Generic profit singularities in time averaged optimization. The case of a control space with a regular boundary

For one-parameter smooth families of pairs of control systems and profit densities on the circle, we consider the problem of maximizing the averaged profit in the infinite horizon from the singularity theory point of view. Namely, we study the generic classification of the optimal averaged profit as function of the parameter. This approach to the problem was introduced in 2002 by V. I. Arnold and all generic classifications in related problems obtained since then, assume a compact control space without boundary. The existence of a boundary in the control space is usual in real problems and so it is worthwhile to be considered. In this work, we consider the existence of a regular boundary in the control space and study the case of one-dimensional parameter. We present all generic singularities of the optimal averaged profit as function of the parameter and conclude that three new singularities appear.