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Free Final Time Adaptive Mesh Covariance Steering via Sequential Convex Programming

Joshua Pilipovsky

Abstract

In this paper we develop a sequential convex programming (SCP) framework for free-final-time covariance steering of nonlinear stochastic differential equations (SDEs) subject to both additive and multiplicative diffusion. We cast the free-final-time objective through a time-normalization and introduce per-interval time-dilation variables that induce an adaptive discretization mesh, enabling the simultaneous optimization of the control policy and the temporal grid. A central difficulty is that, under multiplicative noise, accurate covariance propagation within SCP requires retaining the first-order diffusion linearization and its coupling with time dilation. We therefore derive the exact local linear stochastic model (preserving the multiplicative structure) and introduce a tractable discretization that maintains the associated diffusion terms, after which each SCP subproblem is solved via conic/semidefinite covariance-steering relaxations with terminal moment constraints and state/control chance constraints. Numerical experiments on a nonlinear double-integrator with drag and velocity-dependent diffusion validate free-final-time minimization through adaptive time allocation and improved covariance accuracy relative to frozen-diffusion linearizations.

Free Final Time Adaptive Mesh Covariance Steering via Sequential Convex Programming

Abstract

In this paper we develop a sequential convex programming (SCP) framework for free-final-time covariance steering of nonlinear stochastic differential equations (SDEs) subject to both additive and multiplicative diffusion. We cast the free-final-time objective through a time-normalization and introduce per-interval time-dilation variables that induce an adaptive discretization mesh, enabling the simultaneous optimization of the control policy and the temporal grid. A central difficulty is that, under multiplicative noise, accurate covariance propagation within SCP requires retaining the first-order diffusion linearization and its coupling with time dilation. We therefore derive the exact local linear stochastic model (preserving the multiplicative structure) and introduce a tractable discretization that maintains the associated diffusion terms, after which each SCP subproblem is solved via conic/semidefinite covariance-steering relaxations with terminal moment constraints and state/control chance constraints. Numerical experiments on a nonlinear double-integrator with drag and velocity-dependent diffusion validate free-final-time minimization through adaptive time allocation and improved covariance accuracy relative to frozen-diffusion linearizations.
Paper Structure (27 sections, 6 theorems, 126 equations, 3 figures, 2 tables, 1 algorithm)

This paper contains 27 sections, 6 theorems, 126 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. NOTATION
  3. PROBLEM STATEMENT
  4. PROBLEM REFORMULATION
  5. Time Scaling
  6. Linearization
  7. Discretization
  8. Moment Propagation
  9. Policy parameterization
  10. Chance Constraints
  11. Convex reduction
  12. SEQUENTIAL CONVEX PROGRAMMING
  13. NUMERICAL EXAMPLE
  14. Additive Noise
  15. Multiplicative Noise
  16. ...and 12 more sections

Key Result

Theorem 1

Fix the time step $k\in\{0,\ldots,N-1\}$ and consider the exact mild update $\boldsymbol{x}_{k+1}^{\rm ex}$ in eq:exact-discrete. Let denote the frozen state update, and let $\boldsymbol{x}_{k+1}^{\rm ap}$ denote the projected (fully approximated) state update in eq:approx-discrete-general. Denote the associated errors $\boldsymbol{e}_{k+1}^{(x)}\coloneqq \boldsymbol{x}_{k+1}^{\rm ex}-\boldsymbol

Figures (3)

  • Figure 1: FFT-iCS convergence metrics.
  • Figure 2: Converged FFT-iCS optimal state and control trajectories for $\eta=1$.
  • Figure 3: State $3\sigma$ standard deviation under (solid) linear covariance propagation, and (dashed) empirical nonlinear SDE Monte Carlo.

Theorems & Definitions (17)

  • Remark 1
  • Theorem 1: One-step diffusion discretization error
  • proof
  • Remark 2
  • Theorem 2: One-step moment propagation
  • proof
  • Lemma 1: Orthogonal projection
  • proof
  • Remark 3
  • Lemma 2: Local conditional moment bounds
  • ...and 7 more