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Maximum principle for infinite-horizon optimal control problems with dominating discount. (English) Zbl 1266.49003

The dynamics of the system is given by a nonlinear nonautonomous vector differential equation \[ x'(t) = f(t, x(t), u(t)) \;\;(t \geq 0)\, , \;\;\;\;x(0) = x_0 \, , \] where the control \(u(t)\) takes values in a set \(U\) in \(m\)-dimensional space and the trajectory \(x(t)\) is an \(n\)-dimensional vector. The functional to be maximized is \[ J(x(\cdot), u(\cdot)) = \int_0^\infty e^{- \rho t} g(t, x(t), u(t)) dt \] where \(\rho\) is called the discount rate. There is no constraint on the trajectory at infinity. A peculiarity involving Pontryagin’s maximum principle for this kind of infinite horizon problem is, the problem may be “abnormal” at infinity. This means, there may exist optimal trajectories \(\bar x(t)\) with costates \((\psi_0, \psi(t))\) such that \(\psi_0 = 0.\) Furthermore, transversality conditions of the type \(\lim_{t \to \infty} \psi(t) = 0\) or \(\lim_{t \to \infty}\langle \psi(t), \bar x(t)\rangle = 0\) may fail to hold.
Using ideas that originate in the work of Aubin and Clarke the authors show that if the discount rate is large enough (so that it “dominates” other parameters in the system) transversality conditions hold even in a stronger form.

MSC:

49J15 Existence theories for optimal control problems involving ordinary differential equations
49K15 Optimality conditions for problems involving ordinary differential equations
91B62 Economic growth models
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