DNSS point

DNSS point

DNSS 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]

Contents

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 ρ > 0 is the discount rate, x(t) and u(t) are the state and control variables, respectively, at time t, functions φ and f are assumed to be continuously differentiable with respect to their arguments and they do not depend explicitly on time t, and Ω 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 x0 is called a DNSS point if the system starting from it exhibits multiple optimal solutions or equilibria. Thus, at least in the neighborhood of x0, the system moves to one equilibrium for x > x0 and to another for x < x0. In this sense, x0 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 x0 = 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 x0 < 0, the optimal path is x(t) = -1 + e^{-t\left(x_{0}+1 \right)} and for x0 > 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. ^ 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): 332354. 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): 563579. 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): 615627. doi:10.1007/BF00934454. 
  4. ^ a b Skiba, A.K. (1978). "Optimal Growth with a Convex-Concave Production Function". Econometrica 46 (3): 527539. 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 0387280928 and ISBN 0792386086. 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): 387391. 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

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