Temperature Control of PDE Constrained Optimization Problems Governed by Kobayashi--Warren--Carter Type Models of Grain Boundary Motions

In this paper, we consider a class of optimal control problems governed by state-equations of Kobayashi–Warren–Carter type. The control is given by physical temperature. The focus is on problems in dimensions less than equal to 4. The results are divided in four Main Theorems, concerned with: solvability and parameter-dependence of state-equations and optimal control problems; the first order necessary optimality conditions for these regularized optimal control problems. Subsequently, we derive the limiting systems and optimality conditions and study their well-posedness. ∗The work of the third author supported by Grant-in-Aid for Scientific Research (C) No. 16K05224 and No. 20K03672, JSPS. The work of the forth author supported by Grant-in-Aid for Scientific Research (C) No. 20K03665 , JSPS. In addition, the work of the first and the third authors is partially supported by the Air Force Office of Scientific Research (AFOSR) under Award NO: FA9550-19-1-0036 and NSF grants DMS-1818772 and DMS-1913004. AMS Subject Classification: 35K51, 49J20, 49K20, 74N05.

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