Industrial application(s) of LDA (latent Dirichlet allocation)?
I can't answer for the systems in already existing companies, but I can definitely share an application of LDA in NLP.

Latent Dirichlet Allocation is a popular technique use for topic modelling in Natural Language Processing. The idea is to have corpus of natural langue text with lots of documents and the goal is to get the distribution of the words appearing in the corpus each (Distribution) being termed as a topic. 

For visual analysis you can view this note book here. It makes use of the NLP library built for LDA, called pyLDAvis.

So lets say you have a corpus, and you want to build a tagger which labels the document(input text) into one of the N Classes, but you don't have labelled data. How do you start? One way is to use LDA to perform Topic Modelling, get the topics, then name the topics to N Classes. Label the data using LDA model, Use the loosely labeled data to build a classifier.