AbstractWe 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.
Mild solutions to semilinear stochastic partial differential equations with locally monotone coefficients
