A dual control problem and application to marketing |
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| Institution: | 1. Academy of Chinese Energy Strategy, China University of Petroleum, Beijing, PR China;2. Guanghua School of Management, Peking University, No. 5 Yiheyuan Road Haidian District, Beijing, PR China;3. Lee Business School, University of Nevada Las Vegas, 4505 Maryland Parkway, Las Vegas, NV 89154-6009, United States;4. College of Business Administration and Public Policy, California State University Dominguez Hills, 1000 East Victoria St., Carson, CA 90747, United States;1. State Key Laboratory of Supramolecular Structure and Materials, Jilin University, Changchun 130012, PR China;2. School of Chemistry, Dalian University of Technology, Dalian 116023, PR China;3. School of Science and Technology, Kwansei Gakuin Universty, 2-1 Gakuen, Sanda, Hyogo 660-1337, Japan;1. Key Laboratory of Functional Organometallic Materials of College of Hunan Province, College of Chemistry and Materials Science, Hengyang Normal University, Hengyang, 421008, PR China;2. Key Laboratory of Chemical Biology and Traditional Chinese Medicine Research (Ministry of Education), College of Chemistry and Chemical Engineering, Hunan Normal University, Changsha, 410081, PR China;1. Chair of Macromolecular Chemistry, Institute of Chemistry, Division of Technical and Macromolecular Chemistry, Faculty of Natural Science II (Chemistry, Physics and Mathematics), Martin-Luther-University Halle-Wittenberg, Von-Danckelmann-Platz 4, Halle D-06120, Germany;2. Faculty of Physics, Dynamics of Condensed Systems, University of Vienna, Strudlhofgasse 4, Vienna 1090, Austria;1. Department of Fine Chemistry, Seoul National University of Science and Technology, Seoul 139-743, Republic of Korea;2. Department of Interdisciplinary Bio IT Materials, Seoul National University of Science and Technology, Seoul 139-743, Republic of Korea;3. Department of Chemical and Biomolecular Engineering, Seoul National University of Science and Technology, Seoul 139-743, Republic of Korea;4. Convergence Program of Biomedical Engineering and Biomaterials, Seoul National University of Science and Technology, Seoul 139-743, Republic of Korea |
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| Abstract: | The general approach to adaptive and dual control is to formulate an optimal stochastic control problem. However, for such an approach only mathematical representations of the solution are available which allow little insight into the structure of the optimal controller. Here, an alternative deterministic approach is presented based upon determining a control in which a disturbance attenuation function remains bounded for all allowable (L2 functions) disturbances. The disturbance attenuation function is composed of the ratio of an L2 function of the desired outputs over an L2 function of the disturbance inputs. This disturbance attenuation problem is converted to a differential game. For this game, the optimal control law, in a closed-form, is obtained by performing a minmax operation with respect to a quadratic cost function subjected to a bilinear system. The resulting controller is time-varying and depends nonlinearly on the state and the parameter estimates vector, and on an associated Riccati-type matrix. We provide insights into the structure of the resulting dual controller and illustrate the method by two examples. One of the examples is an application to marketing, to set promotional spending of a company, considering that the effect of promotional effort on sales is unknown. |
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