Journal
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
Volume 51, Issue 3, Pages 2005-2035Publisher
SIAM PUBLICATIONS
DOI: 10.1137/120875739
Keywords
nonlinear hyperbolic systems; boundary conditions; stability; Lyapunov function; backstepping; method of characteristics; integral equation; Goursat problem
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Funding
- ERC advanced grant (CPDENL) of the 7th Research Framework Programme (FP7) [266907]
- Belgian Programme on Interuniversity Attraction Poles [IAP V/22]
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In this work, we consider the problem of boundary stabilization for a quasilinear 2 x 2 system of first-order hyperbolic PDEs. We design a new full-state feedback control law, with actuation on only one end of the domain, which achieves H-2 exponential stability of the closed-loop system. Our proof uses a backstepping transformation to find new variables for which a strict Lyapunov function can be constructed. The kernels of the transformation are found to verify a Goursat-type 4 x 4 system of first-order hyperbolic PDEs, whose well-posedness is shown using the method of characteristics and successive approximations. Once the kernels are computed, the stabilizing feedback law can be explicitly constructed from them.
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