Bilinear Controllability of Semilinear Partial Differential Equations

Washington State University, Pullman WA

Investigators

Abstract

0204037 Khapalov In the modeling of controlled distributed parameter systems "additive" boundary and internal locally distributed controls are typically used. (Examples of such controls can be a source in a heat/mass-transfer process or a piezoceramic actuator placed on a beam.) In terms of applications it appears that such controls can adequately model only those controlled processes that do not change their principal physical characteristics due to the control actions. They rather describe the effect of various externally added "alien" sources and/or forces on the process at hand. This limitation, however, excludes a vast array of new and not quite new technologies, such as, for example, "smart materials" and numerous biomedical, chemical and nuclear chain reactions, which are able to change their principal parameters under certain purposefully induced conditions ("catalysts"). The intent of this proposal is to address the just-outlined issues in the context of global controllability of semilinear partial differential equations (PDEs) through the introduction and study of multiplicative (or "bilinear") controls. These controls enter the system equations as coefficients. Accordingly they can change at least some of the principal parameters of the process at hand, such as, for example, the natural frequency response of a beam or the rate of a chemical reaction. (In the former case this can be caused, e.g., by the embedded "smart" alloys and, in the latter case, by various catalysts and/or by the speed at which the reaction ingredients are mechanically mixed.) The focus of this proposal is on the development of a new methodology for the study of global controllability of the semilinear reaction-diffusion-convection equation and of the wave and beam equations in the framework of bilinear controls. We are particularly interested in the effect such multiplicative controls may have on the issue of controllability of highly nonlinear PDEs, in which case the classical additive controls often appear to be inadequate.