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  "body": "\n## Summary\n\nOptimal control of partial differential equations studies how to choose a control, such as a heat source or a boundary temperature.\nThe aim is that the solution of a [[partial differential equation|GC5BDZ6QBC]] is as good as possible for a given cost.\nThe state lives in an infinite-dimensional function space, such as a [[Sobolev space|M4CEZ5K9HP]].\nIt is the infinite-dimensional branch of [[optimal control|T3V6ZNAMDN]] and is also called PDE-constrained optimization.\nIts main tools are existence theory in function spaces, optimality systems with adjoint equations, and numerical methods such as the [[adjoint state method|YD8SA6WS01]].\n\n## Explanations\n\n### L1 human\n\nThink of a metal plate heated by adjustable burners underneath.\nYou want the plate to reach a given temperature profile, while spending little energy on the burners.\nThe temperature obeys the heat equation, a partial differential equation, and the burner settings are the control.\nChoosing the best burner settings is an optimal control problem for a partial differential equation.\n",
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