Stabilization of a Class of Large-Scale Systems of Linear Hyperbolic PDEs via Continuum Approximation of Exact Backstepping Kernels
arxiv(2024)
摘要
We establish that stabilization of a class of linear, hyperbolic partial
differential equations (PDEs) with a large (nevertheless finite) number of
components, can be achieved via employment of a backstepping-based control law,
which is constructed for stabilization of a continuum version (i.e., as the
number of components tends to infinity) of the PDE system. This is achieved by
proving that the exact backstepping kernels, constructed for stabilization of
the large-scale system, can be approximated (in certain sense such that
exponential stability is preserved) by the backstepping kernels constructed for
stabilization of a continuum version (essentially an infinite ensemble) of the
original PDE system. The proof relies on construction of a convergent sequence
of backstepping kernels that is defined such that each kernel matches the exact
backstepping kernels (derived based on the original, large-scale system), in a
piecewise constant manner with respect to an ensemble variable; while showing
that they satisfy the continuum backstepping kernel equations. We present a
numerical example that reveals that complexity of computation of stabilizing
backstepping kernels may not scale with the number of components of the PDE
state, when the kernels are constructed on the basis of the continuum version,
in contrast to the case in which they are constructed on the basis of the
original, large-scale system. In addition, we formally establish the connection
between the solutions to the large-scale system and its continuum counterpart.
Thus, this approach can be useful for design of computationally tractable,
stabilizing backstepping-based control laws for large-scale PDE systems.
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