Infinite horizon multiobjective optimal control of stochastic cooperative linear-quadratic dynamic difference games. (English) Zbl 1472.93201
Summary: This article is concerned with the infinite horizon stochastic cooperative linear-quadratic (LQ) dynamic difference game in both the regular and the indefinite cases. Firstly, due to the constraints imposed on the weighting matrices and the linearity of the dynamic system, the costs are shown to be convex spontaneously for the regular stochastic cooperative LQ difference game, which yields the equivalence between the minimization of the weighted sum of costs and the Pareto optimal control. Secondly, the Pareto optimal control is derived for the regular game on the ground of the solution to the weighted algebraic Riccati equation (WARE) under exact observability, and then Pareto solutions are identified via the optimal feedback gain matrices and the solution to the weighted algebraic Lyapunov equation (WALE). Moreover, a new criterion which is also necessary and sufficient is developed to guarantee the costs to be convex for the indefinite case, and the Pareto optimality is investigated based on the solutions to the weighted generalized algebraic Riccati equation (WGARE) and the weighted generalized algebraic Lyapunov equation (WGALE) combining with the semidefinite programming (SDP). Finally, the fishery management game in the economy is presented to illustrate the obtained results.
MSC:
| 93E20 | Optimal stochastic control |
| 49N10 | Linear-quadratic optimal control problems |
| 91A12 | Cooperative games |
| 91A15 | Stochastic games, stochastic differential games |
| 91A25 | Dynamic games |
Keywords:
stochastic cooperative linear-quadratic dynamic difference game; stochastic multiobjective optimal controlReferences:
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