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

Pfeiffer Laurent

doi:10.3934/mcrf.2020008

Article Contents

2020, Volume 10, Issue 3: 493-526. Doi: 10.3934/mcrf.2020008

This issue Previous Article Sparse optimal control for the heat equation with mixed control-state constraints Next Article State-constrained semilinear elliptic optimization problems with unrestricted sparse controls

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

Pfeiffer Laurent

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

Early access: December 2019

Published: July 2020

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Abstract

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.

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Mathematics Subject Classification: Primary: 90C15, 93E20; Secondary: 49J53, 49K99.

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Figure 1. Convergence results for the Test Case 1

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Figure 2. Numerical results

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Figure 3. Convergence results for the Test Case 2

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