
Sections 6.1 and 6.2 discuss memory-related control gains, but do not report a low-energy condition of Q=100 or say that Π_zx and Π_zz vanish under it. Searches for Q=100, energy availability, and observational information encoded in memory found no passage addressing the claimed condition. (LLM, conf=0.74)
Experimental — accuracy not yet measuredAt low energy availability (Q=100), the organism estimates the environmental state solely based on the current observation, corresponding to memoryless estimation.
“As a result, the organism estimates the environmental state x t solely based on the current observation y t , corresponding to memoryless estimation.”
From Theoretical Analysis of Resource-Induced Phase Transitions in Estimation Strategies (2026) · cites this paper as [33]
Section 4.3 describes controllers estimating the state from their own memory, not an organism estimating it solely from a current observation. Searches for low energy availability, Q=100, and memoryless estimation found no passage addressing the claimed condition or behavior; the paper's numerical experiments (Section 6.1) instead concern controller performance. (LLM, conf=0.76)
Experimental — accuracy not yet measuredIncreasing Q induces a monotonic transition from memoryless to memory-based estimation.
“While increasing Q induces a monotonic transition from memoryless to memory-based estimation [Fig. 2(a)]”
From Theoretical Analysis of Resource-Induced Phase Transitions in Estimation Strategies (2026) · cites this paper as [33]
Section 5.4 discusses a term denoted Qᵢ that may improve estimation, while Section 6.1 compares control strategies with different degrees of estimation. Neither passage says that increasing Q produces a monotonic transition from memoryless to memory-based estimation (searched for the transition, Q, and Figure 2(a)). (LLM, conf=0.76)
Experimental — accuracy not yet measuredAs E increases, the current observation becomes less reliable, and memory-based estimation becomes favorable.
“As E increases, the current observation becomes less reliable, and memorybased estimation becomes favorable.”
From Theoretical Analysis of Resource-Induced Phase Transitions in Estimation Strategies (2026) · cites this paper as [33]
Section 6.1 says that controlling memories plays an important role in estimation, but it does not compare memory-based estimation with reliance on a current observation as E increases. Searches for E, observation reliability, and observation noise found no passage in our copy establishing the claimed trend; the available passages have no page numbers. (LLM, conf=0.72)
Experimental — accuracy not yet measuredWhen E becomes too large, memory control becomes ineffective, and memoryless estimation again becomes optimal.
“However, when E becomes too large, memory control becomes ineffective, and memoryless estimation again becomes optimal.”
From Theoretical Analysis of Resource-Induced Phase Transitions in Estimation Strategies (2026) · cites this paper as [33]
Section 3.1 discusses the cost and optimization of memory control, and Section 6.1 reports that the memory-aware control performed best in its numerical experiment. Neither passage addresses what happens when E becomes large or says that memoryless estimation then becomes optimal (searched: “memoryless,” “E becomes large,” and high memory-control cost). (LLM, conf=0.72)
Experimental — accuracy not yet measuredIncreasing the observation noise E leads to a nonmonotonic transition.
“increasing the observation noise E leads to a nonmonotonic transition [Fig. 2(b)].”
From Theoretical Analysis of Resource-Induced Phase Transitions in Estimation Strategies (2026) · cites this paper as [33]
Section 4.3 identifies Figure 2b as a depiction of controllers sharing memories, not a plot of observation noise. Sections 6.1–6.2 describe numerical experiments, but the passages found do not report a nonmonotonic transition as observation noise E increases (searched for the transition, noise E, and Figure 2b). (LLM, conf=0.78)
Experimental — accuracy not yet measuredAfter further increase of Q, the optimal solution discontinuously switches from Π_zx = Π_zz = 0 to Π_zx, Π_zz ≠ 0.
“After further increase of Q , the optimal solution discontinuously switches from Π zx = Π zz = 0 to Π zx , Π zz = 0.”
From Theoretical Analysis of Resource-Induced Phase Transitions in Estimation Strategies (2026) · cites this paper as [33]
Section 6.2 (page unavailable) says that some memory-related elements of Π are not always zero, but does not describe how Π_zx or Π_zz changes as Q increases. Searches for a discontinuous switch and a Q-dependent transition found no passage establishing the claimed change from zero to nonzero. (LLM, conf=0.72)
Experimental — accuracy not yet measuredAt low E, memoryless estimation is optimal because the current observation contains sufficient information.
“At low E , memoryless estimation is optimal since the current observation contains sufficient information.”
From Theoretical Analysis of Resource-Induced Phase Transitions in Estimation Strategies (2026) · cites this paper as [33]
The Introduction says controllers generally rely on observation histories, and Section 6.1 reports that accounting for estimation improves performance. Neither passage addresses a low-E regime or says that the current observation alone is sufficient. Searches for memoryless estimation and low-E observation sufficiency found no passage establishing the claimed exception. (LLM, conf=0.74)
Experimental — accuracy not yet measuredThe phase boundaries with respect to Q, M, and F collapse onto a single curve when plotted against Q/MF, indicating a scaling relation among these parameters.
“Moreover, the phase boundaries with respect to Q , M , and F collapse onto a single curve when plotted against Q/MF [Fig. 2(c)], indicating a scaling relation among these parameters.”
From Theoretical Analysis of Resource-Induced Phase Transitions in Estimation Strategies (2026) · cites this paper as [33]
Section 5.2 derives control equations, and Section 6 discusses numerical results for decentralized control; neither addresses phase boundaries collapsing when plotted against Q/MF. Searches for “phase boundaries Q/MF” and “phase transition Figure 2(c) scaling relation Q M F” found no passage supporting the claim in our copy. (LLM, conf=0.75)
Experimental — accuracy not yet measured
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The self-check needs the paper’s full text.
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Self-citation
- Authors
- Tottori, T., & Kobayashi, T. J.
- Affiliations
- The University of Tokyo (JP)
- Institute of Industrial Science, The University of Tokyo (JP)
- Corresponding author
- Tottori, Takehiro
- Venue
- Entropy
- Publisher
- MDPI AG
- Published
- 2023-05-12
- Volume
- 25
- Issue
- 5
- Pages
- 791
- ISSN
- 1099-4300 (ISSN-L)
- PMC
- PMC10217446
- OpenAlex
- W4376274848
- Cited by
- 7
- References
- 52
- Citations by year
- 2026: 2 · 2025: 3 · 2023: 2
- FWCI
- 0.75
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Licence
- Locations
- Entropy (publishedVersion, cc-by)
- PubMed Central (publishedVersion, cc-by)
- PubMed (publishedVersion)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
- Funders
- JSPS Research Fellowship (21J20436, 19H05799, JPMJCR2011)
- JSPS KAKENHI (21J20436, 19H05799, JPMJCR2011)
- JST CREST (21J20436, 19H05799, JPMJCR2011)
- Topics
- Physical Sciences › Engineering › Control and Systems Engineering › Advanced Control Systems Optimization
- Physical Sciences › Computer Science › Computer Networks and Communications › Distributed Control Multi-Agent Systems
- Physical Sciences › Computer Science › Artificial Intelligence › Target Tracking and Data Fusion in Sensor Networks
- Keywords
- decentralized stochastic control, LQG control, sequential Bayesian inference, Kalman filter, controller interaction
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