Analysis of Optimal Control Problems for Hybrid Systems with One State Variable

28 Pages Posted: 24 Jul 2019 Last revised: 29 Jul 2020

See all articles by Puduru Viswanadha Reddy

Puduru Viswanadha Reddy

Indian Institute of Technology Madras

Johannes M. Schumacher

University of Amsterdam - Department of Quantitative Economics (KE)

Jacob C. Engwerda

retiree

Date Written: July 20, 2020

Abstract

We study a class of discounted autonomous infinite horizon optimal control problems where the state dynamics may undergo discontinuous changes when the state crosses from one region of the state space to another. Assuming the state and control space are one-dimensional, we provide structural results of the candidate optimal solutions. In particular, we show that candidate optimal state trajectories are monotonic. Further, we prove a theorem on absence of limit cycles, generalizing the Bendixson criterion to a hybrid context. Using these results, we derive necessary and sufficient conditions for the existence of tipping points for this class of problems. A tipping point is an initial condition starting from which there exist multiple candidate optimal solutions with different qualitative behaviors. Next, we specialize these results for piecewise linear quadratic models to obtain an analytic characterization of tipping points in terms of the model parameters.

Keywords: regime switching models, hybrid systems, infinite horizon optimal control, multiple equilibria, Skiba points

JEL Classification: C61

Suggested Citation

Reddy, Puduru Viswanadha and Schumacher, J.M. (Hans) and Engwerda, Jacob C., Analysis of Optimal Control Problems for Hybrid Systems with One State Variable (July 20, 2020). Available at SSRN: https://ssrn.com/abstract=3424771 or http://dx.doi.org/10.2139/ssrn.3424771

Puduru Viswanadha Reddy

Indian Institute of Technology Madras ( email )

Department of Electrical Engineering
IIT Madras
Chennai, 600036
India

J.M. (Hans) Schumacher (Contact Author)

University of Amsterdam - Department of Quantitative Economics (KE) ( email )

Roetersstraat 11
Amsterdam, 1018 WB
Netherlands

Jacob C. Engwerda

retiree ( email )

P.O. Box 90153
Tilburg, 5000 LE
Netherlands
+31 13 466 2174 (Phone)
+31 13 466 3280 (Fax)

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