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In this paper, we study the maximum principle of mean field type control problems when the volatility function depends on the state and its measure and also the control, by using our recently developed method in [Bensoussan, A., Huang, Z. and Yam, S. C. P. [2023] Control theory on Wasserstein space: A new approach to optimality conditions, Ann. Math. Sci. Appl.; Bensoussan, A., Tai, H. M. and Yam, S. C. P. [2023] Mean field type control problems, some Hilbert-space-valued FBSDEs, and related equations, preprint (2023), arXiv:2305.04019; Bensoussan, A. and Yam, S. C. P. [2019] Control problem on space of random variables and master equation, ESAIM Control Optim. Calc. Var. 25, 10]. Our method is to embed the mean field type control problem into a Hilbert space to bypass the evolution in the Wasserstein space. We here give a necessary condition and a sufficient condition for these control problems in Hilbert spaces, and we also derive a system of forward–backward stochastic differential equations. 

In this paper, we study the maximum principle of mean field type control problems when the volatility function depends on the state and its measure and also the control, by using our recently developed method in [Bensoussan, A., Huang, Z. and Yam, S. C. P. [2023] Control theory on Wasserstein space: A new approach to optimality conditions, Ann. Math. Sci. Appl.; Bensoussan, A., Tai, H. M. and Yam, S. C. P. [2023] Mean field type control problems, some Hilbert-space-valued FBSDEs, and related equations, preprint (2023), arXiv:2305.04019; Bensoussan, A. and Yam, S. C. P. [2019] Control problem on space of random variables and master equation, ESAIM Control Optim. Calc. Var. 25, 10]. Our method is to embed the mean field type control problem into a Hilbert space to bypass the evolution in the Wasserstein space. We here give a necessary condition and a sufficient condition for these control problems in Hilbert spaces, and we also derive a system of forward–backward stochastic differential equations. 

We study the deterministic control problem in the Wasserstein space, following the recent works of Bonnet and Frankowska, but with a new approach. One of the major advantages of our approach is that it reconciles the closed loop and the open loop approaches, without the technicalities of the traditional feedback control methodology. It allows also to embed the control problem in the Wasserstein space into a control problem in a Hilbert space, similar to the lifting method introduced by P. L. Lions, used already in our previous works. The Hilbert space is different from that proposed by P. L. Lions, and it allows to recover the control problem in the Wasserstein space as a particular case. 

Abstract Large constellations of small satellites will significantly increase the number of objects orbiting the Earth. Satellites burn up at the end of service life during reentry, generating aluminum oxides as the main byproduct. These are known catalysts for chlorine activation that depletes ozone in the stratosphere. We present the first atomic‐scale molecular dynamics simulation study to resolve the oxidation process of the satellite's aluminum structure during mesospheric reentry, and investigate the ozone depletion potential from aluminum oxides. We find that the demise of a typical 250‐kg satellite can generate around 30 kg of aluminum oxide nanoparticles, which may endure for decades in the atmosphere. Aluminum oxide compounds generated by the entire population of satellites reentering the atmosphere in 2022 are estimated at around 17 metric tons. Reentry scenarios involving mega‐constellations point to over 360 metric tons of aluminum oxide compounds per year, which can lead to significant ozone depletion.