Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Note on Existence of Solutions to Control Problems of Semilinear
  Partial Differential Equations   In this paper, we study optimal control problems of semilinear elliptic and
parabolic equations. A tracking cost functional, quadratic in the control and
state variables, is considered. No control constraints are imposed. We prove
that the corresponding state equations are well-posed for controls in $L^2$.
However, it is well-known that in the $L^2$ framework the mappings involved in
the control problem are not Frechet differentiable in general, which makes any
analysis of the optimality conditions challenging. Nevertheless, we prove that
every $L^2$ optimal control belongs to $L^\infty$, and consequently standard
optimality conditions are available.

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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-th
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The categories of this paper are: quant-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: math-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-lat
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The categories of this paper are: supr-con
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The categories of this paper are: comp-gas
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The categories of this paper are: econ