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Optimal Control of Mean Field Equations with Monotone Coefficients and Applications in Neuroscience
- Source :
- Applied Mathematics & Optimization. 84:1925-1968
- Publication Year :
- 2021
- Publisher :
- Springer Science and Business Media LLC, 2021.
-
Abstract
- We are interested in the optimal control problem associated with certain quadratic cost functionals depending on the solution $X=X^\alpha$ of the stochastic mean-field type evolution equation in $\mathbb R^d$ $dX_t=b(t,X_t,\mathcal L(X_t),\alpha_t)dt+\sigma(t,X_t,\mathcal L(X_t),\alpha_t)dW_t,$ $X_0\sim \mu$ given, under assumptions that enclose a sytem of FitzHugh-Nagumo neuron networks, and where for practical purposes the control $\alpha_t$ 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 $\pi(X_t)\leq0$. 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.<br />Comment: 32 pages; 11 figures
- Subjects :
- 93E20, 92B20, 65K10
Control and Optimization
Applied Mathematics
Probability (math.PR)
Sigma
Numerical Analysis (math.NA)
Type (model theory)
Optimal control
Lipschitz continuity
Combinatorics
Monotone polygon
Maximum principle
Mathematics::Probability
Optimization and Control (math.OC)
Mean field equation
FOS: Mathematics
Mathematics - Numerical Analysis
ddc:510
Martingale (probability theory)
Mathematics - Optimization and Control
Mathematics - Probability
Mathematics
Subjects
Details
- ISSN :
- 14320606 and 00954616
- Volume :
- 84
- Database :
- OpenAIRE
- Journal :
- Applied Mathematics & Optimization
- Accession number :
- edsair.doi.dedup.....ceef6f550f57d38fbd9b8fa136a7f3d5