Turnpike Theory Properties in Optimal Mathematical Control with Applications

Abstract

Optimal control is a mathematical method to fined values for a systems variables, so that these values lead the system to follow an optimal path or curve that achieves the maximum or minimum values for a characteristic or cost function. The Turnpikephenomenon appears in several variational and optimal control problems, arising in engineering and economic growth. We say that a problem has a turnpike property when the optimal solutions converge to certain path during most of time, this path is known as the turnpike of problem. In this research we discussed number of recent results concerning turnpike properties in the calculus of variations and optimal control problems.The optimal trajectory is shown to remains exponentially close to the steady-state solution of an associated static optimal control problem ,but also corresponding adjoint vector of the Pontryagin maximum principle. We provide a general version of a turnpike theorem, valuable for nonlinear dynamics without any specific assumption ,and for very general terminal conditions. We characterized turnpike properties of the dynamics in terms of the system matrices related to the linear quadratic problem. These characterizations lead to new necessary conditions for the turnpike properties under consideration, and thus eventually to necessary and sufficient conditions in terms of spectral criteria and matrix inequalities.