Optimal control of mean field equations with monotone coefficients and applications in neuroscience
by Antoine Hocquet, Alexander Vogler
Year:
2021
Publication:
Applied Mathematics and Optimization
Abstract:
We are interested in the optimal control problem associated with certain quadratic cost functionals depending on the solution X = Xα of the stochastic mean-field type evolution equation in Rd
dXt = b(t, Xt,L(Xt), αt)dt+σ (t, Xt,L(Xt), αt)dWt , X0 ∼ μ (μ given), (1)
under assumptions that enclose a system of FitzHugh–Nagumo neuron networks, and where for practical purposes the control αt is deterministic. To do so, we assume that we are given a drift coefficient that satisfies a one-sided Lipschitz condition, and that the dynamics (2) satisfies an almost sure boundedness property of the form π(Xt) ≤ 0. The mathematical treatment we propose follows the lines of the recent monograph of Carmona and Delarue for similar control problems with Lipschitz coefficients. After addressing the existence of minimizers via a martingale approach, we show a maximum principle for (2), and numerically investigate a gradient algorithm for the approximation of the optimal control.
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Antoine Hocquet (PhD)
Antoine holds a PhD in applied mathematics from École Polytechnique and spent eight years as a postdoctoral researcher at TU Berlin, working on stochastic PDEs, rough paths theory, and mean-field models, areas where rigorous mathematics try matching messy real-world phenomena. During this period he authored 15+ peer-reviewed publications and taught graduate-level courses in numerical analysis and stochastic methods. Drawn by the challenge of making mathematics work rather than just exist, he then moved into machine learning, building a portfolio spanning image segmentation, reinforcement learning, and LLM-powered document processing. At dida, he works as a machine learning scientist.