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 $A$.
Highlights
We consider the optimal boundary control problem for a parabolic equation with unbounded coefficients
The characteristic feature of this problem is the fact that the stream matrix A(x) is skew-symmetric and its coefficients belongs to L2-space
The aim of this work is to consider optimal boundary control problem (OCP) with a well prescribed skew-symmetric L2matrix A and, using the direct method in the Calculus of variations, to show that this problem admits a unique solution possessing a special singular properties. We prove that this solution cannot be attained through a sequence of optimal solutions to regularized OCP for boundary value problem with skew-symmetric matrices Ak ∈ L∞(Ω; S3) such that Ak → A strongly in L2(Ω; S3)
Summary
We consider the optimal boundary control problem for a parabolic equation with unbounded coefficients. The situation can change dramatically for the matrices A with unremovable singularity In such cases, boundary value problem may admit infinitely many weak solutions which can be divided into two classes: approximable and non-approximable solutions [5], [12], and [13]. We prove that this solution cannot be attained through a sequence of optimal solutions to regularized OCP for boundary value problem with skew-symmetric matrices Ak ∈ L∞(Ω; S3) such that Ak → A strongly in L2(Ω; S3) This result shows that a numerical analysis of optimal control problems for parabolic equations with unbounded coefficients is a non-trivial matter and it requires the elaboration of special approaches
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