A direct approach for the solution of nonlinear optimal control problems with multiple delays subject to mixed state-control constraints

• Published in: Applied Mathematical Modelling Vol. 53; pp. 189 - 213
• Main Authors: Marzban, H.R., Pirmoradian, H.
• Format: Journal Article
• Language: English
• Published: New York Elsevier Inc 01.01.2018; Elsevier BV
• Subjects: Block-pulse functions; Computer simulation; Hybrid functions; Interpolation; Lagrange interpolation; Lagrange multiplier; Multiple delays; Nonlinear control; Nonlinear optimal control; Operational matrix; Optimal control; Optimization; Penalty function; Reliability; Trajectory control; Validity; Operational matrix; Multiple delays; Hybrid functions; Lagrange interpolation; Nonlinear optimal control; Block-pulse functions
• ISSN: 0307-904X; 1088-8691; 0307-904X
• DOI: 10.1016/j.apm.2017.08.025

•Nonlinear constrained optimal control problems involving multiple delays are investigated.•Our approach is based upon a hybrid of block-pulse functions and Lagrange interpolation.•The operational matrix of delay corresponding to the proposed framework is constructed.•An upper bound on the error with respect to the maximum norm is established.•Several examples are included to verify the validity and reliability of the procedure. This paper deals with the numerical investigation of nonlinear optimal control problems with multiple delays in which the state trajectory and control input are subject to mixed state-control constraints. A direct approach based on a hybrid of block-pulse functions and Lagrange interpolation is proposed. The constrained optimal control problem is first reformulated as an unconstrained optimization one using a penalty function technique. The resulting optimization problem is then solved by means of the Lagrange multipliers procedure. The proposed framework is an extension and also a modification of the conventional Lagrange interpolation. Combining block-pulse functions and Lagrange interpolation allows one to simultaneously make use the advantages of the two mentioned bases. The operational matrices of delay and derivative associated with the hybrid functions are presented. An upper error bound for the proposed hybrid functions with respect to the maximum norm is obtained. Simulation studies are provided to verify the validity and reliability of the developed procedure.