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Application of a nonlinear stabilizer for localizing search of optimal trajectories in control problems with infinite horizon

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  • The research is focused on an algorithm for constructing solutions of optimal control problems with infinite time horizon arising, for example, in economic growth models.   
           There are several significant difficulties which complicate solution of the problem, such as: (1) stiffness of Hamiltonian systems generated by the Pontryagin maximum principle; (2) non-stability of equilibrium points; (3) lack of initial conditions for adjoint variables.
         The analysis of the Hamiltonian system implemented in this paper for optimal control problems with infinite horizon provides results effective for construction of optimal solutions, namely, (1) if a steady state exists and satisfies regularity conditions then there exists a nonlinear stabilizer for the Hamiltonian system; (2) a nonlinear stabilizer generates the system with excluding adjoint variables whose trajectories (according to qualitative analysis of the corresponding differential equations) approximate solutions of the original Hamiltonian system in a neighborhood of the steady state;(3) trajectories of the stabilized system serve as first approximations and localize search of optimal trajectories.    
        The results of numerical experiments are presented by modeling of an economic growth system with investment in capital and enhancement of the labor efficiency.
    Mathematics Subject Classification: Primary: 93C10, 34H05, 93D15.


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