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  "body": "\n## Summary\n\nThe costate equation is the differential equation satisfied by the multipliers, or costates, of an optimal control problem along an optimal trajectory.\nIt reads $p' = -\\partial H / \\partial x$, where $H$ is the [[Hamiltonian of the problem|RACVWG5QTK]].\nTogether with the state equation $x' = \\partial H / \\partial p$ it forms a Hamiltonian system in [[Pontryagin's maximum principle|7RPV2T29CF]].\nIt is affine in $p$, and its linear part is the adjoint of the linearized state equation.\n\n## Explanations\n\n### L1 human\n\nThe costate $p(t)$ tells how much the optimal remaining cost would change if the state at time $t$ were nudged.\nIt is like a price that changes with time.\nThe costate equation says how this price evolves, and it is usually solved backward from the final time.\nFor example, if nothing in the cost or dynamics depends on the state, the price stays constant.\n",
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