Optimal control of divergent control systems

  • Dean A. Carlson
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


The study of Lagrange problems defined on an unbounded interval begins in the arena of mathematical economics when Ramsey [36] 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 [18], [19], Faedo [22], [23], and Darbo [21]; 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.


Optimal Control Problem State Constraint Optimal Trajectory Admissible Pair Infinite Horizon 
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© Birkhäuser Boston 1999

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  • Dean A. Carlson

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