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A robust time-inconsistent linear-quadratic problem
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DOI：10.1002/rnc.5426.png)
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This paper is concerned with a deterministic time-inconsistent linear-quadratic optimal control with model uncertainty, where nonexponential time discounting appears in the objective functional. If the objective functional is viewed as an output/observation on the lifetime horizon, the disturbance attenuation of this time-inconsistent linear-quadratic problem is modelled and solved under the philosophy of minimax optimization, and resorts to a time-inconsistent zero-sum linear-quadratic dynamic game with the controller being a minimizing player and the disturbance being an imaginary maximizing player. The definition of robust mixed equilibrium solution is introduced, whose existence is fully characterized. An example is constructed to show that the mixed equilibrium solution exists for all the initial pairs even though neither the open-loop equilibrium control nor the feedback equilibrium strategy exists for some initial pairs. Furthermore, for a time-consistent version of the zero-sum linear-quadratic game, another example is presented to indicate the case that the existence of open-loop time-consistent equilibrium does not imply the existence of open-loop precommitted equilibrium.
Keywords:
dynamic game
precommitted policy
stochastic linear‐ quadratic problem
time inconsistency
time‐ consistent policy
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