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A Nash equilibrium approach for multiobjective optimal control problems with elliptic partial diﬀerential equations

Dreves A.

Control and Cybernetics

2016

Vol. 45, no. 4

457--482

We consider the generalized Nash equilibrium as a solution concept for multiobjective optimal control problems governed by elliptic partial diﬀerential equations with constraints not only for the control but also for the state variables. In the ﬁrst part, we present a constructive proof of the existence of a generalized Nash equilibrium via an approximating sequence of suitable ﬁnite dimensional discretizations. In the second part, we propose a variant of a potential reduction algorithm for the numerical solution of these discretized problems. In contrast to the existing numerical approaches ours does not require the computation of the control–to–state mapping. Instead we introduce diﬀerent state variables and guarantee that they become equal at a solution. We prove suﬃcient conditions for the convergence of our algorithm to a solution. Furthermore, some numerical results showing the applicability are provided.