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  "body": "\n## Summary\n\nThe Riccati differential equation of optimal control is a quadratic matrix differential equation for a symmetric matrix function $P(t)$.\nIt is solved backward in time from a terminal condition.\nIts solution gives the optimal feedback law and the optimal cost of the [[linear-quadratic regulator|HR720TJZZR]].\nIt is the matrix analogue of the scalar [[Riccati equation|0VT2QK6X6Q]], and its steady states solve the [[algebraic Riccati equation|GGZV75ZZMV]].\n\n## Explanations\n\n### L1 human\n\nFor a linear system with a quadratic cost, the optimal remaining cost from state $x$ at time $t$ is a quadratic form $x^T P(t) x$.\nThe matrix $P(t)$ answers the question \"how expensive is it to be at $x$ at time $t$?\".\nThe Riccati equation tells how this matrix changes as one moves backward from the final time.\nOnce $P(t)$ is known, the best control is a simple linear function of the current state.\n",
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