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We consider a single non-holonomic Dubins-like robot traveling with aconstant longitudinal speed in an a priori unknown and unsteady planarenvironment. The robot should detect, locate, and track the boundary of adynamic environmental scalar field. The field is measured by an on-board sensorin a point-wise fashion at the robot's location. The focus is on unsteadyboundaries that evolve over time in an arbitrary fashion, includingdeformations, i.e., changes of shapes and sizes. We present a sliding modecontrol method for localizing and tracking such boundaries: the robot issteered to the boundary and circulates in its close proximity afterwards. Theproposed control algorithm does not require estimation of the spatial gradientof the field and is non-demanding with respect to both computation and motion.The paper offers the proofs of technical facts required for rigorousjustification of non-local convergence of the proposed control law, as well astheoretical illustrations of its performance in specific scenarios.

Proofs of Technical Results Justifying and Illustrating an Algorithm of Navigation for Monitoring Unsteady Environmental Boundaries