DNSS point

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DNSS points, also known as Skiba 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]


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


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]


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]


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 .


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


  1. ^ a b 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. ^ a b 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. ^ a b 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. ^ a b Skiba, A.K. (1978). "Optimal Growth with a Convex-Concave Production Function". Econometrica. 46 (3): 527–539. doi:10.2307/1914229. JSTOR 1914229. 
  5. ^ a b c d e f 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