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A case study in ensemble optimal control for Bayesian input design

Ludovic Sacchelli, Alessandro Scagliotti

TL;DR

The paper addresses input design for uncertainty reduction in linear, Gaussian Bayesian parameter estimation problems by formulating an optimal control problem that minimizes posterior uncertainty. It contrasts classical designs using a fixed θ with an ensemble approach that averages the cost over the prior, requiring a generalized Pontryagin maximum principle for Gaussian-distributed parameters. A detailed PMP analysis reveals bang-bang and singular arcs in both the classical and ensemble settings, with ensemble controls derived from expectations under the prior. Numerical experiments on a damped oscillator illustrate that the ensemble design better respects state constraints and yields favorable posterior estimates, validating the approach and highlighting robustness to parameter uncertainty. Overall, the work provides a theoretical and computational framework for ensemble-based Bayesian input design, including a tailored PMP for Gaussian linear systems and practical insights for experiment design under uncertainty.

Abstract

We discuss the problem of input design for uncertainty reduction in a parameter estimation procedure. Assuming a linear continuous-time control system with noisy measurements, we formulate an objective of variance reduction in a Bayesian Gaussian setting as an optimal control problem and analyze it from a geometric control perspective. The resulting cost functional depends on the unknown parameter, we compare the optimal control approach with a non-standard alternative inspired by ensemble control, where the cost is averaged over the prior distribution after computation, rather than before. This requires the statement of a generalized Pontryagin's maximum principle adapted to Gaussian distributions.

A case study in ensemble optimal control for Bayesian input design

TL;DR

The paper addresses input design for uncertainty reduction in linear, Gaussian Bayesian parameter estimation problems by formulating an optimal control problem that minimizes posterior uncertainty. It contrasts classical designs using a fixed θ with an ensemble approach that averages the cost over the prior, requiring a generalized Pontryagin maximum principle for Gaussian-distributed parameters. A detailed PMP analysis reveals bang-bang and singular arcs in both the classical and ensemble settings, with ensemble controls derived from expectations under the prior. Numerical experiments on a damped oscillator illustrate that the ensemble design better respects state constraints and yields favorable posterior estimates, validating the approach and highlighting robustness to parameter uncertainty. Overall, the work provides a theoretical and computational framework for ensemble-based Bayesian input design, including a tailored PMP for Gaussian linear systems and practical insights for experiment design under uncertainty.

Abstract

We discuss the problem of input design for uncertainty reduction in a parameter estimation procedure. Assuming a linear continuous-time control system with noisy measurements, we formulate an objective of variance reduction in a Bayesian Gaussian setting as an optimal control problem and analyze it from a geometric control perspective. The resulting cost functional depends on the unknown parameter, we compare the optimal control approach with a non-standard alternative inspired by ensemble control, where the cost is averaged over the prior distribution after computation, rather than before. This requires the statement of a generalized Pontryagin's maximum principle adapted to Gaussian distributions.

Paper Structure

This paper contains 13 sections, 10 theorems, 71 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Input design problem
  3. Setting
  4. The Bayesian input design problem
  5. Optimal control point of view
  6. Classical optimal control
  7. Description of the optimal flow
  8. Discussion of second order optimality conditions
  9. Ensemble control approach
  10. Pontryagin's maximum principle statement
  11. Ensemble case
  12. Numerical experiments
  13. Conclusion

Key Result

Lemma 3.1

Let $(u_k)_k$ be a sequence such that $u_k \to u$ in the $L^\infty$ weak-$*$ topologyRecall that $u_k \to u$ in the $L^\infty$ weak-$*$ topology if we have convergence against $L^1$ test functions.. Then

Figures (1)

  • Figure 1: Left: the computed controls $u_{\hat{\theta}}$ (blue), with $u_{\mu_{N}}$ (red) and $u_{\mu_{\mathrm{prior}}}$ (black). Right: the graph of $t\mapsto |x^{\theta_{\mathrm{true}}}_u(t)|^2$ under each of these policies.

Theorems & Definitions (14)

  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 3.1
  • Proposition 3.2
  • Remark 3.4
  • Lemma 3.5
  • Lemma 4.2
  • Theorem 4.5
  • Proposition 4.6
  • ...and 4 more