One numerical approach to optimal control the linear heat conduction processes
Abstract
It is proposed the generalized mathematical formulation of the problem about the optimal control for the heat conduction processes representing by the partial differential equation. The proposed formulation not includes the necessary clarifications about the conditions which must be satisfied by the current and required temperature fields. But, during the generalized solving of the formulated problem, it is established that the current and required temperature fields must be agreed with the mathematical model of the heat conduction so that to have possibilities to provide uniquely these temperature fields by means the control vector. To solve the problem about the optimal control for the heat conduction processes it is developed the numerical approaches based on reducing to the especially built ordinary differential equations and
minimization problem. This reducing is based on discretisation the heat conduction by using the grid method and on defining the unknown control vector as the numerical solution of the especially built Cauchy problem. To satisfy the all limitations it is proposed to build the permissible velocity of the unknown control vector considering with the requirements of necessary switching in some moments of the time. The particular example of using the proposed generalized approaches is considered to illustrate their application technique. It is shown that the proposed generalized mathematical formulation is fully corresponded with the considered particular example. In this considered particular example, the resolving Cauchy problem can be built and the switching time can be found in the depending on the grid node choosing. It is shown that the transient time can be decrease almost twice due to optimizing the control in the particular example at least. All these results will allow giving the clear representation of the proposed approaches and the technique of their using to solve the engineering problems about the optimal control of the heat conduction processes in different industrial systems.
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