Optimal Control of Mean Field Equations with Monotone Coefficients and Applications in Neuroscience

We are interested in the optimal control problem associated with certain quadratic cost functionals depending on the solution $$X=X^\alpha $$ X = X α of the stochastic mean-field type evolution equation in $${\mathbb {R}}^d$$ R d $$\begin{aligned} dX_t=b(t,X_t,{\mathcal {L}}(X_t),\alpha _t)dt+\sigma (t,X_t,{\mathcal {L}}(X_t),\alpha _t)dW_t\,, \quad X_0\sim \mu (\mu \text { given),}\qquad (1) \end{aligned}$$ d X t = b ( t , X t , L ( X t ) , α t ) d t + σ ( t , X t , L ( X t ) , α t ) d W t , X 0 ∼ μ ( μ given), ( 1 ) under assumptions that enclose a system of FitzHugh–Nagumo neuron networks, and where for practical purposes the control $$\alpha _t$$ α t 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 $$\pi (X_t)\le 0$$ 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 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.