pith. sign in

arxiv: 2604.19994 · v1 · submitted 2026-04-21 · 🧮 math.OC · cs.SY· eess.SY

Covariance Steering of Discrete-Time Markov Jump Linear Systems with Multiplicative Noise

Fangji Wang , Siddhartha Ganguly , Panagiotis Tsiotras This is my paper

Pith reviewed 2026-05-10 01:39 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords covariance steeringMarkov jump linear systemsmultiplicative noiselifted formulationsemidefinite programmingchance constraintsdiscrete-time control
0
0 comments X

The pith

A lifted second-moment formulation converts covariance steering for Markov jump linear systems with multiplicative noise into an equivalent SDP.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how to steer the mean and covariance trajectories of discrete-time Markov jump linear systems whose noise intensity depends on both the current state and the applied control. It establishes that the original steering task is equivalent to a lifted problem whose variables are second-moment matrices that already encode both means and covariances. This equivalence justifies representing admissible controls as mode-dependent linear feedback augmented by feedforward terms and independent random components, which is required once multiplicative noise is present. The reformulation produces a lossless relaxation in moment space and a convex semidefinite program for the unconstrained case, together with convex surrogate approximations for chance constraints on the state and input.

Core claim

We prove that the resulting lifted problem is equivalent to the original covariance steering problem formulation. This leads to a lossless relaxation in moment variables and an SDP reformulation for the unconstrained case. Without loss of generality, feasible controls may be represented by mode-dependent linear feedback together with feedforward and independent random components, and a purely affine state-feedback law does not in general suffice.

What carries the argument

The lifted-state formulation that embeds mean and covariance information into a unified second-moment description of the closed-loop dynamics.

If this is right

  • Controls for multiplicative-noise MJLS must include independent random components in addition to mode-dependent linear feedback.
  • The unconstrained covariance steering problem admits a convex SDP reformulation via the lifted moments.
  • Chance constraints on state and control admit tractable convex surrogates.
  • An iterative reference-update procedure reduces conservatism of the chance-constrained solutions.
  • The method applies directly to finance models that switch between regimes with state- and control-dependent volatility.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same lifting may simplify covariance steering for other linear systems whose noise multiplies the state or input.
  • Receding-horizon implementations become feasible once the per-step SDP can be solved in real time.
  • The equivalence suggests analogous lifts could be tried for continuous-time or infinite-horizon MJLS problems.

Load-bearing premise

That admissible controls can always be written, without loss of generality, as mode-dependent linear feedback plus feedforward plus independent random components.

What would settle it

A concrete instance in which the SDP solution, when substituted back into the original nonlinear moment equations, fails to reach the prescribed terminal covariance, or an example where a purely affine feedback law achieves the target covariance under nonzero multiplicative noise.

Figures

Figures reproduced from arXiv: 2604.19994 by Fangji Wang, Panagiotis Tsiotras, Siddhartha Ganguly.

Figure 1
Figure 1. Figure 1: Results for 𝛾 (𝑚) = 1 for each 𝑚. In this case, we enforce 𝑥6 to stay within the norm ball of radius 0.6 centered at (0.5, 1) with high probability. (a)–(c) show the sampled state trajectories at itera￾tions 𝑚 = 1, 5, 10, respectively. The red and blue ellipsoids are the 0.997-confidence ellipsoids at 𝑘 = 2, 4, 6, 8 induced by the Gaussian dis￾tributions N(𝜇𝑘 ( 𝑗), Σ𝑘 ( 𝑗)), 𝑗 = 1, 2, respectively; the red… view at source ↗
Figure 2
Figure 2. Figure 2: Results for iteratively reduced 𝛾-values. In this case, in addi￾tion to the ball constraint on 𝑥6, we also enforced 𝑥𝑘,1 ⩾ 0 and |𝑢𝑘,1| ⩽ 9 with high probability for each 𝑘. (a) and (b) show the sampled state trajectories for the unconstrained and constrained problems, respectively, at the last iteration. The cyan and orange lines in (c) are the upper and lower 0.05-quantiles of the sampled 𝑢𝑘,1, respectiv… view at source ↗
Figure 3
Figure 3. Figure 3: Results for the hedging example. (a) shows sampled state trajectories together with the intermediate exposure limits. (b) shows the upper and lower 0.02-quantiles of the sampled control inputs over the horizon, together with the prescribed control bounds. The system parameters are chosen so that the option introduces slight Delta cross￾coupling and both hedging instruments become less effective in the dist… view at source ↗
read the original abstract

We study a finite-horizon covariance steering problem for discrete-time Markov jump linear systems (MJLS) with both state- and control-dependent multiplicative noise. The objective is to minimize a quadratic running cost while steering the system from given mode-conditioned initial means and covariances to a prescribed terminal mean and covariance. We first show that, without loss of generality, feasible controls may be represented by mode-dependent linear feedback together with feedforward and independent random components, and we highlight that, in contrast to the case without multiplicative noise, a purely affine state-feedback law does not in general suffice. To this end, we introduce a lifted-state formulation that embeds the mean and covariance information into a unified second-moment description, and we prove that the resulting lifted problem is equivalent to the original covariance steering problem formulation. This leads to a lossless relaxation in moment variables and an SDP reformulation for the unconstrained case. We further study chance-constrained covariance steering with ball and half-space constraints on the state and control, derive tractable sufficient convex surrogates, and establish an iterative reference-update scheme to reduce conservatism. Numerical experiments on a finance application illustrate our results.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The paper studies finite-horizon covariance steering for discrete-time Markov jump linear systems with state- and control-dependent multiplicative noise. The objective is to minimize a quadratic cost while steering from given mode-conditioned initial means/covariances to a prescribed terminal mean and covariance. The authors assert that, without loss of generality, controls may be restricted to mode-dependent linear feedback plus feedforward and independent random components (noting that purely affine feedback does not suffice under multiplicative noise). They introduce a lifted second-moment formulation, prove equivalence to the original problem, obtain a lossless relaxation in moment variables, and derive an SDP for the unconstrained case. They further develop tractable convex surrogates for chance constraints on state and control (ball and half-space) together with an iterative reference-update scheme to reduce conservatism, and illustrate the approach on a finance application.

Significance. If the central equivalence and lossless claims hold, the work provides a tractable convex framework for covariance steering in MJLS with multiplicative noise, extending prior results on linear systems and additive noise. The SDP reformulation for the unconstrained case and the iterative scheme for chance constraints are potentially useful for applications such as portfolio optimization or switched systems with uncertainty. The explicit contrast with the no-multiplicative-noise case and the lifting to unified second-moment dynamics are technically interesting contributions.

major comments (3)
  1. [Section on control parameterization and lifted formulation (early sections, prior to the equivalence theorem)] The WLOG control parameterization (mode-dependent linear feedback + feedforward + independent random components) is load-bearing for the equivalence claim and the subsequent lossless relaxation. The manuscript must provide a self-contained proof that this class attains all achievable second-moment trajectories under multiplicative noise; the coupling of noise realizations to both state and control may prevent an independent additive random component from reproducing all conditional covariances attainable by history-dependent or nonlinear policies. Cite the specific proposition or lemma establishing this parameterization and verify it against the closed-loop second-moment equations.
  2. [Lifted formulation and equivalence section] § on lifted-state formulation and equivalence proof: the claim that the lifted problem is equivalent to the original covariance steering problem (yielding a lossless relaxation) requires explicit verification that the lifting map preserves all original constraints and that the mode-dependent multiplicative noise terms are correctly embedded in the second-moment dynamics. Provide the explicit lifting map, the resulting dynamics for the augmented second-moment matrix, and a proof that every feasible moment trajectory in the original problem is recovered.
  3. [Chance-constrained covariance steering section] Chance-constrained section: the sufficient convex surrogates for ball and half-space constraints are presented as tractable, but their conservatism relative to the original chance constraints should be quantified (e.g., via explicit bounds or tightness examples). The iterative reference-update scheme is claimed to reduce conservatism; demonstrate convergence or improvement guarantees, and clarify how the surrogates interact with the lifted SDP.
minor comments (3)
  1. [Preliminaries] Notation for mode-dependent quantities (e.g., A_i, B_i, noise covariances) should be introduced consistently and early; ensure all indices are defined before first use.
  2. [Numerical experiments] In the numerical experiments, report the specific SDP solver tolerances, iteration counts for the reference-update scheme, and any observed gaps between surrogate and true constraint satisfaction.
  3. [Abstract and chance-constraint section] The abstract mentions 'post-hoc reference updates for conservatism'; clarify whether these are part of the main algorithm or an optional post-processing step.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, indicating the revisions we will make to strengthen the presentation.

read point-by-point responses

  1. Referee: The WLOG control parameterization (mode-dependent linear feedback + feedforward + independent random components) is load-bearing for the equivalence claim and the subsequent lossless relaxation. The manuscript must provide a self-contained proof that this class attains all achievable second-moment trajectories under multiplicative noise; the coupling of noise realizations to both state and control may prevent an independent additive random component from reproducing all conditional covariances attainable by history-dependent or nonlinear policies. Cite the specific proposition or lemma establishing this parameterization and verify it against the closed-loop second-moment equations.

    Authors: We agree that a fully self-contained justification of the control parameterization is necessary for the equivalence claim. The manuscript states that this class is without loss of generality and contrasts it with the additive-noise case, but we acknowledge the proof could be expanded for clarity. In the revised version we will add an explicit lemma (new Lemma 2) that derives the closed-loop second-moment equations under the proposed parameterization, shows that the independent random component can reproduce any attainable conditional covariance by solving for the appropriate second-moment matrix, and verifies equivalence to general policies by direct substitution into the multiplicative noise terms. The lemma will be cited at the start of the lifted formulation section. revision: yes

  2. Referee: § on lifted-state formulation and equivalence proof: the claim that the lifted problem is equivalent to the original covariance steering problem (yielding a lossless relaxation) requires explicit verification that the lifting map preserves all original constraints and that the mode-dependent multiplicative noise terms are correctly embedded in the second-moment dynamics. Provide the explicit lifting map, the resulting dynamics for the augmented second-moment matrix, and a proof that every feasible moment trajectory in the original problem is recovered.

    Authors: We appreciate the request for explicit details. The manuscript introduces the lifted second-moment formulation and states equivalence, but we will expand the relevant theorem (Theorem 1) to include: (i) the precise lifting map that augments each mode-conditioned state with its second-moment matrix in a block-diagonal structure; (ii) the closed-form linear dynamics for the augmented second-moment matrix that correctly embed the state- and control-dependent multiplicative noise via Kronecker products; and (iii) a bidirectional proof showing that every feasible original trajectory maps to a feasible lifted trajectory (and conversely) while preserving the quadratic cost and terminal constraints. These additions will make the lossless relaxation fully transparent. revision: yes

  3. Referee: Chance-constrained section: the sufficient convex surrogates for ball and half-space constraints are presented as tractable, but their conservatism relative to the original chance constraints should be quantified (e.g., via explicit bounds or tightness examples). The iterative reference-update scheme is claimed to reduce conservatism; demonstrate convergence or improvement guarantees, and clarify how the surrogates interact with the lifted SDP.

    Authors: We agree that additional analysis of conservatism and the iterative scheme would improve the paper. In the revision we will add: (i) a new subsection quantifying conservatism via tightness examples (e.g., for isotropic Gaussians the ball surrogate becomes exact at a computable radius) and explicit sub-optimality bounds derived from the support function of the chance set; (ii) a convergence result for the reference-update iteration showing monotonic decrease of the surrogate cost and feasibility of the original chance constraints after finitely many steps under compactness; and (iii) an explicit statement that the surrogates enter the lifted SDP as additional linear matrix inequalities on the second-moment variables. Updated numerical results on the finance example will illustrate the improvement. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation proceeds from dynamics to lifted equivalence without self-definition or input reduction

full rationale

The paper starts from the given MJLS dynamics with multiplicative noise, asserts and proves a WLOG control parameterization (mode-dependent linear feedback plus feedforward and independent random terms), then constructs the lifted second-moment state and shows equivalence to the original covariance steering problem by direct substitution into the moment equations. This yields the lossless relaxation and SDP without any fitted parameters being renamed as predictions, without self-citations bearing the central load, and without the equivalence being true by construction of the inputs. The derivation is self-contained against the system equations and does not reduce the claimed equivalence to a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard stochastic control assumptions and convex relaxation properties rather than new postulates. No free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Feasible controls admit a mode-dependent linear feedback plus feedforward and independent random components without loss of generality.
    Stated explicitly as the starting point for the lifted formulation.
  • domain assumption The lifted second-moment vector exactly captures the original mean-covariance steering objective.
    Claimed equivalence between lifted and original problems.

pith-pipeline@v0.9.0 · 5511 in / 1343 out tokens · 34665 ms · 2026-05-10T01:39:22.583185+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

56 extracted references · 56 canonical work pages

  1. [1]

    Chance-Constrained Covariance Steering for Discrete-Time

    Shrivastava, Shaurya and Oguri, Kenshiro , journal=. Chance-Constrained Covariance Steering for Discrete-Time. 2025 , NOTE =

    work page 2025

  2. [2]

    arXiv preprint arXiv:2406.14740 , year=

    Reachability and controllability analysis of the state covariance for linear stochastic systems , author=. arXiv preprint arXiv:2406.14740 , year=

    work page arXiv

  3. [3]

    62nd IEEE Conference on Decision and Control (CDC) , pages=

    Data-driven covariance steering control design , author=. 62nd IEEE Conference on Decision and Control (CDC) , pages=. 2023 , NOTE =

    work page 2023

  4. [4]

    Ono and B

    M. Ono and B. C. Williams , booktitle=. Iterative Risk Allocation: A new approach to robust Model Predictive Control with a joint chance constraint , year=

    work page

  5. [5]

    J. W. Knaup and P. Tsiotras , booktitle=. Computationally Efficient Covariance Steering for Systems Subject to Parametric Disturbances and Chance Constraints , year=

    work page

  6. [6]

    Nakashima and S

    H. Nakashima and S. Ganguly and K. Morimoto and K. Kashima , booktitle=. Formation Shape Control using the. 2025 , volume=

    work page 2025

  7. [7]

    Okamoto and M

    K. Okamoto and M. Goldshtein and P. Tsiotras , journal=. Optimal Covariance Control for Stochastic Systems Under Chance Constraints , year=

    work page

  8. [8]

    Ye and Z

    L. Ye and Z. Zhao and F. Liu , journal =. Stochastic. 2024 , NOTE =

    work page 2024

  9. [9]

    Systems & Control Letters , volume =

    Constrained quadratic control for. Systems & Control Letters , volume =. 2025 , author =

    work page 2025

  10. [10]

    O. L. V. Costa and W. L. de Paulo , journal=. Indefinite quadratic with linear costs optimal control of. 2007 , publisher=

    work page 2007

  11. [11]

    Fast Switching Detector-Based

    Graciani Rodrigues, CC and Todorov, Marcos G and Fragoso, Marcelo D , journal=. Fast Switching Detector-Based. 2021 , publisher=

    work page 2021

  12. [12]

    Schuurmans and P

    M. Schuurmans and P. Patrinos , journal=. A general framework for learning-based distributionally robust. 2023 , NOTE =

    work page 2023

  13. [13]

    Chitraganti and S

    S. Chitraganti and S. Aberkane and C. Aubrun and G. Valencia-Palomo and V. Dragan , NOTE =. On control of discrete-time state-dependent jump linear systems with probabilistic constraints: A receding horizon approach , journal =. 2014 , issn =

    work page 2014

  14. [14]

    Barbieri and O

    F. Barbieri and O. L. V. Costa , journal=. Optimal control with constrained total variance for. 2018 , publisher=

    work page 2018

  15. [15]

    O. L. V. Costa and R. P. Marques and M. D. Fragoso , year=. Discrete-time

    work page

  16. [16]

    O. L. V. Costa and A. de Oliveira , journal=. Optimal mean--variance control for discrete-time linear systems with. 2012 , publisher=

    work page 2012

  17. [17]

    O. L. V. Costa and G. R. A. M. Benites , booktitle=. Linear minimum mean square filter for discrete-time linear systems with multiplicative noise , year=

    work page

  18. [18]

    O. L. V. Costa and G. R. A. M. Benites , journal=. Robust mode-independent filtering for discrete-time. 2013 , publisher=

    work page 2013

  19. [19]

    O. L. V. Costa and G. R. A. M. BenitesM , journal=. Linear minimum mean square filter for discrete-time linear systems with. 2011 , publisher=

    work page 2011

  20. [20]

    Dombrovskii and T

    V. Dombrovskii and T. Pashinskaya , journal=. Design of model predictive control for constrained. 2020 , publisher=

    work page 2020

  21. [21]

    O. L. V. Costa and R. T. Okimura , journal=. Discrete-time mean variance optimal control of linear systems with. 2009 , publisher=

    work page 2009

  22. [22]

    Kim , journal=

    C.-J. Kim , journal=. Dynamic linear models with. 1994 , publisher=

    work page 1994

  23. [23]

    IEEE Transactions on Aerospace and Electronic Systems , volume=

    Interacting multiple model methods in target tracking: a survey , author=. IEEE Transactions on Aerospace and Electronic Systems , volume=. 2002 , NOTE =

    work page 2002

  24. [24]

    Huo and Z

    S. Huo and Z. Wang and G. Liu and F. Li and H. Shen , journal=. Model-free frequency control of power systems with unknown. 2024 , NOTE =

    work page 2024

  25. [25]

    Logothetis and V

    A. Logothetis and V. Krishnamurthy , journal=. Expectation maximization algorithms for MAP estimation of jump. 2002 , NOTE =

    work page 2002

  26. [26]

    Saijai and A

    J. Saijai and A. Abdo and W. Damlakhi and S. X. Ding , booktitle=. A fault detection scheme for discrete-time. 2011 , volume=

    work page 2011

  27. [27]

    A. A. G. Siqueira and M. H. Terra , journal=. Nonlinear and. 2004 , NOTE =

    work page 2004

  28. [28]

    IEEE Transactions on Automatic Control , volume=

    Stochastic receding horizon control of constrained linear systems with state and control multiplicative noise , author=. IEEE Transactions on Automatic Control , volume=

    work page

  29. [29]

    American Control Conference , pages=

    Portfolio optimization applications of stochastic receding horizon control , author=. American Control Conference , pages=

    work page

  30. [30]

    Shin and J

    M. Shin and J. H. Lee and J. A. Primbs , booktitle=. Constrained stochastic

    work page

  31. [31]

    Asia-Pacific Financial Markets , volume=

    A stochastic receding horizon control approach to constrained index tracking , author=. Asia-Pacific Financial Markets , volume=. 2008 , publisher=

    work page 2008

  32. [32]

    International Journal of Control , volume=

    Covariance control theory , author=. International Journal of Control , volume=. 1987 , publisher=

    work page 1987

  33. [33]

    24th IEEE Conference on Decision and Control , pages=

    Covariance control discrete systems , author=. 24th IEEE Conference on Decision and Control , pages=. 1985 , NOTE =

    work page 1985

  34. [34]

    IEEE Transactions on Automatic Control , volume=

    An improved covariance assignment theory for discrete systems , author=. IEEE Transactions on Automatic Control , volume=. 2002 , NOTE =

    work page 2002

  35. [35]

    Automatica , volume=

    Minimum-energy covariance controllers , author=. Automatica , volume=. 1997 , publisher=

    work page 1997

  36. [36]

    Automatica , volume=

    Finite-horizon covariance control for discrete-time stochastic linear systems subject to input constraints , author=. Automatica , volume=. 2018 , publisher=

    work page 2018

  37. [37]

    SIAM Journal on Control and Optimization , volume=

    Convex optimization for finite-horizon robust covariance control of linear stochastic systems , author=. SIAM Journal on Control and Optimization , volume=. 2021 , publisher=

    work page 2021

  38. [38]

    IEEE 58th Conference on Decision and Control (CDC) , pages=

    Input hard constrained optimal covariance steering , author=. IEEE 58th Conference on Decision and Control (CDC) , pages=. 2019 , NOTE =

    work page 2019

  39. [39]

    IEEE Transactions on Automatic Control , volume=

    Optimal covariance steering for discrete-time linear stochastic systems , author=. IEEE Transactions on Automatic Control , volume=. 2024 , NOTE =

    work page 2024

  40. [40]

    I. M. Balci and E. Bakolas , journal=. Exact. 2022 , NOTE =

    work page 2022

  41. [41]

    Sial and A

    T. Sial and A. Halder , year=. Fixed Horizon Linear Quadratic Covariance Steering in Continuous Time with. 2510.21944 , archivePrefix=

    work page arXiv

  42. [42]

    Nakashima and S

    H. Nakashima and S. Ganguly and K. Kashima , booktitle=. Data-driven. 2025 , volume=

    work page 2025

  43. [43]

    Chen and T

    Y. Chen and T. T. Georgiou and M. Pavon , journal=. Optimal steering of a linear stochastic system to a final probability distribution, Part. 2015 , NOTE =

    work page 2015

  44. [44]

    Chen and T

    Y. Chen and T. T. Georgiou and M. Pavon , journal=. Optimal Steering of a Linear Stochastic System to a Final Probability Distribution—Part. 2018 , volume=

    work page 2018

  45. [45]

    IEEE Transactions on Automatic Control , volume=

    Optimal covariance steering for continuous-time linear stochastic systems with multiplicative noise , author=. IEEE Transactions on Automatic Control , volume=. 2024 , NOTE =

    work page 2024

  46. [46]

    American Control Conference (ACC) , pages=

    Distributionally robust covariance steering with optimal risk allocation , author=. American Control Conference (ACC) , pages=. 2023 , NOTE =

    work page 2023

  47. [47]

    Das and S

    S. Das and S. Ganguly and A. Aravind and D. Chatterjee , journal=. Data-driven distributionally robust. 2024 , publisher=

    work page 2024

  48. [48]

    A. R. R. Narvaez and E. F. Costa , journal=. Average reachability of continuous-time. 2016 , NOTE =

    work page 2016

  49. [49]

    Annual Review of Financial Economics , volume=

    Regime changes and financial markets , author=. Annual Review of Financial Economics , volume=. 2012 , publisher=

    work page 2012

  50. [50]

    Wang and J.-F

    B.-C. Wang and J.-F. Zhang , journal=. Distributed output feedback control of. 2013 , publisher=

    work page 2013

  51. [51]

    Karatzas and S

    I. Karatzas and S. E. Shreve , TITLE =. 1991 , PAGES =

    work page 1991

  52. [52]

    1976 , PAGES =

    Rudin, Walter , TITLE =. 1976 , PAGES =

    work page 1976

  53. [53]

    Lee and G

    J.-W. Lee and G. E. Dullerud , journal=. Uniform stabilization of discrete-time switched and. 2006 , publisher=

    work page 2006

  54. [54]

    Grimmett and D

    G. Grimmett and D. Stirzaker, David , year=. Probability and

    work page

  55. [55]

    Parmar and D

    C. Parmar and D. Chatterjee , journal=. Robust density steering for uncertain. 2026 , publisher=

    work page 2026

  56. [56]

    J. C. Hull , year=. Options,

    work page