Modelling of extremal fuzzy dynamic systems: Part V. Optimization of continuous controllable extremal fuzzy processes and the choice of decisions

Abstract

This work deals with the problems of optimization of continuous controllable extremal fuzzy processes that have been presented in parts I, II and III of the current paper. A separate consideration is given to the case where an extremal fuzzy control process acting on an extremal fuzzy dynamic system (EFDS) (i) depends and (ii) does not depend on an EFDS state.

Sufficient and necessary conditions for the existence of an optimal control are proved. Applying Bellman's optimality principle and assuming that the gain–loss process exists for an EFDS, a variant of the fuzzy integral representation of an optimal control is given for continuous EFDSs. This variant employs the instrument of extended extremal fuzzy composition measures constructed in parts I and II of this work.

Finally, the questions of defining a fuzzy gain relation for an EFDS are considered, taking into account the available expert knowledge on the EFDS subject matter.

Keywords:

• Extremal fuzzy dynamic system
• Extremal fuzzy control process
• Gain–loss process
• R. Bellman's optimality principle
• Extremal fuzzy composition measures
• Sugeno integral

Acknowledgements

The author is grateful to reviewers for their helpful recommendations.