Optimal control of the process of cultivation in the conditions of infection

The article presents a way of solving of the optimal control problem of antibiotic feeding under condition of infection, consisting in the selection of the optimal control in the field of admissible control, with the aim of achieving a compromise between the losses in production due to the presence of foreign microflora, and the cost of its suppression due to the application of antibiotic. The presence of other microorganisms in the finished product, in particular of the “wild“ ones, considerably impairs the quality indicators of the final product (in particular, it reduces the storage time). In peculiar conditions of production it is possible to improve the quality of target product due to elimination of infection, including, when used antibiotics in the process of cultivation, but due to the lack of efficient algorithms and control systems of their supply the question is still open. We use the system of Lotka-Volterra adapted for microbiological process as a mathematical model adequately describing the situation of competitive interaction of two populations of microorganisms (useful and “wild“ ones) due to the consumption of one resource. The aim is to find a control law U(t) belonging to the field of admissible control. The control that affords minimum to the optimization criterion in accordance with the principle of maximum is defined by the condition of the maximum of Hamilton function and the resulting canonical system of equations. The modified conjugated system of equations in matrix form is obtained. The solution of system of differential-different equations in the analytical form is found using the method of coordinate transformation. As a result an optimal control law is found (with regard to the selected criterion). This is the control law of application of the antibiotic, allowing to control the concentration of foreign microflora in the process of cultivation of microorganisms and accounting for the specific equation describing the speed of growth of “wild“ microorganisms.