DNSS point

DNSS points, also known as Skiba points, arise in optimal control problems that exhibit multiple optimal solutions. A DNSS point$-$named 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

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

s.t.

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

where $\rho > 0$ is the discount rate, $x(t)$ and $u(t)$ are the state and control variables, respectively, at time $t$, functions $\varphi$ and $f$ are assumed to be continuously differentiable with respect to their arguments and they do not depend explicitly on time $t$, and $\Omega$ is the set of feasible controls and it also is explicitly independent of time $t$. Furthermore, it is assumed that the integral converges for any admissible solution $\left(x(.), u(.)\right)$. In such a problem with one-dimensional state variable $x$, the initial state $x_{0}$ is called a DNSS point if the system starting from it exhibits multiple optimal solutions or equilibria. Thus, at least in the neighborhood of $x_0$, the system moves to one equilibrium for $x > x_0$ and to another for $x < x_0$. In this sense, $x_0$ 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.^[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 $\varphi\left(x,u\right) =xu,$ $f\left(x,u\right) = -x + u,$ and $\Omega = \left[-1, 1\right]$. It is shown in Grass et al.^[5] that $x_{0} = 0$ is a DNSS point for this problem because the optimal path $x(t)$ can be either $\left(1-e^{-t}\right)$ or $\left(-1+e^{-t}\right)$. Note that for $x_{0} < 0$, the optimal path is $x(t) = -1 + e^{-t\left(x_{0}+1 \right)}$ and for $x_{0} > 0$, the optimal path is $x(t) = 1 + e^{-t\left(x_{0}-1 \right)}$.

Extensions

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

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.
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.
4. Skiba, A.K. (1978). "Optimal Growth with a Convex-Concave Production Function". Econometrica 46 (3): 527–539. doi:10.2307/1914229. JSTOR 1914229.
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.
6. 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
7. 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 |
8. 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.
9. 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