A Constraining Hyperplane Technique for State Variable Constrained Optimal Control Problems

1973 ◽  
Vol 95 (4) ◽  
pp. 380-389 ◽  
Author(s):  
K. Martensson
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Control Variable ◽  
Dynamic Programming Algorithm ◽  
Inequality Constraints ◽  
Programming Algorithm ◽  
Control Problems ◽  
State Control ◽  
Constrained Optimal Control ◽  
State Variable

A new approach to the numerical solution of optimal control problems with state-variable inequality constraints is presented. It is shown that the concept of constraining hyperplanes may be used to approximate the original problem with a problem where the constraints are of a mixed state-control variable type. The efficiency and the accuracy of the combination of constraining hyperplanes and a second-order differential dynamic programming algorithm are investigated on problems of different complexity, and comparisons are made with the slack-variable and the penalty-function techniques.

scholarly journals Constants within Error Estimates for Legendre-Galerkin Spectral Approximations of Control-Constrained Optimal Control Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Jianwei Zhou
Keyword(s):  
Optimal Control ◽  
Error Estimate ◽  
Optimal Control Problems ◽  
Control Variable ◽  
Control Problems ◽  
Constrained Optimal Control ◽  
One Dimensional ◽  
Error Indicator ◽  
Spectral Approximations ◽  
Explicit Formulae

Explicit formulae of constants within the aposteriorierror estimate for optimal control problems are investigated with Legendre-Galerkin spectral methods. The constrained set is put on the control variable. For simpleness, one-dimensional bounded domain is taken. Meanwhile, the corresponding aposteriorierror indicator is established with explicit constants.

scholarly journals The Constants in A Posteriori Error Indicator for State-Constrained Optimal Control Problems with Spectral Methods

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Jianwei Zhou
Keyword(s):  
Optimal Control ◽  
Spectral Methods ◽  
Optimal Control Problems ◽  
Posteriori Error ◽  
Control Problems ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Constrained Optimal Control ◽  
Error Indicator ◽  
State Variable

We employ Legendre-Galerkin spectral methods to solve state-constrained optimal control problems. The constraint on the state variable is an integration form. We choose one-dimensional case to illustrate the techniques. Meanwhile, we investigate the explicit formulae of constants within a posteriori error indicator.

Multi-item Optimal control problem with fuzzy costs and constraints using Fuzzy variational principle

2019 ◽  
Vol 53 (3) ◽  
pp. 1061-1082
Author(s):  
Jotindra Nath Roul ◽  
Kalipada Maity ◽  
Samarjit Kar ◽  
Manoranjan Maiti
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Variational Principles ◽  
Control Variable ◽  
Equality Constraint ◽  
Raw Material ◽  
Control Problems ◽  
Constrained Optimal Control ◽  
State Variable

An imperfect multi-item production system is considered against time dependent demands for a finite time horizon. Here production is defective. Following [Khouja and Mehrez J. Oper. Res. Soc. 45 (1994) 1405–1417], unit production cost depends on production, raw-material and maintenance costs. Produced items have same fixed life-time. Warehouse capacity is limited and used as a constraint. Available space, production, stock and different costs are assumed as crisp or imprecise. With the above considerations, crisp and fuzzy constrained optimal control problems are formulated for the minimization of total cost consisting of raw-material, production and holding costs. These models are solved using conventional and fuzzy variational principles with equality constraint condition and no-stock as end conditions. For the first time, the inequality space constraint is converted into an equality constraint introducing a pseudo state variable following Bang Bang control. [Roul et al., J. Intell. Fuzzy Syst. 32 (2017) 565–577], as stock is mainly controlled by production, for the control problems production is taken as the control variable and stock as state variable. The reduced optimal control problem is solved by generalised reduced gradient method using Lingo-11.0. The models are illustrated numerically. For the fuzzy model, optimum results are obtained as fuzzy numbers expressed by their membership functions. From fuzzy results, crisp results are derived using α-cuts.

Indirect Solution of Inequality Constrained and Singular Optimal Control Problems Via a Simple Continuation Method

2013 ◽  
Vol 136 (2) ◽  
Author(s):  
Brian C. Fabien
Keyword(s):  
Optimal Control ◽  
Original Problem ◽  
Optimal Control Problems ◽  
Necessary Conditions ◽  
Continuation Method ◽  
Singular Control ◽  
Inequality Constraints ◽  
Control Problems ◽  
Quadratic Loss ◽  
State Variable

This paper develops a simple continuation method for the approximate solution of optimal control problems. The class of optimal control problems considered include (i) problems with bounded controls, (ii) problems with state variable inequality constraints (SVIC), and (iii) singular control problems. The method used here is based on transforming the state variable inequality constraints into equality constraints using nonnegative slack variables. The resultant equality constraints are satisfied approximately using a quadratic loss penalty function. Similarly, singular control problems are made nonsingular using a quadratic loss penalty function based on the control. The solution of the original problem is obtained by solving the transformed problem with a sequence of penalty weights that tends to zero. The penalty weight is treated as the continuation parameter. The paper shows that the transformed problem yields necessary conditions for a minimum that can be written as a boundary value problem involving index-1 differential–algebraic equations (BVP-DAE). The BVP-DAE includes the complementarity conditions associated with the inequality constraints. It is also shown that the necessary conditions for optimality of the original problem and the transformed problem differ by a term that depends linearly on the algebraic variables in the DAE. Numerical examples are presented to illustrate the efficacy of the proposed technique.

Sign in / Sign up

Export Citation Format

Share Document