An analogue of Pontryagin's maximum principle in problems of minimization of multiple integrals

2017 ◽  
Vol 81 (5) ◽  
pp. 973-984
Author(s):  
M I Zelikin
Keyword(s):  
Maximum Principle ◽  
Pontryagin’S Maximum Principle ◽  
Pontryagin's Maximum Principle ◽  
Multiple Integrals

scholarly journals Optimal Speed Profile Determination with Fixed Trip Time in the Electric Train Operation of the Cat Linh-Ha Dong Metro Line based on Pontryagin's Maximum Principle

2020 ◽  
Vol 10 (6) ◽  
pp. 6488-6493
Author(s):  
T. T. T. A. Anh ◽  
N. V. Quyen
Keyword(s):  
Energy Consumption ◽  
Maximum Principle ◽  
Pontryagin’S Maximum Principle ◽  
Speed Profile ◽  
Saving Energy ◽  
Pontryagin's Maximum Principle ◽  
Improve Energy Efficiency ◽  
Optimal Speed ◽  
Trip Time ◽  
Train Operation

The significant energy consumption for railway electric transportation operation poses a great challenge in outlining saving energy solutions. Speed profile optimization based on optimal control theory is one of the most common methods to improve energy efficiency without the railway infrastructure investment costs. The paper proposes an optimization method based on Pontryagin's Maximum Principle (PMP), not only to find optimal switching points in three operation phases: accelerating, coasting, braking, and from these switching points being able to determine the optimal speed profile, but also to ensure fixed-trip time. In order to determine trip time abiding by the scheduled timetables by applying nonlinear programming puts the Lagrange multiplier λ in the objective function regarded as a time constraint condition. The correctness and energy effectiveness of this method have been verified by the simulation results with data collected from the electrified trains of the Cat Linh-Ha Dong metro line in Vietnam. The saving energy levels are compared in three scenarios: electrified train operation tracking the original speed profile (energy consumption of the route: 144.64kWh), train operation tracking the optimal speed profile without fixed-trip time (energy consumption of the route: 129.18kWh), and train operation tracking the optimal speed profile and fixed trip time (energy consumption of the route: 132.99kWh) in an effort to give some useful choices for operating metro lines.

Using Pontryagin's Maximum Principle to Solve One-Dimensional Optimization Problems, with and without Constraints, on an Iterative Analog Computer

SIMULATION ◽  
1965 ◽  
Vol 4 (6) ◽  
pp. 382-389 ◽  
Author(s):  
Hans L. Steinmetz
Keyword(s):  
Maximum Principle ◽  
Differential Equations ◽  
Time Domain ◽  
Pontryagin’S Maximum Principle ◽  
Analog Computer ◽  
Solution Time ◽  
Pontryagin's Maximum Principle ◽  
Computer Technique ◽  
One Dimensional ◽  
Time Required

An analog computer technique is presented which enables application of Pontryagin's maximum prin ciple to the problem of optimizing control systems. The key problem in using Pontryagin's maximum principle is the extremization of the Hamiltonian function at every instant of time. Since the analog computer is an excellent differential equation solver, it is of advantage to convert this task into a dynamic problem. The technique used to do this is based upon the steepest ascent method. The method is applied to a one-dimensional control problem; higher-di mensional control problems can be treated using the same approach. The argument that an analog computer can solve differential equations with only one independent variable, corresponding to machine time, is true only in a technical sense. In practice it is feasible for cer tain types of problems to integrate one set of differ ential equations sufficiently fast enough so that, while integrating another set of differential equations at a much slower rate, the solution error associated with this approach remains within acceptable limits. When using the analog computer in this way, one time domain always corresponds to the solution time required for solving the differential equations de scribing the system; a second time domain corre sponds to the solution time required for solving an auxiliary set of differential equations which has no direct relationship with the system. Technological improvements and innovations made in the analog computer field during the recent past have contributed to the successful application of this approach.

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