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Download file# Distributed parameter theory in optimal control.

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posted on 19.11.2015, 08:55 by M. J. GregsonThe main result of this work is the solution of open loop optimal control problems for counterflow diffusion processes, which occur very widely in chemical and mechanical engineering. In these processes two fluids pass each other moving in opposite directions separated by a membrane which is permeable to heat or a chemical solute. The membrane may also take the form of a liquid-gas interface. Subject to certain simplifying assumptions, the equations describing such processes are 01 (x,t), 02 (x,t) are the temperatures, or concentrations of solute, of the two fluids and u(t), v(t) are time dependent flow rates. k is a transfer coefficient which is assumed constant, and C1, C2 are thermal or solute capacities of the fluids per unit length of tube. h is an equilibrium constant; h = 1 for heat transfer. Possible controls are the inlet temperature or concentration of one stream and the flow rates, while possible objectives are the regulation of the outlet temperature or concentration of the other stream, or the maximisation of heat or solute transfer. Subsidiary results are the optimal control of simpler but related hyperbolic systems. One of these is the restricted counterflow problem in which the controlling stream is assumed to be so massive that it is unaffected by giving up heat or solute to the controlled stream, i.e. the system is described by the equations ; Another is the furnace equation in which u and w are possible controls. Different classes of problem arise according to whether the multiplicative controls u and v are subject to rigid constraints (frequently leading to "bang-bang" controls), or whether they are constants, functions of x and t, or functions of t only. Variational methods based on the maximum principle of A.I. Egorov are employed. Analytic solutions and numerical solutions using finite differences are obtained to the various problems. The simplifying assumptions made are probably too severe for many of the results to be directly applicable to industry. However the qualitative features of the optimal control of these processes are explained, and it is not too difficult to build more complex models.