On non-variational solutions to optimal boundary control problems for parabolic equations

S.O. Gorbonos; P.I. Kogut

doi:10.15421/241405

On non-variational solutions to optimal boundary control problems for parabolic equations

S.O. Gorbonos (Oles Honchar Dnipropetrovsk National University)
P.I. Kogut (Oles Honchar Dnipropetrovsk National University)

Abstract

We study an optimal boundary control problem (OCP) associated to the linear parabolic equation $$$y_t - \mathrm{div}(\nabla y + A(x) \nabla y) = f$$$. The characteristic feature of this equation is the fact that the matrix $$$A(x) = [a_{ij}(x)]_{i,j=1,...,N}$$$ is skew-symmetric, $$$a_{ij}(x) = -a_{ji}(x)$$$ and belongs to $$$L^2$$$-space (rather than $$$L^{\infty}$$$). We show that under special choice of matrix $$$A$$$ and distribution $$$f$$$, a unique solution to the original OCP inherits a singular character of the original matrix $$$A$$$ and it can not be attainable by the solutions of the similar OCPs with $$$L^{\infty}$$$-approximations of matrix $$$A$$$.

Keywords

parabolic equation; optimal control; variational solution; unbounded coefficients; skew-symmetric matrix

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