Investigation of liquid product optimal heating problem with a liquid coolant in a jacketed tank

Two problems of optimal control of heating of a liquid product in a heating tank with a liquid coolant have been posed and solved on the Pontryagin maximum principle. In the first task, the problem of the speed of heating the product to a given temperature is solved. In the second task, we are looking for a control in which the product is heated for a given time to a given temperature with minimal coolant consumption. The description of the control object with a flow diagram, assumptions simplifying the compilation of the model, a mathematical model of the control object, formulations of optimal control problems and the solutions obtained are given. In the problem of heating speed, analytical expressions are obtained to determine the temperature of the product, the coolant and the control time. An example of solving the problem of heating speed is given with an illustration of the dynamic characteristics of the object variables and the optimal control found. In the problem of minimizing coolant consumption, the solution is obtained numerically. The problem of finding optimal trajectories of variables and control is formulated as a boundary value problem with missing initial conditions, the search for which is realized using the difference method of solving the boundary value problem. Based on the found initial conditions, the Cauchy problem of an extended system of equations is solved, and the optimal trajectory of changes in product temperature and coolant flow is obtained. An example of solving the problem of minimizing coolant consumption is given with a demonstration of the dynamic characteristics of the variables of the object and the optimal control found.