Research on Optimal Control for Electron Beam Processing Systems and Its Application to Industry

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In many fields of industry, the processing of materials, specially, the cutting and welding of metals, is very important. Recently, a technique to cut and weld metals has made a considerable progress and the processing technique by the high power electron beam has been developed. In this research, the mathematical formulation of the high power beam processing system has been studied. Since the evaporation boundary of the material constructs a free boundary, the electron beam processing system lead us to the free boundary problem. As a typical example of the free boundary problem, a Stefan problem is often cited, however, the free boundary problem in the electron beam processing system is different from the Stefan problem. Because the condition on the free boundary contains the power term corresponding to the power density of the electron beam. One of the powerful analytical method to solve the free boundary problem is variational inequality. Then, the electron beam processing system was formulated by the variational inequality. Taking the influence of impurities contained in the material into consideration, the model is formulated by the stochastic variational inequality.To obtain the information of the state of the material under processing is very important in the electron beam processing. In this research, we have constructed the estimator system of the state under the processing. Results obtained in this research are as follows :(1) The mathematical model of the high power beam processing system was formulated by the stochastic variational inequality of a new type. And the mathematical security of the proposed model was guaranteed.(2) We construct the boundary observation mechanism to perform the estimation of the state of the material under processing.(3) We had proposed the method to estimatie the state of the material under processing. And the efficiency of the estimator system was guaranteed through the simulation experiments.