Adaptive discounted optimization for multidimensional random-walk targets in a two-stage stochastic transport model: Coverage and finiteness of first-encounter times

Mohamed Abd Allah El-Hadidy; Alaa Awad Alzulaibani; Mohamed Abd Allah El-Hadidy; Alaa Awad Alzulaibani

doi:10.3934/math.20261335

AIMS Mathematics

2026, Volume 11, Issue 10: 34215-34251. doi: 10.3934/math.20261335

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Research article

Adaptive discounted optimization for multidimensional random-walk targets in a two-stage stochastic transport model: Coverage and finiteness of first-encounter times

Mohamed Abd Allah El-Hadidy ^{1
,
,},
Alaa Awad Alzulaibani ²

1.
Mathematics Department, Faculty of Science, Tanta University, Tanta 31527, Egypt
2.
Department of Mathematics and Statistics, College of Science in Yanbu, Taibah University, Yanbu Governorate, Saudi Arabia

Received: 19 August 2026 Revised: 30 September 2026 Accepted: 09 October 2026 Published: 21 October 2026
MSC : 49J20, 60G50, 60J10, 60J45, 60K35, 90B40

This paper developed an adaptive discounted two-stage stochastic transport model for locating and tracking a target governed by a multidimensional random walk. In the first stage, a discounted effort–reward criterion is incorporated into the regional search process to balance localization time, traveled distance, and detection probability. In the second stage, coordinated searchers track the target along intersecting trajectories while the target continues to move according to the multidimensional random-walk process. The tracking discount is adjusted according to the Stage-1 shell uncertainty, thereby linking the two stages within a unified framework. Coverage-preserving tracking distances are constructed to maintain recurrent exploration. Sufficient conditions are established for the existence of an optimal policy and complete coverage, while finiteness of the expected first-encounter time is obtained under an additional uniform positive finite-block encounter-probability assumption. The undiscounted model is recovered as the discount parameters approach 1. Numerical results illustrate the effects of discounting, uncertainty, and dimensionality on search effort and encounter performance.
- adaptive discounting,
- two-stage stochastic transport,
- multidimensional random walk,
- first-encounter time,
- coverage preservation
Citation: Mohamed Abd Allah El-Hadidy, Alaa Awad Alzulaibani. Adaptive discounted optimization for multidimensional random-walk targets in a two-stage stochastic transport model: Coverage and finiteness of first-encounter times[J]. AIMS Mathematics, 2026, 11(10): 34215-34251. doi: 10.3934/math.20261335

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Abstract

This paper developed an adaptive discounted two-stage stochastic transport model for locating and tracking a target governed by a multidimensional random walk. In the first stage, a discounted effort–reward criterion is incorporated into the regional search process to balance localization time, traveled distance, and detection probability. In the second stage, coordinated searchers track the target along intersecting trajectories while the target continues to move according to the multidimensional random-walk process. The tracking discount is adjusted according to the Stage-1 shell uncertainty, thereby linking the two stages within a unified framework. Coverage-preserving tracking distances are constructed to maintain recurrent exploration. Sufficient conditions are established for the existence of an optimal policy and complete coverage, while finiteness of the expected first-encounter time is obtained under an additional uniform positive finite-block encounter-probability assumption. The undiscounted model is recovered as the discount parameters approach 1. Numerical results illustrate the effects of discounting, uncertainty, and dimensionality on search effort and encounter performance.

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