The basic attractor of the viscous Moore-Greitzer equation

The operation of a jet engine is described and a number of current research areas outlined. An equation for the three dimensional air flow through jet engine compression systems is derived. This equation is reduced to a parabolic nonlinear nonlocal equation on the unit circle coupled with two ordinary differential equations. It is proven that this evolution equation, called the viscous Moore Greitzer (vMG) equation, has a unique solution.; The existence of a global attractor for the vMG equation is proven and a bound for the Hausdorff and fractal dimensions of the attractor established. The basic attractor is defined and it is shown that the basic attractor of the vMG equation consists of design flow, surge and stall. Explicit solutions for stall are found and their stability analyzed analytically as well as numerically. The global attractor is constructed for a certain parameter range.; A low order model which captures the flow on the basic attractor is constructed and simulations of this model compared with simulations of the vMG equation. Numerical results indicate that this model captures quantitatively as well as qualitatively the essential dynamics of the PDE model. Some extensions of this low order model are discussed.; Attractor controllability and basic controllability are defined. It is confirmed that in the linearized viscous Moore-Greitzer model stall disturbances are uncontrollable by throttle control. Basic control is defined and a basic throttle control is constructed for the vMG equation. The performance of this control is compared numerically to a backstepping control construction by A. Banaszuk, the author and I. Mezic. These numerical results indicate clearly that the basic control, by utilizing the asymptotic dynamics of the equation, requires much less control effort. It is shown that the vMG equation is not basically controllable with throttle control, but if one in addition has nonaxisymmetric control, then it is basically controllable.