DNSS point

DNSS points, also known as Skiba points, arise in optimal control problems that exhibit multiple optimal solutions. A DNSS point${\displaystyle -}$named alphabetically after Deckert and Nishimura,[1] Sethi,[2][3][4] and Skiba[5]${\displaystyle -}$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.[6] [7]

Definition

Of particular interest here are discounted infinite horizon optimal control problems that are autonomous.[8] These problems can be formulated as

${\displaystyle \max _{u(t)\in \Omega }\int _{0}^{\infty }e^{-\rho t}\varphi \left(x(t),u(t)\right)dt}$

s.t.

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

where ${\displaystyle \rho >0}$ is the discount rate, ${\displaystyle x(t)}$ and ${\displaystyle u(t)}$ are the state and control variables, respectively, at time ${\displaystyle t}$, functions ${\displaystyle \varphi }$ and ${\displaystyle f}$ are assumed to be continuously differentiable with respect to their arguments and they do not depend explicitly on time ${\displaystyle t}$, and ${\displaystyle \Omega }$ is the set of feasible controls and it also is explicitly independent of time ${\displaystyle t}$. Furthermore, it is assumed that the integral converges for any admissible solution ${\displaystyle \left(x(.),u(.)\right)}$. In such a problem with one-dimensional state variable ${\displaystyle x}$, the initial state ${\displaystyle x_{0}}$ is called a DNSS point if the system starting from it exhibits multiple optimal solutions or equilibria. Thus, at least in the neighborhood of ${\displaystyle x_{0}}$, the system moves to one equilibrium for ${\displaystyle x>x_{0}}$ and to another for ${\displaystyle x<x_{0}}$. In this sense, ${\displaystyle x_{0}}$ 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.[6] and Zeiler et al.[9] present examples that exhibit DNSS curves.

Some references on the application of DNSS points are Caulkins et al.[10] and Zeiler et al.[11]

History

Suresh P. Sethi identified such indifference points for the first time in 1977.[2] Further, Skiba,[5] Sethi,[3][4] 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.,[6] recognizes (alphabetically) the contributions of these authors.

These indifference points have been referred to earlier as Skiba points or DNS points in the literature.[6]

Example

A simple problem exhibiting this behavior is given by ${\displaystyle \varphi \left(x,u\right)=xu,}$ ${\displaystyle f\left(x,u\right)=-x+u,}$ and ${\displaystyle \Omega =\left[-1,1\right]}$. It is shown in Grass et al.[6] that ${\displaystyle x_{0}=0}$ is a DNSS point for this problem because the optimal path ${\displaystyle x(t)}$ can be either ${\displaystyle \left(1-e^{-t}\right)}$ or ${\displaystyle \left(-1+e^{-t}\right)}$. Note that for ${\displaystyle x_{0}<0}$, the optimal path is ${\displaystyle x(t)=-1+e^{-t\left(x_{0}+1\right)}}$ and for ${\displaystyle x_{0}>0}$, the optimal path is ${\displaystyle x(t)=1+e^{-t\left(x_{0}-1\right)}}$.

Extensions

For further details and extensions, the reader is referred to Grass et al.[6]

References

1. Deckert, D.W.; Nishimura, K. (1983). "A Complete Characterization of Optimal Growth Paths in an Aggregated Model with Nonconcave Production Function". Journal of Economic Theory. 31 (2): 332–354. doi:10.1016/0022-0531(83)90081-9.
2. Sethi, S.P. (1977). "Nearest Feasible Paths in Optimal Control Problems: Theory, Examples, and Counterexamples". Journal of Optimization Theory and Applications. 23 (4): 563–579. doi:10.1007/BF00933297. S2CID 123705828.
3. Sethi, S.P. (1979). "Optimal Advertising Policy with the Contagion Model". Journal of Optimization Theory and Applications. 29 (4): 615–627. doi:10.1007/BF00934454. S2CID 121398518.
4. Sethi, S.P., "Optimal Quarantine Programmes for Controlling an Epidemic Spread," Journal of Operational Research Society, 29(3), 1978, 265-268. JSTOR 3009454 SSRN 3587573
5. Skiba, A.K. (1978). "Optimal Growth with a Convex-Concave Production Function". Econometrica. 46 (3): 527–539. doi:10.2307/1914229. JSTOR 1914229.
6. 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.
7. Sethi, S.P., Optimal Control Theory: Applications to Management Science and Economics, Third Edition, Springer Nature Switzerland AG, 2019. (565 pages – ISBN 978-3-319-98236-6) Springer Link.
8. Sethi, S. P.; Thompson, G. L. (2000). Optimal Control Theory: Applications to Management Science and Economics (Second ed.). Springer. ISBN 0-387-28092-8. Slides are available at http://www.utdallas.edu/~sethi/OPRE7320presentation.html
9. 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 |
10. Caulkins, J. P.; Feichtinger, G.; Grass, D.; Tragler, G. (2009). "Optimal control of terrorism and global reputation: A case study with novel threshold behavior". Operations Research Letters. 37 (6): 387–391. doi:10.1016/j.orl.2009.07.003.
11. 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