# American Institute of Mathematical Sciences

September  2020, 10(3): 493-526. doi: 10.3934/mcrf.2020008

## Optimality conditions in variational form for non-linear constrained stochastic control problems

 Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria

Dedicated to Prof. Dr. Fréderic Bonnans on the occasion of his 60th birthday

Received  February 2018 Revised  December 2018 Published  December 2019

Optimality conditions in the form of a variational inequality are proved for a class of constrained optimal control problems of stochastic differential equations. The cost function and the inequality constraints are functions of the probability distribution of the state variable at the final time. The analysis uses in an essential manner a convexity property of the set of reachable probability distributions. An augmented Lagrangian method based on the obtained optimality conditions is proposed and analyzed for solving iteratively the problem. At each iteration of the method, a standard stochastic optimal control problem is solved by dynamic programming. Two academical examples are investigated.

Citation: Laurent Pfeiffer. Optimality conditions in variational form for non-linear constrained stochastic control problems. Mathematical Control & Related Fields, 2020, 10 (3) : 493-526. doi: 10.3934/mcrf.2020008
##### References:

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##### References:
Convergence results for the Test Case 1
Numerical results
Convergence results for the Test Case 2
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