Optimal control of mean field equations with monotone coefficients and applications in neuroscience
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We are interested in the optimal control problem associated with certain quadratic cost functionals depending on the solution X=X^α of the stochastic mean-field type evolution equation in ℝ^d dX_t=b(t,X_t,ℒ(X_t),α_t)dt+σ(t,X_t,ℒ(X_t),α_t)dW_t, X_0∼μ given, under assumptions that enclose a sytem of FitzHugh-Nagumo neuron networks, and where for practical purposes the control α_t is deterministic. To do so, we assume that we are given a drift coefficient that satisfies a one-sided Lipshitz condition, and that the dynamics is subject to a (convex) level set constraint of the form π(X_t)≤0. The mathematical treatment we propose follows the lines of the recent monograph of Carmona and Delarue for similar control problems with Lipshitz coefficients. After addressing the existence of minimizers via a martingale approach, we show a maximum principle and then numerically investigate a gradient algorithm for the approximation of the optimal control.
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