This paper investigates the optimal control of dynamic systems subject to fuzzy uncertainties in the initial conditions. By applying Zadeh’s Extension Principle, we interpret the evolution of the fuzzy membership function as the solution to a partial differential equation (PDE) of advection type. This formulation allows us to transform the problem of controlling a fuzzy differential equation into a deterministic optimal control problem for the associated PDE. We derive the first-order necessary optimality conditions — which constitute an extension of the Pontryagin Maximum Principle to the fuzzy setting — obtained via Lagrange multipliers applied to the equivalent PDE formulation, and illustrate the approach with worked examples, demonstrating how to steer the possibility distribution of the system state towards a desired target.
Cecconello et al. (Fri,) studied this question.