Mathematics Colloquium: Multiplicative Controls in Applied PDEs
4:10 pm, Neill Hall 5W
Abstract: In the traditional mathematical control theory for partial differential equations the typical controls act as additive terms-boundary or internal, locally distributed or point ones. In the context of applications, they describe the affect of various external forces. Accordingly, they cannot be used to model the processes which can change their principal physical characteristics due to the control actions, such as, for example, various chain reaction-like processes in biomedicine, the nuclear and chemical engineering, which can change their reaction rate when controlled by means of so-called ``catalysts.'' In addition, from the mathematical viewpoint, the additive controls turned out to be not effective to deal with superlinear terms and multiple solutions (possibly) generated by them. In this talk we discuss the new methodology and results developed in our recent works in the area of controllability of various nonlinear pde's, governed by multiplicative (or bilinear) controls, that is, entering the system equations as coefficients. In particular, we also address the issue of controllability for swimming phenomena.