Proofs of the Technical Results Justifying an Algorithm of Extremum Seeking Navigation in Dynamic Environmental Fields

We consider a single kinematically controlled robot with a bounded control range. The robot travels in a two-dimensional region supporting an unknown unsteady scalar field. A single sensor provides the field value at the current location of the robot. The paper is concerned with a reactive navigation strategy to drive the robot to the time-varying location where the field attains its spatial maximum, with subsequent keeping the robot close to this location. This strategy is based on a novel paradigm of kinematic control different from conventionally trying to align the velocity vector with the field gradient, does not employ estimation of the entire field gradient or derivative-dependent quantities, like the rate at which the available field reading evolves over time, and is non-demanding with respect to both computation and motion. The paper presents the proofs of the technical facts required to justify convergence of the proposed algorithm.