How to construct a DFA which accepts all the strings endig with aa and no two consecutive b (bb) I have to construct a DFA which ends with aa and does not contain any pair of b-s. A = {a,b}, {aa,baa,aaa,abaa,babaa,...}. I know how to construct them as separate DFAs but not together merged.
If you know how to construct DFAs for two languages separately, you can use systematic methods to build the DFA which recognises the intersection of the languages (e.g. https://math.stackexchange.com/questions/147457/intersection-of-two-deterministic-finite-automata). However to achieve a minimalist and elegant result (as well as for learning purposes) you may want to try and find ad-hoc methods.
NB: You can always apply a DFA minimization procedure to 'clean up' the automaton obtained via the above general method (https://en.m.wikipedia.org/wiki/DFA_minimization).