Answer: Unfortunately, based on your description I am not certain what is happening to your plants. Perovskia is generally considered to be pest and disease free. However, they are in the mint family so I think it is possible they could be suffering from powdery mildew, although this would discolor the foliage as well as the stems. To treat, try a baking powder based spray such as Remedy and also clean up and dispose of the damaged plant debris now and then do a thorough clean up in the fall. Put it in the trash, not the compost pile.
If only the stems are turning white and collapsing or rotting, then it might be a rot problem such as Sclerotium rolfsii or Sclerotinia sclerotiorum -- these are notable for a white mass of mycelium at the base of the plant around the crown and into the soil. Clean up the damaged portion of the plant and dispose of it in the trash, and also remove and replace the top soil where you see the mycelium around the plant.
Clean up is important in preventing reinfection. In addition, make sure the plants are growing in a well drained location with adequate spacing for good air circulation. Do not water them unless there is an extended period of drought -- they require a well drained soil and are extremely tolerant of dry weather. If you water, apply it to the soil surface and water deeply but infrequently, avoid wetting the foliage.
Both of the above problems can affect mints, so it might be one of them. However, since I have not seen the plants, I would strongly suggest you consult with your local Penn State cooperative extension to get a more specific diagnosis. Based on knowing that, they can help you decide what to do. If a chemical control is needed, they will have the most up to date information on what to use and how/when to use it for best results.
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