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We develop a global perturbation technique for obtaining approximate but analytical expressions of stabilizing solutions to Hamilton-Jacobi equations when stable directions near equilibria exist. To this end, we approximately compute the stable manifolds of the equilibria in the corresponding Hamiltonian systems and use their relation to the stabilizing solutions. Our basic idea for the computation is similar to an extension of Melnikov's method, which is a well known technique for detecting homoclinic orbits to equilibria or periodic orbits. We illustrate the developed technique for the Duffing oscillator as an example. A numerical computation for the stable manifold is also given and compared with the theoretical result to demonstrate the validity of the theoretical techinique.