Most of the optimal decision making problems studied in economics and ecology are complex in nature. These complexities generally arise while modeling the inherent behavior of the dynamic environment, which includes agents interacting with the system. Modeling with systems described by the integration of continuous and discrete dynamics capture some of these situations. A change of discrete state of the system is called a switch. If a decision maker influences a switch then it is said to be controlled/external, whereas an internal switch generally results when the continuous state variable satisfies some constraints. Threshold effects are autonomous switches that happen when the continuous state variable hits a boundary. Optimal control of dynamic systems with threshold effects received considerable interest in control engineering and more recently in environmental economics.
In this talk we introduce a class of discounted autonomous infinite horizon optimal control problems with threshold effects. We derive some additional results, which follow as a consequence of switched maximum principle, by restricting the state and control variables to dimension one.
The main objective of this work is to highlight, with an example, the key observations when smooth nonlinear models are approximated with simple discontinuous functions. We approximate the classical shallow lake model using simple and hysteresis switching by using deterministic hard thresholds. Assuming symmetry in agents' actions, we solve the associated optimal management problem. This approximation leads to a multi-modal optimal vector field. In particular, the optimal behavior is explained by simple dynamics within each mode and a complex jump rule near the switching surface. The dynamic behavior of the switched vector field is similar, qualitatively, to the smooth version. More specifically, we observe that the variation of switching surface induces bifurcations in this vector field. Further, we notice that agents' mode of play has an impact on bifurcations; that is, cooperation can lead to 'clean' steady state (oligotrophic state) and noncooperation can lead to 'turbid' steady state (eutrophic state). However, we notice that there are some key differences. The bifurcations in the present analysis are governed by simple rules, a set of inequalities, which can be checked/verified once the parameters are given. In the previous works these bifurcation scenarios were established numerically though their existence was proved using tools from ordinary differential equations. Finally, we discuss some open issues that arise when modeling with threshold effects.