Multiplicative controls and controllability of linear and semilinear PDE's.

Prof. Alexander Khapalov

In the modeling of controlled distributed parameter systems the boundary  and internal locally distributed controls are typically used. These controls enter the model equations as additive terms and, in the context of applications, describe the effect of externally added ``alien'' sources or forces. Accordingly, additive controls are not suitable for the study of a vast array of applications associated with the processes that can change their principal characteristics due to the control actions (e.g., the ``smart materials''  or chain reaction-like processes in biomedicine, nuclear or chemical engineering).

  This course is dedicated to the study of (global) controllability properties of linear and semilinear  pde's by means of multiplicative (or bilinear) controls, which enter the system equations as  coefficients.  These controls can change the principal parameters of the process at hand, such as, for example, the natural frequency response of a string (or a beam) or the rate of a chemical reaction. In the former case this can be achieved, for example,  by making use of embedded ``smart’’ alloys and in the latter case by a ``catalyst''.

  The study of controllability of pde's governed by multiplicative controls requires different methods (compared to the case of additive controls). Indeed, even if the controlled system at hand is linear, its solution depends on a coefficient (a ``multiplicative control'') in a highly nonlinear way. This course focuses on a new controllability methodology developed in recent years for linear and semilinear parabolic and hyperbolic pde's governed by multiplicative controls (e.g., associated with a variable reaction rate or a variable load and/or damping gain).

  Another class of problems we intend to discuss  are the models of objects (either of mechanical nature, e.g., a robotic fish,  or of living organisms) that can ``swim" in a fluid (as opposed to a body that is being  pushed/pulled by external forces). Such models deal with coupled nonlinear systems which include the fluid equations and the equations describing the position of swimming object in the fluid. The swimming motion is the result of the action of internal forces generated by this object, changing, in particular, its geometric shape, against the surrounding fluid. The magnitudes of these forces are modeled as coefficients (i.e., multiplicative controls). Models like this are of interest in biology and in engineering applications dealing with propulsion systems in fluids.