  Prosocial incentive can promote cooperation, but providing incentive is
costly. Institutions in human society may prefer to use an incentive strategy
which is able to promote cooperation at a reasonable cost. However, thus far
few works have explored the optimal institutional incentives which minimize
related cost for the benefit of public cooperation. In this work, in
combination with optimal control theory we thus formulate two optimal control
problems to explore the optimal incentive strategies for institutional reward
and punishment respectively. By using the approach of Hamilton-Jacobi-Bellman
equation for well-mixed populations, we theoretically obtain the optimal
positive and negative incentive strategies with the minimal cumulative cost
respectively. Additionally, we provide numerical examples to verify that the
obtained optimal incentives allow the dynamical system to reach the desired
destination at the lowest cumulative cost in comparison with other given
incentive strategies. Furthermore, we find that the optimal punishing strategy
is a cheaper way for obtaining an expected cooperation level when it is
compared with the optimal rewarding strategy.
