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  "body": "\n## Definition\n\nLet $Y$, $Z$ and $V$ be real [[Banach spaces|06GZ956657]], called the state space, the target space of the equation, and the control space.\nLet $e \\colon Y \\times V \\to Z$ be a map that encodes a [[partial differential equation|GC5BDZ6QBC]] with its boundary conditions.\nThus $e(y, u) = 0$ means that $y$ solves the equation with control $u$.\nLet $V_{\\mathrm{ad}} \\subseteq V$ be a nonempty set of admissible controls, and let $J \\colon Y \\times V \\to \\R$ be a cost functional.\nAn *optimal control problem for a partial differential equation* is the problem\n$$\\text{minimize } J(y, u) \\quad \\text{subject to } e(y, u) = 0, \\; u \\in V_{\\mathrm{ad}}.$$\nA pair $(y^*, u^*)$ with $e(y^*, u^*) = 0$ and $u^* \\in V_{\\mathrm{ad}}$ is *optimal* if $J(y^*, u^*) \\le J(y, u)$ for all such pairs $(y, u)$.\n\n## Remarks\n\nExample: let $\\Omega \\subseteq \\R^d$ be a bounded open set, let $y_d \\in L^2(\\Omega)$, $\\alpha \u003e 0$ and $a \u003c b$.\nMinimize $\\frac12 \\norm{y - y_d}_{L^2(\\Omega)}^2 + \\frac{\\alpha}{2} \\norm{u}_{L^2(\\Omega)}^2$ over $u \\in L^2(\\Omega)$ with $a \\le u \\le b$ almost everywhere, where $y \\in H^1_0(\\Omega)$ is the weak solution of $-\\Delta y = u$ in $\\Omega$, $y = 0$ on $\\partial \\Omega$.\nThis fits the definition with $Y = H^1_0(\\Omega)$, $V = L^2(\\Omega)$, $Z = H^{-1}(\\Omega)$ and $e(y, u) = -\\Delta y - u$.\nHere the problem has a unique solution, because the control-to-state map is linear and bounded.\nSo the reduced cost is strictly [[convex|02AS45T3H9]], continuous and coercive on the closed convex set $V_{\\mathrm{ad}}$.\nNon-example: if the state equation is an ordinary differential equation in $\\R^n$, the problem is an ordinary [[optimal control|T3V6ZNAMDN]] problem, not one for a partial differential equation.\nWhen $e(y, u) = 0$ has a unique solution $y = S(u)$ for each $u$, the problem reduces to minimizing $f(u) = J(S(u), u)$ over $V_{\\mathrm{ad}}$.\n",
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