On expansions for nonlinear systems, error estimates and convergence issues

Beauchard, Karine; Borgne, Jérémy Le; Marbach, Frédéric

doi:10.48550/arxiv.2012.15653

Cited by 3 publications

(7 citation statements)

References 49 publications

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“…Then (2.10) and (2.11) follow from the multivariate CBHD formula [1,Proposition 2.34]. Technically, this proposition is stated for finite products.…”

Section: Magnus Formula In Interaction Picturementioning

confidence: 95%

A unified approach of obstructions to small-time local controllability for scalar-input systems

Beauchard¹,

Marbach²

2022

Preprint

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We propose a unified approach to determine and prove obstructions to small-time local controllability of scalar-input control systems. Our approach relies on a recent Magnus-type representation formula of the state, a new Hall basis of the free Lie algebra over two generators and an appropriate use of Sussmann's infinite product to compute the Magnus expansion.First, we recover the necessary conditions, due to Sussmann [19] and Stefani [18], concerning the strongest obstruction at each even order of the control. We also recover our classification of quadratic obstructions of [3], involving the regularity of the control, and Kawski's tight necessary condition of [11] concerning the second quadratic drift.Then, we prove and generalize a conjecture of 1986 due to Kawski [10] on a new family of loose necessary conditions, linked with quadratic drifts. In the particular case of the third quadratic drift, we state and prove a tight necessary condition, which is new.Eventually, as a further illustration of the approach, we derive an entirely new obstruction, linked with a bracket of the sixth order with respect to the control and for which the functional measuring the amplitude of the drift is not directly a Sobolev norm of the control.

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“…Then (2.10) and (2.11) follow from the multivariate CBHD formula [1,Proposition 2.34]. Technically, this proposition is stated for finite products.…”

Section: Magnus Formula In Interaction Picturementioning

confidence: 95%

A unified approach of obstructions to small-time local controllability for scalar-input systems

Beauchard¹,

Marbach²

2022

Preprint

Self Cite

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“…This idea was introduced in [25] and later used in [9,16] for the Schrödinger equation. It was also used in finite dimension in [7] to study the quadratic behavior of differential systems or in [4] to give refined error estimates for various expansions of scalar-input affine control systems. By the Duhamel formula, the solution of the auxiliary system (63) with X(0) = ϕ 1 satisfies…”

Section: Quadratic and Cubic Behaviorsmentioning

confidence: 99%

“…This follows the work of [9] where the authors already denied L ∞ -STLC in the case n = 1. Let us stress that such a result entails that, under (H lin ), (4) and (H quad ), the Schrödinger equation (1) is not H 3 -STLC.…”

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confidence: 99%

Small-time local controllability of the bilinear Schrödinger equation, despite a quadratic obstruction, thanks to a cubic term

Bournissou¹

2022

Preprint

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We consider a 1D linear Schrödinger equation, on a bounded interval, with Dirichlet boundary conditions and bilinear control. We study its controllability around the ground state when the linearized system is not controllable. More precisely, we study to what extent the nonlinear terms of the expansion can recover the directions lost at the first order.In the works [9,16], for any positive integer n, assumptions have been formulated under which the quadratic term induces a drift in the nonlinear dynamics, quantified by the H −n -norm of the control. This drift is an obstruction to the small-time local controllability (STLC) under a smallness assumption on the controls in regular spaces.In this paper, we prove that for controls small in less regular spaces, the cubic term can recover the controllability lost at the linear level, despite the quadratic drift. The proof is inspired by Sussman's method to prove the sufficiency of the S(θ) condition for STLC of ODEs. However, it uses a different global strategy relying on a new concept of tangent vector, better adapted to the infinite-dimensional setting of PDEs. Given a target, we first realize the expected motion along the lost direction by using control variations for which the cubic term dominates the quadratic one. Then, we correct the other components exactly, by using the STLC in projection result of [15], with simultaneous estimates of weak norms of the control. These estimates ensure that the new error along the lost direction is negligible, and we conclude with the Brouwer fixed point theorem.

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“…Our initial motivation, both for symmetric and asymmetric estimates, stems from convergence issues for series of Lie brackets of analytic vector fields (see e.g. the open problem raised in [2, Section 2.4.3] and the conditional result in [2,Section 4.4.3]). From an algebraic point of view, such estimates are linked with the intricacies of the Lie product in a Hall basis.…”

Section: Context and Motivationmentioning

confidence: 99%

“…By item(2) of the second point of Proposition 5.1, the elements of the right-hand side are (evaluations of) distinct elements of B. Thus, [X 0 , ad n X1 (X 2 )] B = n k=0 n k = 2 n .…”

mentioning

confidence: 98%

Growth of structure constants of free Lie algebras relative to Hall bases

Beauchard,

Borgne,

Marbach

2021

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We derive a priori bounds on the size of the structure constants of the free Lie algebra over a set of indeterminates, relative to its Hall bases. We investigate their asymptotic growth, especially as a function of the length of the involved Lie brackets.First, using the classical recursive decomposition algorithm, we obtain a rough upper bound valid for all Hall bases. We then introduce new notions (which we call alphabetic subsets and relative foldings) related to structural properties of the Lie brackets created by the algorithm, which allow us to prove a sharp upper bound for the general case. We also prove that the length of the relative folding provides a strictly decreasing indexation of the recursive rewriting algorithm. Moreover, we derive lower bounds on the structure constants proving that they grow at least geometrically in all Hall bases.Second, for the celebrated historical length-compatible Hall bases and the Lyndon basis, we prove tighter sharp upper bounds, which turn out to be geometric in the length of the brackets.Third, we construct two new Hall bases, illustrating two opposite behaviors in the twoindeterminates case. One is designed so that its structure constants have the minimal growth possible, matching exactly the general lower bound, linked with the Fibonacci sequence. The other one is designed so that its structure constants grow super-geometrically.Eventually, we investigate asymmetric growth bounds which isolate the role of one particular indeterminate. Despite the existence of super-geometric Hall bases, we prove that the asymmetric growth with respect to each fixed indeterminate is uniformly at most geometric in all Hall bases.

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On expansions for nonlinear systems, error estimates and convergence issues

Cited by 3 publications

References 49 publications

A unified approach of obstructions to small-time local controllability for scalar-input systems

A unified approach of obstructions to small-time local controllability for scalar-input systems

Small-time local controllability of the bilinear Schrödinger equation, despite a quadratic obstruction, thanks to a cubic term

Growth of structure constants of free Lie algebras relative to Hall bases

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