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Control theory and splitting methods

Karine Beauchard, Adrien Laurent, Frédéric Marbach

TL;DR

This work builds a formal bridge between numerical splitting methods and control theory for forward-only dynamics, recasting splittings as trajectories of a control system with Dirac impulses. By leveraging free Lie algebras, Lazard elimination, and Magnus/BCH theory, it establishes both constructive and non-constructive results for high-order splittings across real, complex, and mixed coefficient regimes, and clarifies fundamental order barriers via obstructions tied to specific Lie brackets. Key contributions include: (i) arbitrary-order existence results for $(R,R)$ and $(R^+,C)$ splittings; (ii) precise obstructions (notably $W_1$, $W_2$, $M_4$) that limit order in the real-valued setting; (iii) high-order schemes using commutator flows and degeneracies; (iv) complex-coefficient splittings and corresponding controllability insights; and (v) a cohesive algebraic framework linking controllability and splitting theory. The findings deepen theoretical understanding of non-reversible dynamics numerically and guide the design of high-order forward-splitting schemes with principled limitations and capabilities.

Abstract

Our goal is to highlight some of the deep links between numerical splitting methods and control theory. We consider evolution equations of the form $\dot{x} = f_0(x) + f_1(x)$, where $f_0$ encodes a non-reversible dynamic, so that one is interested in schemes only involving forward flows of $f_0$. In this context, a splitting method can be interpreted as a trajectory of the control-affine system $\dot{x}(t)=f_0(x(t))+u(t)f_1(x(t))$, associated with a control~$u$ which is a finite sum of Dirac masses. The general goal is then to find a control such that the flow of $f_0 + u(t) f_1$ is as close as possible to the flow of $f_0+f_1$. Using this interpretation and classical tools from control theory, we revisit well-known results concerning numerical splitting methods, and we prove a handful of new ones, with an emphasis on splittings with additional positivity conditions on the coefficients. First, we show that there exist numerical schemes of any arbitrary order involving only forward flows of $f_0$ if one allows complex coefficients for the flows of $f_1$. Equivalently, for complex-valued controls, we prove that the Lie algebra rank condition is equivalent to the small-time local controllability of a system. Second, for real-valued coefficients, we show that the well-known order restrictions are linked with so-called "bad" Lie brackets from control theory, which are known to yield obstructions to small-time local controllability. We use our recent basis of the free Lie algebra to precisely identify the conditions under which high-order methods exist.

Control theory and splitting methods

TL;DR

This work builds a formal bridge between numerical splitting methods and control theory for forward-only dynamics, recasting splittings as trajectories of a control system with Dirac impulses. By leveraging free Lie algebras, Lazard elimination, and Magnus/BCH theory, it establishes both constructive and non-constructive results for high-order splittings across real, complex, and mixed coefficient regimes, and clarifies fundamental order barriers via obstructions tied to specific Lie brackets. Key contributions include: (i) arbitrary-order existence results for and splittings; (ii) precise obstructions (notably , , ) that limit order in the real-valued setting; (iii) high-order schemes using commutator flows and degeneracies; (iv) complex-coefficient splittings and corresponding controllability insights; and (v) a cohesive algebraic framework linking controllability and splitting theory. The findings deepen theoretical understanding of non-reversible dynamics numerically and guide the design of high-order forward-splitting schemes with principled limitations and capabilities.

Abstract

Our goal is to highlight some of the deep links between numerical splitting methods and control theory. We consider evolution equations of the form , where encodes a non-reversible dynamic, so that one is interested in schemes only involving forward flows of . In this context, a splitting method can be interpreted as a trajectory of the control-affine system , associated with a control~ which is a finite sum of Dirac masses. The general goal is then to find a control such that the flow of is as close as possible to the flow of . Using this interpretation and classical tools from control theory, we revisit well-known results concerning numerical splitting methods, and we prove a handful of new ones, with an emphasis on splittings with additional positivity conditions on the coefficients. First, we show that there exist numerical schemes of any arbitrary order involving only forward flows of if one allows complex coefficients for the flows of . Equivalently, for complex-valued controls, we prove that the Lie algebra rank condition is equivalent to the small-time local controllability of a system. Second, for real-valued coefficients, we show that the well-known order restrictions are linked with so-called "bad" Lie brackets from control theory, which are known to yield obstructions to small-time local controllability. We use our recent basis of the free Lie algebra to precisely identify the conditions under which high-order methods exist.
Paper Structure (54 sections, 40 theorems, 134 equations)

This paper contains 54 sections, 40 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Order theory for splitting methods
  3. Splitting methods
  4. Order theory
  5. Order theory for semigroups
  6. Commutator flows and degeneracies
  7. Link with control theory
  8. Definitions and notations
  9. Formal brackets and evaluated Lie brackets
  10. Splitting methods
  11. Controllability
  12. Main results
  13. Arbitrary order (R,R) splitting methods
  14. Arbitrary order (R+,C) splitting methods
  15. The first obstruction to controllability and $(\mathbb{R}^+,\mathbb{R})$ splitting methods
  16. ...and 39 more sections

Key Result

Theorem 1.9

For every $N\in\mathbb{N}^*$, there exists an $(\mathbb{R},\mathbb{R})$ splitting method of order $N$ with a number of flows at most $2 \dim (\mathcal{L}^N(X))-1$ (which is bounded aboveSee subsec:alg for a definition of $\mathcal{L}^N(X)$. By Witt's formula zbMATH03116668, $\dim \mathcal{L}^N(X) =

Theorems & Definitions (89)

  • Definition 1.1: Formal brackets
  • Definition 1.2
  • Definition 1.3: Lie bracket of vector fields
  • Definition 1.4: Evaluated Lie bracket
  • Definition 1.5: Order of a splitting method with $(\mathbb{A},\mathbb{B})$ coefficients
  • Definition 1.6: Splitting method relative to vector fields
  • Definition 1.7: Splitting method involving commutator flows
  • Definition 1.8: Small-state Small-Time-Local-Controllability
  • Theorem 1.9
  • Theorem 1.10
  • ...and 79 more