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  "body": "\n## Definition\n\nLet $U \\subseteq \\R^m$ be nonempty, $T \u003e 0$, and let $f \\colon [0, T] \\times \\R^n \\times U \\to \\R^n$ and $L \\colon [0, T] \\times \\R^n \\times U \\to \\R$ be continuous with continuous partial derivatives in $x$.\nLet $H(t, x, u, p, \\lambda_0) = p \\cdot f(t, x, u) - \\lambda_0 L(t, x, u)$ be the [[Hamiltonian|RACVWG5QTK]], and fix $\\lambda_0 \\ge 0$.\nLet $u \\colon [0, T] \\to U$ be a measurable bounded control with trajectory $x$ of the [[control system|ZPDTVZQN5K]] $x' = f(t, x, u)$.\nThe *costate equation* along $(x, u)$ is the linear differential equation\n$$p'(t) = -\\frac{\\partial H}{\\partial x}(t, x(t), u(t), p(t), \\lambda_0) = -\\Big(\\frac{\\partial f}{\\partial x}(t, x(t), u(t))\\Big)^{T} p(t) + \\lambda_0 \\nabla_x L(t, x(t), u(t))$$\nfor an [[absolutely continuous|ZT6F4C2YHK]] function $p \\colon [0, T] \\to \\R^n$, required to hold for almost every $t \\in [0, T]$.\nA solution $p$ is a *costate* or *adjoint* vector along $(x, u)$.\n\n## Remarks\n\nExample: $x' = u$ in $\\R$ with running cost $\\int \\frac12 (x^2 + u^2) \\, dt$ and $\\lambda_0 = 1$.\nThen $H = p u - \\frac12 (x^2 + u^2)$ and $\\partial H / \\partial x = -x$, so the costate equation is $p' = x$.\nNon-example: $p' = +\\partial H / \\partial x$ is not the costate equation, because the sign is wrong.\nWith it, the quantity $p \\cdot \\delta x$ for a solution $\\delta x$ of the linearized equation $\\delta x' = (\\partial f / \\partial x) \\delta x$ would not be constant when $\\lambda_0 = 0$.\nThe costate equation is fixed by a boundary condition, the transversality condition.\nFor a free final state and an endpoint cost $\\varphi(x(T))$ to be minimized, it is $p(T) = -\\lambda_0 \\nabla \\varphi(x(T))$.\n",
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