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In the paper, for the system which possesses both an attractor and a stablefixed point, we first formulate new stable control problems to find theasymptotically stable control function which realizes to transit a state movingaround the attractor to the stable fixed point. Then by using the ordinarydifferential equation based on the infinite horizon optimal control model withnegative discounts, we give one of answers for the stable control problem in atwo-dimensional case. Furthermore, under some conditions, we verify that thephase space can be separated to some open connected components depending on theasymptotic behavior of the orbit starting from the initial point in theircomponents. This classification of initial points suggests that it is enable torobustly achieve a stable control. Moreover, we illustrate some numericalresults for the stable control obtained by applying our focused system for theBonhoeffer-van der Pol model.

Asymptotically stable control problems by infinite horizon optimal control with negative discounting