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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References


Adams R. Sobolev spaces, Academic Press, New York, 1975.

Buttazzo G., Kogut P.I. "Weak optimal controls in coefficients for linear elliptic problems", Revista Matematica Complutense, 2011; 24: pp. 83-94.

Gorbonos S.O., Kogut P.I. "Variational solutions of an optimal control problem with unbounded coefficient", Visnyk DNU. Series: Mathematical Modelling, DNU, Dnipropetrovsk, 2013; 5(8): pp. 69-83. (in Ukrainian)

Ivanenko V.I., Mel’nik V.S. Variational Methods in Optimal Control Problems for Systems with Distributed Parameters, Naukova Dumka, Kyiv, 1988. (in Russian)

Fannjiang M.A., Papanicolaou G.C. "Diffusion in turbulence", Probab. Theory and Related Fields, 1996; 105: pp. 279-334.

Fursikov A.V. Optimal Control of Distributed Systems. Theory and Applications, AMS, Providence, RI, 2000.

Kogut P.I. "On Approximation of an Optimal Boundary Control Problem for Linear Elliptic Equation with Unbounded Coefficients", Discrete and Continuous Dynamical Systems — Series A, 2014; 34(5): pp. 2105-2133.

Kogut P.I., Leugering G. Optimal Control Problems for Partial Differential Equations on Reticulated Domains: Approximation and Asymptotic Analysis, Birkhauser, Boston, 2011.

Jin T., Mazya V., Schaftinger J. van. "Pathological solutions to elliptic problems in divergence form with continuous coefficients", C. R. Math. Acad. Sci. Paris, 2009; 347(13-14): pp. 773-778.

Salsa S. Partial Differential Equations in Action: From Modelling to Theory, Springer-Verlag, Milan, 2008.

Serrin J. "Pathological solutions of elliptic differential equations", Ann. Scuola Norm. Sup. Pisa, 1964; 3(18): pp. 385-387.

Zhikov V.V. "Diffusion in incompressible random flow", Functional Analysis and Its Applications, 1997; 31(3): pp. 156-166.

Zhikov V.V. "Remarks on the uniqueness of a solution of the Dirichlet problem for second-order elliptic equations with lower-order terms", Functional Analysis and Its Applications, 2004; 38(3): pp. 173-183.

Vazquez J.L., Zuazua E. "The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential", J. of Functional Analysis, 2000; 173: pp. 103-153.




DOI: https://dx.doi.org/10.15421/241405

  

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ISSN (Online): 2664-5009
ISSN (Print): 2664-4991
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