分类： 数学 >> 控制和优化 提交时间： 2023-01-27
摘要： This paper deals with the uniform exponential stabilities (UESs) of two hybrid control systems consisting of wave equation and a second-order ordinary differential equation. Linear feedback law and local viscosity, and nonlinear feedback law and interior anti-damping are considered, respectively. Firstly, the hybrid system is reduced to a first order port-Hamiltonian system with dynamical boundary conditions and the resulting systems are then discretized by average central-difference scheme. Secondly, the UES of the discrete system is obtained without prior knowledge on the exponential stability of continuous system. The frequency domain characterization of UES for a family of contractive semigroups and discrete multiplier method are utilized to verify main results, respectively. Finally, the convergence analysis of the numerical approximation scheme is performed by the Trotter-Kato Theorem. Most interestingly, the exponential stability of the continuous system is derived by the convergence of energy and UES and this is a new idea to investigate the exponential stability of some complicate systems. The effectiveness of the numerical approximating scheme is verified by numerical simulation.
分类： 数学 >> 控制和优化 提交时间： 2023-01-27
摘要： In this note, Timoshenko beams with interior damping and boundary damping are studied from the viewpoints of control theory and numerical approximation. Especially, the uniform exponential stabilities of the beams are studied. The meaning of uniform exponential stability in this paper is two-fold: The first one is in the classical sense and also is concisely called exponential stability by many authors; The second one is that the semi-discretization systems, which are derived from an exponentially stable continuous beam by some semi-discretization schemes, are uniformly exponentially stable with respect to the discretized parameter. To investigate uniform exponential stability of continuous and discrete systems, five completely different methods, which are stability theory of port-Hamiltonian system, direct method of Lyapunov functional, perturbation theory of $C_{0}$-semigroup, spectral analysis of unbounded operator and frequency standard of exponential stability for contractive semigroup, are involved. Especially, a new method, which is based on the frequency domain characteristics of uniform exponential stability of $C_{0}$-semigroup of contractions, is established to verify the uniform exponential stability of semi-discretization systems derived from coupled system. The effectiveness of the numerical approximating algorithms is verified by numerical simulations.
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