Optimal control of divergent control systems
The study of Lagrange problems defined on an unbounded interval begins in the arena of mathematical economics when Ramsey  formulated a model of economic growth and, through the development of the Euler-Lagrange equations, stated the fist “golden rule of accumulation” now known as Ramsey’s rule. The study of these problems remained dormant until the 1940’s when the students of Tonelli initiated a systematic treatment for free problems in the calculus of variations defined on [0,∞).These works; Cinquini , , Faedo , , and Darbo ; provide a rather complete theory providing existence theorems, necessary conditions, and most importantly (at least for the purposes here) the asymptotic convergence of optimal trajectories to a steady state. In all of these works, the objective or cost function was assumed to be a convergent improper integral. That this may not be the case was realized by Ramsey in his seminal paper. To circumvent this he introduced what we now call the “optimal steady-state problem,” which in Rarnsey’s model was referred to as the “maximal sustainable rate of enjoyment,” or more simply as “Bliss.” In his approach, Ramsey introduces a new problem in which the optimal value is finite whenever it is possible to “control” the system to reach Bliss in a finite time. This approach leads to what we now know as the asymptotic turnpike theory.
KeywordsOptimal Control Problem State Constraint Optimal Trajectory Admissible Pair Infinite Horizon
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