HOMOGENIZATION OF SOME CHEAP CONTROL PROBLEMS

作者：Mahadevan Rajesh^*; Muthukumar T

来源：SIAM Journal on Mathematical Analysis, 2011, 43(5): 2211-2229.

DOI：10.1137/100811581

• homogenization
• optimal control
• Gamma-convergence
• two-scale convergence
• measure data

The study of control problems governed by partial differential equations where the cost of the control is offset by a small parameter epsilon had been initiated by J. L. Lions ["Remarks on 'cheap control,'" in Topics in Numerical Analysis, Academic Press, London, 1973, pp. 203-209], [Some Methods in the Mathematical Analysis of Systems and Their Control, Gordon and Breach, New York, 1981]. These problems were called cheap control problems. He also conceived of situations where the state equation might be singularly perturbed with respect to the same parameter. A different kind of situation which happens quite frequently in applications is that of oscillations in the coefficients of the partial differential equation or the cost at a scale related to this parameter, due to heterogeneities in the domain. In all of these situations, the problem of interest is to obtain the asymptotic behavior of the control problem as epsilon -> 0. This is, in general, a difficult problem. The main difficulty is that the controls converge only very weakly as epsilon -> 0, which, in the context of homogenization (that is, state equations and costs with oscillating coefficients), makes it hard even to guess the asymptotic control problem. Although this problem has been treated previously by Kesavan and J. Saint Jean Paulin ["Low cost control problems," in Trends in Industrial and Applied Mathematics, Kluwer Academic, Dordrecht, The Netherlands, 2002, pp. 251-274] and then by Muthukumar [Asymptotic Behaviour of Some Optimal Control Problems, Ph.D. thesis, The Institute of Mathematical Sciences, University of Madras, Chennai, 2006], Kesavan and Muthukumar [Proc. Indian Acad. Sci. (Math. Sci.), 118 (2008), pp. 133-157], and Muthukumar and Nandakumaran [J. Math. Anal. Appl., 351 (2009), pp. 29-42], only some partial answers could be obtained. In this article, we show how to obtain the complete asymptotic behavior for two broad classes of problems: one involving periodically oscillating coefficients and the other involving controls bounded in the sense of measures.

• 出版日期2011