Sethi-Skiba point

Sethi-Skiba points,¹²³⁴ also known as DNSS points, arise in optimal control problems that exhibit multiple optimal solutions. A Sethi-Skiba point 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.²⁵⁶

Of particular interest here are discounted infinite horizon optimal control problems that are autonomous.² 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 Sethi-Skiba 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.⁵ and Zeiler et al.⁷ present examples that exhibit DNSS curves.

Some references on the applications of Sethi-Skiba points are Caulkins et al.,⁸ Zeiler et al.,⁹ and Carboni and Russu¹⁰

Suresh P. Sethi identified such indifference points for the first time in 1977.¹¹ Further, Skiba,¹² Sethi,¹³¹⁴¹⁵ and Deckert and Nishimura¹⁶ 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 also referred to as Skiba points or DNS points in earlier literature.⁵

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.⁵ that ${\displaystyle x_{0}=0}$ is a Sethi-Skiba 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 ${\displaystyle x(t)=1+e^{-t\left(x_{0}-1\right)}}$.

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