  The search for heavy Higgs bosons is an essential step in the exploration of
the Higgs sector and in probing the Supersymmetric parameter space. This paper
discusses the constraints on the M(A) and tan beta parameters derived from the
bounds on the different decay channels of the neutral H and A bosons accessible
at the LHC, in the framework of the phenomenological MSSM. The implications
from the present LHC results and the expected sensitivity of the 14 TeV data
are discussed in terms of the coverage of the [M(A) - tan beta] plane. New
channels becoming important at 13 and 14 TeV for low values of tan beta are
characterised in terms of their kinematics and the reconstruction strategies.
The effect of QCD systematics, SUSY loop effects and decays into pairs of SUSY
particles on these constraints are discussed in details.
