### Abstract

Difficulties often arise when we apply the gradient type algorithms employing penalty functions to optimal control problems with variable final time. There is another class of optimal control problems for which the necessary conditions for optimality require a zero gradient at the final time. This causes the gradient-type algorithms, in their standard forms, to become incapable of changing the terminal value of the control variable at each iteration and the rate of convergence is adversely affected. In this paper, we first apply a new transformation method developed by Polak [19] which transforms the variable final time problem into a fixed final time problem. Second, an improved gradient-type algorithm is developed to overcome the zero terminal gradient problem. It is shown that, by applying this transformation and improved algorithm to four examples, not only the variable final time and zero terminal gradient problems are solved and the control vector updated in the correct direction but the rate of convergence of the improved algorithm is faster than that of the traditional gradient-type algorithms.

Original language | English (US) |
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Pages (from-to) | 355-362 |

Number of pages | 8 |

Journal | Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME |

Volume | 109 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1987 |

### ASJC Scopus subject areas

- Control and Systems Engineering
- Information Systems
- Instrumentation
- Mechanical Engineering
- Computer Science Applications