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.

Famous quotes containing the word definition:

“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)

“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)

“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)