DNSS Point - Definition

Definition

Of particular interest here are discounted infinite horizon optimal control problems that are autonomous. These problems can be formulated as

$\max_{u(t)\in \Omega}\int_0^{\infty} e^{-\rho t} \varphi\left(x(t), u(t)\right)dt$

s.t.

$\dot{x}(t) = f\left(x(t), u(t)\right), x(0) = x_{0},$

where is the discount rate, and are the state and control variables, respectively, at time, functions and are assumed to be continuously differentiable with respect to their arguments and they do not depend explicitly on time, and is the set of feasible controls and it also is explicitly independent of time . Furthermore, it is assumed that the integral converges for any admissible solution . In such a problem with one-dimensional state variable, the initial state is called a DNSS point if the system starting from it exhibits multiple optimal solutions or equilibria. Thus, at least in the neighborhood of, the system moves to one equilibrium for and to another for . In this sense, is an indifference point from which the system could move to either of the two equilibria.

For two-dimensional optimal control problems, Grass et al. and Zeiler et al. present examples that exhibit DNSS curves.

Some references on the application of DNSS points are Caulkins et al. and Zeiler et al.

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