TY - JOUR

T1 - Finite time stabilization of nonautonomous first-order hyperbolic systems

AU - Kmit, Irina

AU - Lyul’ko, Natalya

N1 - Funding Information: \ast Received by the editors June 5, 2020; accepted for publication (in revised form) June 24, 2021; published electronically September 14, 2021. https://doi.org/10.1137/20M1343610 Funding: The first author was supported by the VolkswagenStiftung project ``From Modeling and Analysis to Approximation."" The second author was supported by the state contract of the Sobolev Institute of Mathematics, project 0314-2019-0012. \dagger Institute of Mathematics, Humboldt University of Berlin, 10117 Berlin, Germany. On leave from the Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine (kmit@mathematik.hu-berlin.de). \ddagger Sobolev Institute of Mathematics, Russian Academy of Sciences, Novosibirsk, Russia 630090, and Novosibirsk State University, Novosibirsk, Russia 630090 (natlyl@mail.ru). Publisher Copyright: © 2021 Society for Industrial and Applied Mathematics

PY - 2021

Y1 - 2021

N2 - We address nonautonomous initial boundary value problems for decoupled linear first-order one-dimensional hyperbolic systems and investigate the phenomenon of finite time stabilization. We establish sufficient and necessary conditions ensuring that solutions stabilize to zero in a finite time for any initial L2-data. In the nonautonomous case we give a combinatorial criterion stating that robust stabilization occurs if and only if the matrix of reflection boundary coefficients corresponds to a directed acyclic graph. An equivalent robust algebraic criterion is that the adjacency matrix of this graph is nilpotent. In the autonomous case we also provide a spectral stabilization criterion, which is nonrobust with respect to perturbations of the coefficients of the hyperbolic system.

AB - We address nonautonomous initial boundary value problems for decoupled linear first-order one-dimensional hyperbolic systems and investigate the phenomenon of finite time stabilization. We establish sufficient and necessary conditions ensuring that solutions stabilize to zero in a finite time for any initial L2-data. In the nonautonomous case we give a combinatorial criterion stating that robust stabilization occurs if and only if the matrix of reflection boundary coefficients corresponds to a directed acyclic graph. An equivalent robust algebraic criterion is that the adjacency matrix of this graph is nilpotent. In the autonomous case we also provide a spectral stabilization criterion, which is nonrobust with respect to perturbations of the coefficients of the hyperbolic system.

KW - Finite time stabilization

KW - Nonautonomous first-order hyperbolic systems

KW - Reflection boundary conditions

KW - Robustness

KW - Stabilization criteria

UR - http://www.scopus.com/inward/record.url?scp=85115223048&partnerID=8YFLogxK

U2 - 10.1137/20M1343610

DO - 10.1137/20M1343610

M3 - Article

AN - SCOPUS:85115223048

VL - 59

SP - 3179

EP - 3202

JO - SIAM Journal on Control and Optimization

JF - SIAM Journal on Control and Optimization

SN - 0363-0129

IS - 5

ER -