Pointwise maximum principle for convex optimal control problems with mixed control-phase variable inequality constraints

1980 ◽  
Vol 30 (2) ◽  
pp. 269-291 ◽  
Author(s):  
M. Köhler
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Inequality Constraints ◽  
Control Problems ◽  
Phase Variable ◽  
Control Phase ◽  
Mixed Control ◽  
Pointwise Maximum Principle ◽  
Pointwise Maximum ◽  
Variable Inequality

A Simple Continuation Method for the Solution of Optimal Control Problems With State Variable Inequality Constraints

2013 ◽  
Author(s):  
Brian C. Fabien
Keyword(s):  
Optimal Control ◽  
Pure State ◽  
Original Problem ◽  
Optimal Control Problems ◽  
Necessary Conditions ◽  
Continuation Method ◽  
Inequality Constraints ◽  
Control Problems ◽  
State Variable ◽  
Variable Inequality

This paper develops a simple continuation method for the approximate solution of optimal control problems with pure state variable inequality constraints. The method is based on transforming the inequality constraints into equality constraints using nonnegative slack variables. The resultant equality constraints are satisfied approximately using a quadratic loss penalty function. 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 necessary conditions for a minimum are written as a boundary value problem involving index-1 differential-algebraic equations (BVP-DAE). The BVP-DAE include the complementarity conditions associated with the inequality constraints. The paper shows that the necessary conditions for optimality of the original problem and the transformed problems are remarkably similar. In particular, the BVP-DAE for each problem differ by a linear term related to the Lagrange multipliers associated with the state variable inequality constraints. Numerical examples are presented to illustrate the efficacy of the proposed technique. Specifically, the paper presents results for; (1) the optimal control of a simplified model of a gantry crane system, (2) the optimal control of a rigid body moving in the vertical plane, and (3) the trajectory optimization of a planar two-link robot. All problems include pure state variable inequality constraints.

A weak maximum principle for optimal control problems with mixed constraints under a constant rank condition

2020 ◽  
Vol 37 (3) ◽  
pp. 1021-1047
Author(s):  
Roberto Andreani ◽  
Valeriano Antunes de Oliveira ◽  
Jamielli Tomaz Pereira ◽  
Geraldo Nunes Silva
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Optimal Control Problems ◽  
Necessary Optimality Conditions ◽  
Inequality Constraints ◽  
Equality Constraints ◽  
Control Problems ◽  
Constant Rank ◽  
Rank Condition ◽  
Weak Maximum Principle

Abstract Necessary optimality conditions for optimal control problems with mixed state-control equality constraints are obtained. The necessary conditions are given in the form of a weak maximum principle and are obtained under (i) a new regularity condition for problems with mixed linear equality constraints and (ii) a constant rank type condition for the general non-linear case. Some instances of problems with equality and inequality constraints are also covered. Illustrative examples are presented.

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