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This paper has proposed Specific class of single input single output (SISO) Mamdani fuzzy logic proportional controllers (FLPC) with triangle input membership functions, output membership functions with the same area, the sum-product composition rule for inference and centroid of area defuzzifier for control of both stable and unstable SISO linear time invariant systems. The proposed FLPC in fact represents memoryless, nonlinear, locally Lipschitz mapping between its input and output. According to the Popov criterion for absolute stability, nonlinear part of the system must satisfy sector conditions. In this paper necessary and sufficient conditions have been determined, which must satisfy FLPC parameters in order for the memoryless function represented by FLPC to satisfy the sector conditions. Subsequently, using the Popov criterion, the sufficient conditions for absolute stability of feedback connected linear system and FLPC have been determined. Based on the results achieved, an optimization method for both parameters and decision rules of FLPC has been proposed using genetic algorithms.