Dynamic Programming and a Distributed Parameter Maximum Principle

1968 ◽  
Vol 90 (2) ◽  
pp. 152-156 ◽  
Author(s):  
W. L. Brogan
Keyword(s):  
Dynamic Programming ◽  
Maximum Principle ◽  
Programming Approach ◽  
Distributed Parameter ◽  
Programming Technique ◽  
Nonhomogeneous Boundary Condition ◽  
The Maximum Principle ◽  
Dynamic Programming Approach ◽  
Nonhomogeneous Boundary ◽  
Definition Of

A proof of a distributed parameter maximum principle is given by using dynamic programming. An example problem involving a nonhomogeneous boundary condition is also treated by using the dynamic programming technique and by extending the definition of the differential operator. It is thus demonstrated that for linear systems the dynamic programming approach is just as powerful as the variational approach originally used to derive the maximum principle.

Stochastic Hamilton's principle and noisy sliding surfaces in the tracking control of manipulators

Robotica ◽  
1995 ◽  
Vol 13 (2) ◽  
pp. 209-213
Author(s):  
Guy Jumarie
Keyword(s):  
Dynamic Programming ◽  
Tracking Control ◽  
Programming Approach ◽  
Hamilton’S Principle ◽  
Sliding Surface ◽  
Hamilton's Principle ◽  
Dynamic Programming Approach ◽  
Sliding Surfaces ◽  
Definition Of

SummaryIn the tracking control of manipulators via the sliding scheme, it may happen that sometimes, because of various inaccuracies, the definition of the actual sliding surface involves errors terms which may be either deterministic or, on the contrary, stochastic. This paper considers this last case and shows how one can estimate the new performances of the system so disturbed. A stochastic Hamilton's principle is applied, by combining the Lagrange parameter technique with results of the dynamic programming approach.

On the dynamic programming approach to Pontriagin's maximum principle

1968 ◽  
Vol 5 (3) ◽  
pp. 679-692 ◽  
Author(s):  
Richard Morton
Keyword(s):  
Dynamic Programming ◽  
Maximum Principle ◽  
The State ◽  
Dynamic Programming Principle ◽  
Programming Approach ◽  
Bellman Function ◽  
State Variables ◽  
Dynamic Programming Approach ◽  
Minimum Value ◽  
Image Position

Suppose that the state variables x = (x1,…,xn)′ where the dot refers to derivatives with respect to time t, and u ∊ U is a vector of controls. The object is to transfer x to x1 by choosing the controls so that the functional takes on its minimum value J(x) called the Bellman function (although we shall define it in a different way). The Dynamic Programming Principle leads to the maximisation with respect to u of and equality is obtained upon maximisation.

On the dynamic programming approach to Pontriagin's maximum principle

1968 ◽  
Vol 5 (03) ◽  
pp. 679-692
Author(s):  
Richard Morton
Keyword(s):  
Dynamic Programming ◽  
Maximum Principle ◽  
The State ◽  
Dynamic Programming Principle ◽  
Programming Approach ◽  
Bellman Function ◽  
State Variables ◽  
Dynamic Programming Approach ◽  
Minimum Value ◽  
Image Position

Suppose that the state variables x = (x 1,…,x n )′ where the dot refers to derivatives with respect to time t, and u ∊ U is a vector of controls. The object is to transfer x to x 1 by choosing the controls so that the functional takes on its minimum value J(x) called the Bellman function (although we shall define it in a different way). The Dynamic Programming Principle leads to the maximisation with respect to u of and equality is obtained upon maximisation.

Dynamic Airspace Configuration Using Approximate Dynamic Programming

2012 ◽  
Vol 2266 (1) ◽  
pp. 31-37 ◽  
Author(s):  
Sameer Kulkarni ◽  
Rajesh Ganesan ◽  
Lance Sherry
Keyword(s):  
Dynamic Programming ◽  
Traffic Control ◽  
Programming Approach ◽  
Programming Technique ◽  
Control Center ◽  
Traffic Demand ◽  
Dynamic Programming Approach ◽  
Dynamic Airspace Configuration ◽  
Maximum Benefit ◽  
Ground Delay

On the basis of weather and high traffic, the Next Generation Air Transportation System envisions an airspace that is adaptable, flexible, controller friendly, and dynamic. Sector geometries, developed with average traffic patterns, have remained structurally static with occasional changes in geometry due to limited forming of sectors. Dynamic airspace configuration aims at migrating from a rigid to a more flexible airspace structure. Efficient management of airspace capacity is important to ensure safe and systematic operation of the U.S. National Airspace System and maximum benefit to stakeholders. The primary initiative is to strike a balance between airspace capacity and air traffic demand. Imbalances in capacity and demand are resolved by initiatives such as the ground delay program and rerouting, often resulting in systemwide delays. This paper, a proof of concept for the dynamic programming approach to dynamic airspace configuration by static forming of sectors, addresses static forming of sectors by partitioning airspace according to controller workload. The paper applies the dynamic programming technique to generate sectors in the Fort Worth, Texas, Air Route Traffic Control Center; compares it with current sectors; and lays a foundation for future work. Initial results of the dynamic programming methodology are promising in terms of sector shapes and the number of sectors that are comparable to current operations.

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