OPTIMAL CONTROL OF A BIOREACTOR WITH MODAL SWITCHING

SUZANNE M. LENHART, THOMAS I. SEIDMAN, JIONGMIN YONG
2001 Mathematical Models and Methods in Applied Sciences  
We consider a model for bioremediation of a pollutant by bacteria in a well-stirred bioreactor. A key feature is the inclusion of dormancy for bacteria, which occurs when the critical nutrient level falls below a critical threshold. This feature gives a discrete component to the system due to the change in dynamics (governed by a system of ordinary differential equations between transitions) at switches to/from dormancy. After setting the problem in an appropriate state space, the control is
more » ... , the control is the rate of injection of the critical nutrient and the functional to be minimized is the pollutant level at the final time and the amount of nutrient added. The existence of an optimal control and a discussion of the transitions between dormant and active states are given. † This phrasing suggests that bacteria are directly consuming the pollutant, which, in turn, might suggest that the growth rate of bacteria would depend on the availability of pollutant. While consistently using this language here, we note that in our scenario the bacteria do not actually consume the pollutant directly, but that the consumption of other nutrients by the bacteria leads to the transformation of the pollutant into a less hazardous substance; this type of bioremediation is more properly called "cometabolism." We note, for example, that paraxylene is degraded by metabolism while trichloroxylene is degraded by cometabolism [14] , [17] . t 0 u, implying boundedness. One then easily
doi:10.1142/s0218202501001185 fatcat:kagtnd4yyrfsrofgmafuslluxq