Abstract
In this paper, we consider a singular optimal control problem with cost function containing a small parameter ε. Using the boundary layer theory developped by Lions in [1], we give some estimates of the singular optimal control u ε in Sobolev space. On the basis of the interior estimate obtained in [4], we analyse the boundary singularity of u ε. According to the generalized Pohozaev identity[3], we obtain the estimation of \(\left\| {\frac{{\partial u_ \in }}{{\partial v}}} \right\|_{L^2 \left( \Gamma \right)}\).
This research is supported by the National Natural Science Foundation of China and Partially by the Institute of Mathematics (Open), Academia Sinica.
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References
Lions, J. L., Perturbations Singulières dans les Problèmes aux limites et en Contrôle Optimal, Springer, 1973.
Hicks, J. N., Note on differentiable geometry, D. Van Nostrand Company, Inc. Toronto, 1965.
Pohozaev, S.I., Eigenfunction of the equation Δu+λf(u)=0, 3 Soviet Math. Doklady 6, 1965, 1408–1411 (translated from the Russian Dokl. Akd. Nauk USSR 165, 1965, 33–36).
Zhang Weitao, Analysis of boundary layer singularity, J. Sys. Sci. & Math. Scis., 4(2), 1984, 81–96.
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© 1991 International Federation for Information Processing
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Zhang, WT., Feng, DX. (1991). Analysis of the boundary singularity of a singular optimal control problem. In: Li, X., Yong, J. (eds) Control Theory of Distributed Parameter Systems and Applications. Lecture Notes in Control and Information Sciences, vol 159. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0004453
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DOI: https://doi.org/10.1007/BFb0004453
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