PaperlayerView exact revision on arXiv

Source-linked AI summary

Linear Exponential Quadratic Gaussian Covariance Steering

Chiran B. Cherian, Yasemin Isik, Abhishek Halder

arXiv:2609.12463v1math.OCcs.AIcs.LGcs.SYeess.SYstat.ML

TL;DR

The paper studies continuous-time LEQG covariance steering between Gaussian endpoints, replacing the risk-neutral average-cost objective with a risk-sensitive one. It derives a linear-feedback solution structure parameterized by a symmetric matrix, and for matched noise and input channels proves a unique solution locally around the risk-neutral optimum.

Abstract

from arXiv · show

We formulate and analyze the linear exponential quadratic Gaussian (LEQG) covariance steering problem in continuous time over a given deadline (finite time horizon). The solution for this problem can be seen as a risk-sensitive Schr\"{o}dinger bridge between Gaussian endpoints in the linear quadratic setting. Unlike the risk-neutral case, the LEQG covariance steering controller--still a linear state feedback--can no longer be written in closed form. We show that the optimal controller is parameterized by a symmetric matrix solving an algebraic equation that encodes the implicit dependence on the risk-sensitivity parameter. We explain how the structure of this optimal controller significantly generalizes the existing results for the risk-neutral case. Building on these results, for the matched noise and input channel case, we prove the existence-uniqueness of solution for the LEQG covariance steering problem in the neighborhood of the known risk-neutral optimal solution. We give an illustrative numerical example.

I. INTRODUCTION

The paper formulates continuous-time LEQG covariance steering between centered Gaussian endpoints over a finite horizon, using a risk-sensitive objective in place of average cost. It develops the optimal-controller structure, proves a local existence-uniqueness result for matched noise and input channels, and illustrates a numerical procedure.

I. INTRODUCTION

  • The problem designs Markovian feedback to steer a controlled state between centered Gaussian endpoint distributions with specified covariances over a finite time horizon.The formulation uses continuous-time linear dynamics and a fixed finite risk-sensitivity parameter.Source: §I · p. 1
  • The analysis assumes uniformly controllable dynamics, bounded continuous system and cost trajectories, positive definite control-cost matrices, and positive definite endpoint covariances.The drift and control matrices are also bounded and continuous over the horizon.Source: §I · p. 1
  • The LEQG objective replaces the risk-neutral average cost with a logarithmic moment-generating function, and θ determines risk sensitivity.Positive θ is risk-averse, negative θ is risk-seeking, and the limit θ → 0 recovers the risk-neutral objective.Source: §I · p. 1
  • The objective penalizes above-average cost realizations more than below-average ones, motivating conservative control when cost variance is large.For large positive θ, larger costs are penalized more aggressively; the resulting controller may perform worse on average but better in the worst case.Source: §I · p. 1
  • The problem admits a probabilistic interpretation as a risk-sensitive Schrödinger bridge between Gaussian endpoints in the linear-quadratic setting.The paper focuses on optimal-solution structure and computation rather than developing the large-deviation viewpoint.Source: §I · pp. 1–2
  • The paper derives the optimal-solution structure, establishes local existence-uniqueness near the risk-neutral solution, and presents a numerical procedure illustrated on a noisy double integrator.The structural results explain how the risk-sensitive controller departs from risk-neutral results.Source: §I · p. 2

A. Necessary Conditions for Optimality

The necessary conditions yield a linear state-feedback controller and a boundary-value formulation in covariance and costate variables. A change of variables gives a Riccati-Riccati system whose risk-sensitive terms lead to new challenges; for matched channels, the paper establishes a locally unique solution near the risk-neutral one.

A. Necessary Conditions for Optimality

  • The optimal controller is linear state feedback, and the necessary conditions produce an ODE boundary-value problem in the covariance and costate pair.The controller follows from Pontryagin’s minimum principle, while its linearity follows from standard linear-quadratic arguments.Source: A. Necessary Conditions for Optimality · p. 2
  • A change of variables rewrites the one-way coupled Lyapunov-Riccati system as a Riccati-Riccati boundary-value problem.The transformed system is expressed in an indeterminate pair and retains covariance boundary conditions.Source: A. Necessary Conditions for Optimality · p. 2
  • The transformed equations generalize the risk-neutral system, whose highlighted risk-sensitive terms vanish at θ = 0.With matched channels, the boundary-value problem reduces to the one in [7, eq. (2)]; the new terms introduce mathematical challenges absent from the risk-neutral case.Source: A. Necessary Conditions for Optimality · p. 2
  • For matched noise and input channels, the paper establishes existence and uniqueness near the known risk-neutral optimal solution.The local result is parameterized through a symmetric matrix and the risk-sensitivity parameter.Source: A. Necessary Conditions for Optimality · pp. 2–3

B. Parametric Solution Structure

The risk-sensitive controller is parameterized through an implicit algebraic condition on a symmetric matrix, rather than an explicit formula in the risk parameter. Under matched channels, a unique local solution branch continues from the risk-neutral solution.

B. Parametric Solution Structure

  • The Riccati ODE solutions are represented using linear Hamiltonian ODEs, whose state-transition matrices supply the blocks used in the parametrization.The representation requires Xt|θ and X̂t|θ to remain invertible over the horizon.Source: B. Parametric Solution Structure · p. 3
  • For θ ≠ 0, the hatted Hamiltonian matrix depends on the unknown initial symmetric matrix, making its transition blocks depend on both θ and Y0|θ.In the risk-neutral case, the corresponding transition blocks depend on θ but not on Y0|θ.Source: B. Parametric Solution Structure · p. 3
  • The controller is determined by solving an implicit algebraic equation for Y0|θ; no general explicit expression Y0|θ = Y0|θ(θ) is available.A suitable unique solution determines Πt|θ and therefore the optimal control.Source: B. Parametric Solution Structure · p. 3
  • Unlike at θ = 0, the risk-sensitive transition matrices do not match, and the mixed terms no longer satisfy the same symplectic identities.The risk-neutral derivation reduces the condition to a quadratic equation; the risk-sensitive case instead requires handling the implicit algebraic equation.Source: B. Parametric Solution Structure · p. 3
  • Under matched channels, a unique C1 solution branch continues locally from the risk-neutral admissible solution as the risk parameter varies.The branch passes through Y^-_{0|0} and satisfies G(θ, h(θ)) = 0 for θ in an open interval containing zero.Source: B. Parametric Solution Structure · p. 4

III. PROOFS AND AUXILIARY RESULTS

The proof establishes a locally unique, continuously differentiable solution branch through the risk-neutral solution. It does so by showing regularity of the defining map and verifying the required Jacobian nonsingularity.

III. PROOFS AND AUXILIARY RESULTS

  • The nonlinear map G is C1, supported by joint C1 dependence of the Riccati solutions on θ and their initial matrix.The proof uses parameter-dependent ODE regularity to establish this smoothness.Source: §III · p. 4
  • Although (25) generalizes (18), the authors do not use that generalized result in the subsequent proofs.Source: §III · p. 5
  • The proof establishes nonsingularity of the relevant Jacobian by showing that the matrix Ω is positive definite.This makes the determinant condition for nonsingularity hold.Source: §III · p. 5
  • A unique C1 solution branch h exists near θ = 0 and passes through the risk-neutral solution.The branch satisfies G(θ, h(θ)) = 0 on an open interval around zero.Source: §III · p. 5

IV. NUMERICAL EXAMPLE

The numerical example applies the LEQG covariance-steering solution to a noisy double integrator. As risk sensitivity increases, the controlled covariance’s log-determinant decreases and optimal cost increases.

IV. NUMERICAL EXAMPLE

  • The example uses a noisy double integrator with randomly generated endpoint covariances satisfying assumptions A1–A3.Source: §IV · p. 5
  • Newton’s method with backtracking solves for Y0|θ, using the risk-neutral solution as its initial guess and sequential Riccati IVPs.The second IVP uses the trajectory from the first because the equations are one-way coupled.Source: §IV · p. 5
  • For fixed t, log det Σ_t|θ decreases as θ increases, while optimal cost increases as the solution shifts from risk-seeking to risk-averse.Source: Figure 2 · p. 6, §IV · p. 6

V. CONCLUDING REMARKS

The paper introduces LEQG covariance steering and emphasizes how its solution structure generalizes risk-neutral results. It proves local existence-uniqueness, while a global extension remains ongoing work.

V. CONCLUDING REMARKS

  • The paper introduces LEQG covariance steering and describes structural generalizations of risk-neutral results.Source: §V · p. 6
  • Theorem 1 proves local existence-uniqueness of the solution.Source: §V · p. 6
  • A global extension is conjectured but remains ongoing work, requiring a C1 extension of h to finite-interval boundaries.Source: §V · p. 6
Loading 2609.12463v1…