scholarly journals Weak limits of powers of Chacon’s automorphism

2013 ◽  
Vol 35 (1) ◽  
pp. 128-141 ◽  
Author(s):  
É. JANVRESSE ◽  
A. A. PRIKHOD’KO ◽  
T. DE LA RUE ◽  
V. V. RYZHIKOV
Keyword(s):  
Weak Mixing ◽  
Koopman Operator ◽  
Weak Closure ◽  
Weak Limits

AbstractWe completely describe the weak closure of the powers of the Koopman operator associated with Chacon’s classical automorphism. We show that weak limits of these powers are the ortho-projector to constants and an explicit family of polynomials. As a consequence, we answer negatively the question of$\alpha $-weak mixing for Chacon’s automorphism.

scholarly journals Linearization in the Large of the Anaerobic Digestion Process Using a Reduced-Order Koopman Operator

2020 ◽  
Vol 53 (2) ◽  
pp. 16840-16845
Author(s):  
Camilo Garcia-Tenorio ◽  
Mihaela Sbarciog ◽  
Eduardo Mojica-Nava ◽  
Alain Vande Wouwer
Keyword(s):  
Anaerobic Digestion ◽  
Koopman Operator ◽  
Reduced Order ◽  
Anaerobic Digestion Process ◽  
Digestion Process

scholarly journals Learning Koopman Operator under Dissipativity Constraints

2020 ◽  
Vol 53 (2) ◽  
pp. 1169-1174
Author(s):  
Keita Hara ◽  
Masaki Inoue ◽  
Noboru Sebe
Keyword(s):  
Koopman Operator

scholarly journals Gain-Preserving Data-Driven Approximation of the Koopman Operator and Its Application in Robust Controller Design

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 949
Author(s):  
Keita Hara ◽  
Masaki Inoue
Keyword(s):  
Controller Design ◽  
Internal Model Control ◽  
Feedback Controller ◽  
Data Driven ◽  
Koopman Operator ◽  
Plant System ◽  
Dimensional Approximation ◽  
Model Control ◽  
Data Driven Modeling ◽  
L2 Gain

In this paper, we address the data-driven modeling of a nonlinear dynamical system while incorporating a priori information. The nonlinear system is described using the Koopman operator, which is a linear operator defined on a lifted infinite-dimensional state-space. Assuming that the L2 gain of the system is known, the data-driven finite-dimensional approximation of the operator while preserving information about the gain, namely L2 gain-preserving data-driven modeling, is formulated. Then, its computationally efficient solution method is presented. An application of the modeling method to feedback controller design is also presented. Aiming for robust stabilization using data-driven control under a poor training dataset, we address the following two modeling problems: (1) Forward modeling: the data-driven modeling is applied to the operating data of a plant system to derive the plant model; (2) Backward modeling: L2 gain-preserving data-driven modeling is applied to the same data to derive an inverse model of the plant system. Then, a feedback controller composed of the plant and inverse models is created based on internal model control, and it robustly stabilizes the plant system. A design demonstration of the data-driven controller is provided using a numerical experiment.

New proofs that weak mixing is generic

1976 ◽  
Vol 32 (3) ◽  
pp. 263-278 ◽  
Author(s):  
Steven Alpern
Keyword(s):  
Weak Mixing

Analyzing Effectiveness of Gang Interventions using Koopman Operator Theory

2020 ◽  
Author(s):  
Sian Wen ◽  
Andy Chen ◽  
Tanishq Bhatia ◽  
Nicholas Liskij ◽  
David Hyde ◽  
...  
Keyword(s):  
Operator Theory ◽  
Koopman Operator ◽  
Gang Interventions

scholarly journals Rigidity of Branching Microstructures in Shape Memory Alloys

2021 ◽  
Author(s):  
Theresa M. Simon
Keyword(s):  
Shape Memory Alloy ◽  
Shape Memory Alloys ◽  
Shape Memory ◽  
Differential Inclusion ◽  
Linearized Strain ◽  
Weak Limits ◽  
All Solutions ◽  
Rank One ◽  
Branched Structures ◽  
Habit Planes

AbstractWe analyze generic sequences for which the geometrically linear energy $$\begin{aligned} E_\eta (u,\chi )\,{:}{=} \,\eta ^{-\frac{2}{3}}\int _{B_{1}\left( 0\right) } \left| e(u)- \sum _{i=1}^3 \chi _ie_i\right| ^2 \, \mathrm {d}x+\eta ^\frac{1}{3} \sum _{i=1}^3 |D\chi _i|({B_{1}\left( 0\right) }) \end{aligned}$$ E η ( u , χ ) : = η - 2 3 ∫ B 1 0 e ( u ) - ∑ i = 1 3 χ i e i 2 d x + η 1 3 ∑ i = 1 3 | D χ i | ( B 1 0 ) remains bounded in the limit $$\eta \rightarrow 0$$ η → 0 . Here $$ e(u) \,{:}{=}\,1/2(Du + Du^T)$$ e ( u ) : = 1 / 2 ( D u + D u T ) is the (linearized) strain of the displacement u, the strains $$e_i$$ e i correspond to the martensite strains of a shape memory alloy undergoing cubic-to-tetragonal transformations and the partition into phases is given by $$\chi _i:{B_{1}\left( 0\right) } \rightarrow \{0,1\}$$ χ i : B 1 0 → { 0 , 1 } . In this regime it is known that in addition to simple laminates, branched structures are also possible, which if austenite was present would enable the alloy to form habit planes. In an ansatz-free manner we prove that the alignment of macroscopic interfaces between martensite twins is as predicted by well-known rank-one conditions. Our proof proceeds via the non-convex, non-discrete-valued differential inclusion $$\begin{aligned} e(u) \in \bigcup _{1\le i\ne j\le 3} {\text {conv}} \{e_i,e_j\}, \end{aligned}$$ e ( u ) ∈ ⋃ 1 ≤ i ≠ j ≤ 3 conv { e i , e j } , satisfied by the weak limits of bounded energy sequences and of which we classify all solutions. In particular, there exist no convex integration solutions of the inclusion with complicated geometric structures.

Is there large weak mixing in heavy nuclei?

1994 ◽  
Vol 49 (6) ◽  
pp. 3297-3300 ◽  
Author(s):  
J. J. Szymanski ◽  
J. D. Bowman ◽  
M. Leuschner ◽  
B. A. Brown ◽  
I. C. Girit
Keyword(s):  
Heavy Nuclei ◽  
Weak Mixing
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