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  "body": "\n## Definition\n\nLet $n, m \\ge 1$ and $T \u003e 0$.\nLet $A$ be a real $n \\times n$ matrix and $B$ a real $n \\times m$ matrix.\nLet $Q$ and $F$ be [[symmetric|0VR671PHE5]] [[positive semidefinite|01MMVW8CKC]] $n \\times n$ matrices, and let $R$ be a symmetric [[positive definite|0KX66CZMBB]] $m \\times m$ matrix.\nThe *Riccati differential equation* of the associated linear-quadratic problem is the terminal value problem\n$$-P'(t) = A^{T} P(t) + P(t) A - P(t) B R^{-1} B^{T} P(t) + Q, \\qquad P(T) = F,$$\nfor a differentiable function $P \\colon (-\\infty, T] \\to \\R^{n \\times n}$ with symmetric values.\n\n## Remarks\n\nUnder these hypotheses the solution exists on all of $(-\\infty, T]$, and $P(t)$ is symmetric positive semidefinite for every $t$.\nExample: $n = m = 1$, $A = 0$, $B = Q = R = 1$ and $F = 0$ give $-P' = 1 - P^2$, $P(T) = 0$.\nThe solution is $P(t) = \\tanh(T - t)$: indeed $P'(t) = -(1 - \\tanh^2(T - t)) = -(1 - P(t)^2)$ and $P(T) = 0$.\nAs $t \\to -\\infty$, $P(t) \\to 1$, the positive solution of the algebraic equation $1 - P^2 = 0$.\nNon-example: the linear equation $-P' = A^{T} P + P A + Q$ is a Lyapunov differential equation, not a Riccati equation, since it lacks the quadratic term.\nIt gives the cost of the zero control; with $A - BK$ in place of $A$ and $Q + K^{T} R K$ in place of $Q$ it gives the cost of a fixed linear feedback $u = -Kx$.\nThe quadratic term of the Riccati equation accounts for optimizing the control.\nFor $n = 1$ the equation is a scalar [[Riccati equation|0VT2QK6X6Q]] $P' = q_0 + q_1 P + q_2 P^2$ with constant coefficients.\n",
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