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Piecewise linear Lyapunov functions are used to design control gain matrices so that closed systems are robustly stable and attractive regions are expanded as large as possible in given polytopic regions. The design problems for these specifications are formulated as bilinear programming problems whose constraints are divided into two groups, namely, linear inequalities and bilinear inequalities. In order to solve the constrained optimization problems, this paper presents a genetic algorithm that differs from traditional ones based on penalty techniques. The genetic algorithm starts from a feasible solution and retains its offspring, or population, within the feasible region. Besides the method creating such a starting point, several techniques are proposed to develop the diversity of population.