A verification theorem and application to the linear quadratic regulator for minimax control problems
In a number of previous publications the author of this paper has developed a Bellman equation and a Pontryagin maximum principle for optimal control problems attempting to minimize the maximum value of a cost criterion (optimal minimax control). The paper extends this theory by providing sufficient conditions for a function to be an optimal minimax control for the general problem that now also includes a terminal cost within the cost criterion. Then attention is turned to the control problem with minimax \textit{quadratic} cost function and \textit{linear} state equation, the minimax analogon to the well-known LQ optimization problem. The Bellman equation is derived for the minimax LQ problem and a lower bound for the value function is given. Unfortunately, for the multidimensional case, an explicit solution of the Bellman equation may not be possible. Only for minimax problems in one dimension the minimum cost function is shown to be a quadratic function of the state and the solution may be found via a nonlinear first-order variational inequality. The numerical solution of a particular scalar minimax problem is presented and compared to the corresponding classical linear quadratic regulator.
scientific article; zbMATH DE number 584070