Searches for non-resonant and resonant Higgs boson pair production are performed in the $\gamma \gamma WW^*$ channel with the final state of $\gamma \gamma \ell \nu jj$ using 36.1  $\hbox {fb}^{-1}$ of proton–proton collision data recorded at a centre-of-mass energy of $\sqrt{s}=$ 13 TeV by the ATLAS detector at the Large Hadron Collider. No significant deviation from the Standard Model prediction is observed. A 95% confidence-level observed upper limit of 7.7 pb is set on the cross section for non-resonant production, while the expected limit is 5.4 pb. A search for a narrow-width resonance X decaying to a pair of Standard Model Higgs bosons HH is performed with the same set of data, and the observed upper limits on $\sigma (pp\rightarrow X)\times B(X\rightarrow HH)$ range between 40.0 and 6.1 pb for masses of the resonance between 260 and 500 GeV, while the expected limits range between 17.6 and 4.4 pb. When deriving the limits above, the Standard Model branching ratios of the $H{\rightarrow }{\gamma \gamma }$ and $H{\rightarrow }WW^*$ are assumed.

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