Invariance of the Generalized Steady-State for 1-D Distributed Parameter Systems

摘要

For large-scale distributed parameter systems the generalized steady-state is often the main part of the system state. Computer simulation of large-scale systems requires partitioning of the definition region into (many) smaller regions. Then applicability of the concept generalized steady-state requires that the generalized steady-stae is invariant under spatial system decomposition as obtained by this partitioning of the definition region. In this paper this invariance is proven for some linear 1-D distributed parameter systems. Spatial decomposition invariance is firstly considered for 1-D diffusion systems. The boundary conditions of the diffusion system are conceived to be corner input signals forcing a uniquely defined 2-corner generalized steady-state onto the diffusion systems. It is proven that this 2-corner generalized steady-state has the property to be invariant under spatial system decomposition. Spatial decomposition invariance of the generalized steady-state is further considered for two other 1-D distributed parameter systems of second order in space: a diffusion system with uniformly distributed damping and a convection-diffusion system. It is pointed out for the convection-diffusion system that the proof used for the diffusion system (without damping) remains valid-because of the structure of this proof- for other 1-D distributed parameter systems.