Randomized Linear Search for a Drifted Brownian Target with Direction-Reversal Costs

This paper introduces a randomized linear-search model for a target moving on the real line according to Brownian motion with drift. The searcher alternates between the two directions through expanding random excursions, with a fixed cost incurred at each change in direction. The performance criterion combines the expected encounter time with the expected number of reversals. A survival-probability representation is derived for the total expected cost, together with sufficient and necessary integrability conditions for its finiteness. The existence of an optimal randomized strategy is also established within a compact parametric class. The symmetric zero-drift case is characterized by a reflection property, whereas nonzero drift may lead to asymmetric search behavior. In addition, the optimal cost is shown to be non-decreasing and concave in the reversal cost. The numerical study illustrates the effects of drift, the initial search direction, the expansion factor, and the reversal cost. Overall, the model provides a unified framework for randomized search for a moving target under operational direction-reversal costs.