Discrete & Continuous Dynamical Systems
January 1995 , Volume 1 , Issue 1
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We investigate the topological and dynamical structure of internally chain recurrent sets for surface flows having particularly simple limit sets, including planar flows with finitely many equilibria. We verify a conjecture of Thieme (1992) concerning the limit sets of planar asymptotically autonomous equations.
The problems of approximate, and exact, controllability of the transient behavior of a system of interconnected, two-dimensional elastic membranes in three dimensional space are considered. The membranes may have differing material properties. Control inputs and outputs are assumed to be restricted to the outer edges of the network and to the junction regions where two or more membranes are joined. The object is to characterize those membrane configurations which are approximately, or exactly, controllable. A class of membrane configurations which may be approximately controlled from the outer edges alone is identified. In particular, any two-membrane network may be approximately controlled from an arbitrarily small open subset of the outer boundary of one of the membranes. It is further proved that under some restrictions on the geometries of the individual membranes and the overall configuration, exactly controllability may be achieved through the action of controls along both the outer boundaries and in the junction regions of the network.
An epidemic model is analyzed which allows for the spatial spread of individuals within a geographical region and the incubation of the disease within infected individuals. The spatial spread of the disease is modelled by diffusion processes. The incubation period of infectives is modelled by infection-age structure. Results are established which provide qualitative prediction of the development of the epidemic in terms of spatially dependent and age dependent parameters.
A two-scale microstructure model of current flow in a medium with continuously distributed capacitance is extended to include nonlinearities in the conductance across the interface between the local capacitors and the global conducting medium. The resulting degenerate system of partial differential equations is shown to be in the form of a semilinear parabolic evolution equation in Hilbert space. It is shown directly that such an equation is equivalent to a subgradient flow and, hence, displays the appropriate parabolic regularizing effects. Various limiting cases are identified and the corresponding convergence results obtained by letting selected parameters tend to infinity.
We prove a version of Pontryagin's maximum principle for linear infinite dimensional control systems (including point target conditions and state constraints). This result covers some examples for which no nonlinear theory is available at present.
The existence of periodic solutions for some planar systems is investigated. Applications are given to positive solutions for a class of Kolmogorov systems generalizing a predator - prey model for the dynamics of two species in a periodic environment.
A time domain feedback control methodology for reducing sound pressure levels in a nonlinear 2-D structural acoustics application is presented. The interior noise in this problem is generated through vibrations of one wall of the cavity (in this case a beam), and control is implemented through the excitation of piezoceramic patches which are bonded to the beam. These patches are mounted in pairs and are wired so as to create pure bending moments which directly affect the manner in which the structure vibrates. Th application of control in this manner leads to an unbounded control input term and the implications of this are discussed. The coupling between the beam vibrations and the interior acoustic response is inherently nonlinear, and this is addressed when developing a control scheme for the problem. Gains for the problem are calculated using a periodic LQR theory and are then fed back into the nonlinear system with results being demonstrated by a set of numerical examples. In particular, these examples demonstrate the viability of the method in cases involving excitation involving a large number of frequencies through both spatially uniform and nonuniform exterior forces.
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