The main aim of this paper is to address the global asymptotic stabilization (GAs) of a class of dissipative partial differential equations (PDEs) via finite-dimensional admissible (regular and bounded) damping (outpu...
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The main aim of this paper is to address the global asymptotic stabilization (GAs) of a class of dissipative partial differential equations (PDEs) via finite-dimensional admissible (regular and bounded) damping (output) controls. To this end, we use the inertial manifold theory to derive infinite-dimensional dissipative control systems given by interconnection of a finite-dimensional control system on the inertial manifold plus an infinite-dimensional zero-input system on the complement. We show that the GAS of such systems is reduced to the finite-dimensional one. Then, we prove that the finite-dimensional systems which are zero input point dissipative (have global attractors K) and those which are B-strictly passive (passivity relative to bounded sets) are connected. Finally, we use the control Lyapunov functions (cLF) theory to design admissible feedback damping controls for the GAS of B-strictly passive systems. (C) 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
The main aim of this paper is to address the global asymptotic stabilization (GAS) of a class of dissipative partial differential equations (pdes) via finite-dimensional admissible (regular and bounded) damping (outpu...
详细信息
The aim of this paper is to apply new results on the boundary stabilisation via energy-shaping of distributed port-Hamiltonian systems to a nonlinear PDE, i.e. a slightly simplified formulation of the shallow water eq...
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The aim of this paper is to apply new results on the boundary stabilisation via energy-shaping of distributed port-Hamiltonian systems to a nonlinear PDE, i.e. a slightly simplified formulation of the shallow water equation. Usually, stabilisation of non-zero equilibria via energy-balancing has been achieved by looking at, or generating, a set of structural invariants (Casimir functions), in closed-loop. This approach is not successful in case of the shallow water equation because at the equilibrium the regulator is supposed to supply an infinite amount of energy (dissipation obstacle). In this paper, it is shown how to construct a controller that behaves as a state-modulated boundary source and that asymptotically stabilises the desired equilibrium. The proposed approach relies on a parametrisation of the dynamics provided by the image representation of the Dirac structure associated to the distributed port-Hamiltonian system. In this way, the effects of the boundary inputs on the state evolution are explicitly shown, and as a consequence the boundary control action that maps the open-loop system into a target one characterised by the desired stability properties, i.e. by a “new” Hamiltonian with an isolated minimum at the equilibrium, is determined.
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