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A revised version of Samuelson’s correspondence principle: Applications of recent results on the asymptotic stability of optimal control to the problem of comparing long run equilibria. (English) Zbl 0587.90026
Models of economic dynamics, Proc. Workshop, Minneapolis/Minn. 1983, Lect. Notes Econ. Math. Syst. 264, 86-116 (1986).
[For the entire collection see Zbl 0579.00022.]
We discuss some recent results on the global asymptotic stability of optimal control and show how they may be used with profit in the problem of comparing long run equilibria. To be more specific, we shall study optimal control problems of the form \[ (1)\text{ maximize }\int^{\infty}_{0}e^{-\rho t}\pi (\dot x(t),x(t),\alpha)dt\quad s.t.\quad x(0)=x_ 0 \] where \(\pi(\dot x(t),x(t),\alpha)\) is instantaneous payoff at time t, which is assumed to be a function of the state of the system at time t, \(x(t)\), the rate of change of the state \(dx/dt=x(t)\) and a vector of parameters, \(\alpha\). Here \(x(t)\in R^ n\), \(\dot x(t)\in R^ n\), \(\alpha \in R^ m\) and \(\rho >0\). The maximum in (1) istaken over the set of absolutely continuous functions \(x(\cdot)\) such that \(x(0)=x_ 0.\)
If \(\pi\) (\(\cdot,...,\alpha)\) is strictly concave in \((x,\dot x)\) then there is at most one optimum path \(x(t,x_ 0,\alpha)\) for each \(x_ 0\), \(\alpha\). If the optimum exists for each \((x_ 0,\alpha)\) then there is a function \(h: R^ n\times R^ m\to R^ n\) called the ”optimal policy function” such that \[ (2)\quad \dot x(t,x_ 0,\alpha)=h[x(t,x_ 0,\alpha),\alpha]. \] Furthermore, h does not depend upon \(x_ 0\) or on t due to the time stationarity of (1). We discuss a set of sufficient conditions on \(\pi\) and \(\rho\) which imply that there is a unique steady state \(\bar x(\alpha)\) such that for all \(x_ 0\) \[ (3)\quad x(t,x_ 0,\alpha)\to \bar x(\alpha),\quad t\to \infty. \] Property (3) is called global asymptotic stability.

MSC:
91B62 Economic growth models
93D20 Asymptotic stability in control theory