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'''PDE-constrained optimization''' is a subset of [[mathematical optimization]] where at least one of the [[Constrained optimization|constraints]] may be expressed as a [[partial differential equation]].<ref>{{Cite journal|last=|first=|date=2014|editor-last=Leugering|editor-first=Günter|editor2-last=Benner|editor2-first=Peter|editor3-last=Engell|editor3-first=Sebastian|editor4-last=Griewank|editor4-first=Andreas|editor5-last=Harbrecht|editor5-first=Helmut|editor6-last=Hinze|editor6-first=Michael|editor7-last=Rannacher|editor7-first=Rolf|editor8-last=Ulbrich|editor8-first=Stefan|title=Trends in PDE Constrained Optimization|url=https://link.springer.com/book/10.1007/978-3-319-05083-6|journal=International Series of Numerical Mathematics|language=en-gb|publisher=Springer|volume=|pages=165|doi=10.1007/978-3-319-05083-6|isbn=978-3-319-05082-9|issn=0373-3149|via=}}</ref> Typical domains where these problems arise include [[aerodynamics]], [[computational fluid dynamics]], [[image segmentation]], and [[Inverse problem|inverse problems]].<ref>{{Cite book|url=https://epubs.siam.org/doi/abs/10.1137/1.9780898718935|title=Real-Time PDE-Constrained Optimization|date=2007-01-01|publisher=Society for Industrial and Applied Mathematics|isbn=978-0-89871-621-4|editor-last=Lorenz T. Biegler|series=Computational Science & Engineering|doi=10.1137/1.9780898718935|editor-last2=Omar Ghattas|editor-last3=Matthias Heinkenschloss|editor-last4=David Keyes|editor-last5=Bart van Bloemen Waanders}}</ref> A standard formulation of PDE-constrained optimization encountered in a number of disciplines is given by:<ref name=":0">{{Cite web|url=https://www.maths.dundee.ac.uk/aathanassoulis/Pearson_May2018.pdf|title=PDE-Constrained Optimization in Physics, Chemistry & Biology: Modelling and Numerical Methods|last=Pearson|first=John|date=May 16, 2018|website=University of Edinburgh|url-status=live|archive-url=|archive-date=|access-date=}}</ref><math display="block">\min_{y,u} \; \frac 1 2 \|y-\widehat{y}\|_{L_2(\Omega)}^2 + \frac\beta2 \|u\|_{L_2(\Omega)}^2, \quad \text{s.t.} \; \mathcal{D}y = u</math>where <math>u</math> is the control variable and <math>\|\cdot\|_{L_{2}(\Omega)}^{2}</math> is the squared [[Euclidean norm]] and is not a norm itself. Closed-form solutions are generally unavailable for PDE-constrained optimization problems, necessitating the development of [[Numerical methods for partial differential equations|numerical methods]].<ref>{{Cite journal|lastlast1=Biros|firstfirst1=George|last2=Ghattas|first2=Omar|date=2005-01-01|title=Parallel Lagrange--Newton--Krylov--Schur Methods for PDE-Constrained Optimization. Part I: The Krylov--Schur Solver|url=https://epubs.siam.org/doi/abs/10.1137/S106482750241565X|journal=SIAM Journal on Scientific Computing|volume=27|issue=2|pages=687–713|doi=10.1137/S106482750241565X|bibcode=2005SJSC...27..687B |issn=1064-8275}}</ref><ref>{{Cite journal|lastlast1=Antil|firstfirst1=Harbir|last2=Heinkenschloss|first2=Matthias|last3=Hoppe|first3=Ronald H. W.|last4=Sorensen|first4=Danny C.|date=2010-08-01|title=Domain decomposition and model reduction for the numerical solution of PDE constrained optimization problems with localized optimization variables|url=https://doi.org/10.1007/s00791-010-0142-4|journal=Computing and Visualization in Science|language=en|volume=13|issue=6|pages=249–264|doi=10.1007/s00791-010-0142-4|s2cid=9412768|issn=1433-0369|url=https://nbn-resolving.org/urn:nbn:de:bvb:384-opus4-10652}}</ref><ref>{{Cite journal|lastlast1=Schöberl|firstfirst1=Joachim|last2=Zulehner|first2=Walter|date=2007-01-01|title=Symmetric Indefinite Preconditioners for Saddle Point Problems with Applications to PDE-Constrained Optimization Problems|url=https://epubs.siam.org/doi/abs/10.1137/060660977|journal=SIAM Journal on Matrix Analysis and Applications|volume=29|issue=3|pages=752–773|doi=10.1137/060660977|issn=0895-4798}}</ref><br />
== Applications ==
* Aerodynamic shape optimization<ref>{{Cite web|url=http://aero-comlab.stanford.edu/Papers/jameson.vki03.pdf|title=Aerodynamic Shape Optimization Using the Adjoint Method|last=Jameson|first=Antony|date=2003|website=Stanford University|url-status=live|archive-url=|archive-date=|access-date=}}</ref><ref>{{Cite journal|lastlast1=Hazra|firstfirst1=S. B.|last2=Schulz|first2=V.|last3=Brezillon|first3=J.|last4=Gauger|first4=N. R.|date=2005-03-20|title=Aerodynamic shape optimization using simultaneous pseudo-timestepping|url=http://www.sciencedirect.com/science/article/pii/S0021999104004061|journal=Journal of Computational Physics|language=en|volume=204|issue=1|pages=46–64|doi=10.1016/j.jcp.2004.10.007|bibcode=2005JCoPh.204...46H |issn=0021-9991|url-access=subscription}}</ref>
* [[Drug delivery]]<ref>{{Cite journal|lastlast1=Somayaji|firstfirst1=Mahadevabharath R.|last2=Xenos|first2=Michalis|last3=Zhang|first3=Libin|last4=Mekarski|first4=Megan|last5=Linninger|first5=Andreas A.|date=2008-01-01|title=Systematic design of drug delivery therapies|url=http://www.sciencedirect.com/science/article/pii/S0098135407001688|journal=Computers & Chemical Engineering|series=Process Systems Engineering: Contributions on the State-of-the-Art|language=en|volume=32|issue=1|pages=89–98|doi=10.1016/j.compchemeng.2007.06.014|issn=0098-1354|url-access=subscription}}</ref><ref>{{Cite journal|lastlast1=Antil|firstfirst1=Harbir|last2=Nochetto|first2=Ricardo H.|last3=Venegas|first3=Pablo|date=2017-10-19|title=Optimizing the Kelvin force in a moving target subdomain|url=https://www.worldscientific.com/doi/abs/10.1142/S0218202518500033|journal=Mathematical Models and Methods in Applied Sciences|volume=28|issue=011|pages=95–130|doi=10.1142/S0218202518500033|issn=0218-2025|arxiv=1612.07763|s2cid=119604277}}</ref>
* [[Mathematical finance]]<ref>{{Cite journal|lastlast1=Egger|firstfirst1=Herbert|last2=Engl|first2=Heinz W.|date=2005|title=Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates|urljournal=Inverse Problems|volume=21|issue=3|pages=1027–1045|doi=https://iopscience.iop.org/article/10.1088/0266-5611/21/3/014/pdf|journalbibcode=Inverse2005InvPr..21.1027E Problems|volume=21|issue=3|pages=1027-1045|vias2cid=11012681 }}</ref>
* [[Epidemiology]]<ref>{{Cite journal |last1=Mehdaoui |first1=Mohamed |last2=Lacitignola |first2=Deborah |last3=Tilioua |first3=Mouhcine |date=2024 |title=Optimal social distancing through cross-diffusion control for a disease outbreak PDE model |url=https://www.sciencedirect.com/science/article/pii/S1007570424000418 |journal=Communications in Nonlinear Science and Numerical Simulation |volume=131 |pages=107855 |doi=10.1016/j.cnsns.2024.107855 |bibcode=2024CNSNS.13107855M |issn=1007-5704|url-access=subscription }}</ref>
=== Optimal control of bacterial chemotaxis system ===
