Finite time stabilization of nonautonomous first-order hyperbolic systems. (English) Zbl 1473.35046
Summary: 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 \(L^2\)-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.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35L50 | Initial-boundary value problems for first-order hyperbolic systems |
| 93D20 | Asymptotic stability in control theory |
| 93D40 | Finite-time stability |
| 37L15 | Stability problems for infinite-dimensional dissipative dynamical systems |
Keywords:
nonautonomous first-order hyperbolic systems; reflection boundary conditions; finite time stabilization; stabilization criteria; robustnessReferences:
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