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Cauchy problem and: \\[v(x,z):=\\inf_{u\\in V} \\{\\int_0^{+\\infty} e^{-\\lambda s} L(y_{x,z,u}(s), u(s))ds \\}\\] where $V$ is a class of admissible controls, we prove that $v$ is the only viscosity solution of an Hamilton-Jacobi-Bellman equation of the form: \\[\\lambda v(x,z)+H(x,z,\\nabla_x v(x,z))+D_z v(x,z), \\dot{z} >=0\\] in the sense of the theory 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