On the optimal control of stationary diffusion processes with inaccessible boundaries and no discounting

1971 ◽  
Vol 8 (3) ◽  
pp. 551-560 ◽  
Author(s):  
R. Morton
Keyword(s):  
Optimal Control ◽  
Dynamic Programming ◽  
Diffusion Processes ◽  
Necessary Conditions ◽  
Optimal Solution ◽  
Multidimensional Case ◽  
Potential Cost ◽  
One Dimensional ◽  
Dynamic Programming Equation ◽  
Stationary Diffusion

SummaryBecause there are no boundary conditions, extra properties are required in order to identify the correct potential cost function. A solution of the Dynamic Programming equation for one-dimensional processes leads to an optimal solution within a wide class of alternatives (Theorem 1), and is completely optimal if certain conditions are satisfied (Theorem 2). Necessary conditions are also given. Several examples are solved, and some extension to the multidimensional case is shown.

On the optimal control of stationary diffusion processes with inaccessible boundaries and no discounting

1971 ◽  
Vol 8 (03) ◽  
pp. 551-560 ◽  
Author(s):  
R. Morton
Keyword(s):  
Optimal Control ◽  
Dynamic Programming ◽  
Diffusion Processes ◽  
Necessary Conditions ◽  
Optimal Solution ◽  
Multidimensional Case ◽  
Potential Cost ◽  
One Dimensional ◽  
Dynamic Programming Equation ◽  
Stationary Diffusion

Summary Because there are no boundary conditions, extra properties are required in order to identify the correct potential cost function. A solution of the Dynamic Programming equation for one-dimensional processes leads to an optimal solution within a wide class of alternatives (Theorem 1), and is completely optimal if certain conditions are satisfied (Theorem 2). Necessary conditions are also given. Several examples are solved, and some extension to the multidimensional case is shown.

On the optimal control of stationary diffusion processes with natural boundaries and discounted cost

1971 ◽  
Vol 8 (3) ◽  
pp. 561-572 ◽  
Author(s):  
R. Morton
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Cost Function ◽  
Diffusion Processes ◽  
Expected Cost ◽  
Discounted Cost ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Stationary Diffusion ◽  
Natural Boundaries

SummaryResults similar to those in [3] are obtained for one-dimensional diffusion processes with discounted cost. The stronger assumption that at least one of the inaccessible boundaries is natural enables us to identify the solution of a differential equation corresponding to the future expected cost function.

On the optimal control of stationary diffusion processes with natural boundaries and discounted cost

1971 ◽  
Vol 8 (03) ◽  
pp. 561-572
Author(s):  
R. Morton
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Cost Function ◽  
Diffusion Processes ◽  
Expected Cost ◽  
Discounted Cost ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Stationary Diffusion ◽  
Natural Boundaries

Summary Results similar to those in [3] are obtained for one-dimensional diffusion processes with discounted cost. The stronger assumption that at least one of the inaccessible boundaries is natural enables us to identify the solution of a differential equation corresponding to the future expected cost function.

scholarly journals An improved heuristic-dynamic programming algorithm for rectangular cutting problem

2020 ◽  
Vol 17 (3) ◽  
pp. 717-735
Author(s):  
Aihua Yin ◽  
Chong Chen ◽  
Dongping Hu ◽  
Jianghai Huang ◽  
Fan Yang
Keyword(s):  
Dynamic Programming ◽  
Dynamic Programming Algorithm ◽  
Optimal Solution ◽  
Computation Time ◽  
Programming Algorithm ◽  
Recursive Method ◽  
Two Dimensional ◽  
One Dimensional ◽  
Block Width ◽  
Cutting Problem

In this paper, the two-dimensional cutting problem with defects is discussed. The objective is to cut some rectangles in a given shape and direction without overlapping the defects from the rectangular plate and maximize some profit associated. An Improved Heuristic-Dynamic Program (IHDP) is presented to solve the problem. In this algorithm, the discrete set contains not only the solution of one-dimensional knapsack problem with small rectangular block width and height, but also the cutting positions of one unit outside four boundaries of each defect. In addition, the denormalization recursive method is used to further decompose the sub problem with defects. The algorithm computes thousands of typical instances. The computational experimental results show that IHDP obtains most of the optimal solution of these instances, and its computation time is less than that of the latest literature algorithms.

scholarly journals A Two-Stage Method for UCAV TF/TA Path Planning Based on Approximate Dynamic Programming

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Hao-xiang Chen ◽  
Ying Nan ◽  
Yi Yang
Keyword(s):  
Optimal Control ◽  
Dynamic Programming ◽  
Path Planning ◽  
Approximate Dynamic Programming ◽  
Optimal Solution ◽  
Planning Problem ◽  
Optimization Approach ◽  
Two Stage ◽  
Control Approach ◽  
Optimal Control Approach

We present a two-stage method for solving the terrain following (TF)/terrain avoidance (TA) path-planning problem for unmanned combat air vehicles (UCAVs). The 1st stage of planning takes an optimization approach for generating a 2D path on a horizontal plane with no collision with the terrain. In the 2nd stage of planning, an optimal control approach is adopted to generate a 3D flyable path for the UCAV that is as close as possible to the terrain. An approximate dynamic programming (ADP) algorithm is used to solve the optimal control problem in the 2nd stage by training an action network to approximate the optimal solution and training a critical network to approximate the value function. Numerical simulations indicate that ADP can determine the optimal control variables for UCAVs; relative to the conventional optimization method, the optimal control approach with ADP shows a better performance under the same conditions.

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