Global stabilization of the Hamiltonian system in the two-sector economic growth model*

In optimal control problems with infinite time horizon, arising in economic growth models, the analytical solution can be derived in specific cases only. This fact is explained, first of all, by nonlinear character of the Hamiltonian system arising in the Pontryagin maximum principle. Another difficulty is connected with the so-called transversality condition which describes the asymptotic behavior of adjoint variables at the infinite time.In the paper, a synthesis of optimal trajectories is carried out. Obtained results allow to conclude that the nonlinear stabilizer ensures global stabilization of the Hamiltonian dynamics. The structure of nonlinear stabilizer is based on the qualitative theory of differential equations. Under assumption on existence of a saddle steady state an “eigen–plane” is constructed by two eigenvectors corresponding to negative eigenvalues. Relations describing “eigen-plane” allow to exclude adjoint variables from (a) the Hamiltonian dynamics and (b) optimal control representations at the steady state neighborhood. So we obtain (a) the stabilized dynamics independent from adjoint variables, and (b) the structure of the nonlinear stabilizer. The provided analysis of the Hamiltonian system argues that the suggested nonlinear stabilizer ensures global stabilization of the Hamiltonian dynamics.