# DNSS point

**DNSS points** arise in optimal control problems that exhibit multiple optimal solutions. A DNSS pointnamed alphabetically after Deckert and Nishimura,^{[1]} Sethi,^{[2]}^{[3]} and Skiba^{[4]}is an indifference point in an optimal control problem such that starting from such a point, the problem has more than one different optimal solutions. A good discussion of such points can be found in Grass et al.^{[5]}

## Definition

Of particular interest here are discounted infinite horizon optimal control problems that are autonomous.^{[6]} These problems can be formulated as

s.t.

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.^{[5]} and Zeiler et al.^{[7]} present examples that exhibit DNSS curves.

Some references on the application of DNSS points are Caulkins et al.^{[8]} and Zeiler et al.^{[9]}

## History

Suresh P. Sethi identified such indifference points for the first time in 1977.^{[2]} Further, Skiba,^{[4]} Sethi,^{[3]} and Deckert and Nishimura^{[1]} explored these indifference points in economic models. The term DNSS (Deckert, Nishimura, Sethi, Skiba) points, introduced by Grass et al.,^{[5]} recognizes (alphabetically) the contributions of these authors.

These indifference points have been referred to earlier as *Skiba points* or *DNS points* in the literature.^{[5]}

## Example

A simple problem exhibiting this behavior is given by and . It is shown in Grass et al.^{[5]} that is a DNSS point for this problem because the optimal path can be either or . Note that for , the optimal path is and for , the optimal path is .

## Extensions

For further details and extensions, the reader is referred to Grass et al.^{[5]}

## References

- ↑
^{1.0}^{1.1}{{#invoke:Citation/CS1|citation |CitationClass=journal }} - ↑
^{2.0}^{2.1}{{#invoke:Citation/CS1|citation |CitationClass=journal }} - ↑
^{3.0}^{3.1}{{#invoke:Citation/CS1|citation |CitationClass=journal }} - ↑
^{4.0}^{4.1}{{#invoke:Citation/CS1|citation |CitationClass=journal }} - ↑
^{5.0}^{5.1}^{5.2}^{5.3}^{5.4}^{5.5}Grass, D., Caulkins, J.P., Feichtinger, G., Tragler, G., Behrens, D.A. (2008).*Optimal Control of Nonlinear Processes: With Applications in Drugs, Corruption, and Terror*. Springer. ISBN 978-3-540-77646-8. - ↑ Sethi, S.P. and Thompson, G.L. (2000).
*Optimal Control Theory: Applications to Management Science and Economics*. Second Edition. Springer. ISBN 0-387-28092-8 and ISBN 0-7923-8608-6. Slides are available at http://www.utdallas.edu/~sethi/OPRE7320presentation.html - ↑ Zeiler, I., Caulkins, J., Grass, D., Tragler, G. (2009). Keeping Options Open: An Optimal Control Model with Trajectories that Reach a DNSS Point in Positive Time.
*SIAM Journal on Control and Optimization*, Vol. 48, No. 6, pp. 3698-3707.| doi =10.1137/080719741 | - ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ I. Zeiler, J. P. Caulkins, and G. Tragler. When Two Become One: Optimal Control of Interacting Drug.
*Working paper,*Vienna University of Technology, Vienna, Austria