Part 1: Hybrid Search

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Modeling Framework

Readers familiar with stochastic (or search) theory can safely omit this section, and jump to results part below.

In the existing literature, such as Phelps et al. [2014], the target is typically modeled with an uncertain initial position but deterministic motion. In these formulations, the initial state is explicitly characterized by a probability density function (PDF), while the subsequent motion follows a known differential equation.

In contrast, we model the target motion as a stochastic process $\lbrace \mathrm{x}(t) \mid t \in [0, t_f]\rbrace$. The evolution of $\mathrm{x}(t)$ is described by a stochastic differential equation (SDE) [see (1) below]. Figure 3 provides a representative comparison between traditional target modeling approaches and the method proposed in this work.

$$ \mathrm{d}\mathrm{x}(t) = \mathrm{v}(\mathrm{x}, t)\, \mathrm{d}t + D(\mathrm{x}, t)\, \mathrm{d} \omega(t), \tag{1} $$

here $\omega(t)$ is a vector Brownian motion process. Examples of the drift term $\mathrm{v}(\mathrm{x}, t)$ and diffusion term $D(\mathrm{x}, t)$ can be found in Zhao and Bewley [2025] and are thus omitted here.

Figure 2. Animated illustration of target motion and searcher trajectories. The red and blue curves represent the searchers, while the yellow, purple, and green paths denote the traditional, non-evasive, and evasive targets, respectively.

It is well known that the PDF of a target whose motion is governed by a SDE satisfies Fokker-Planck equation Jazwinski [2013]. In this work, we extend this classical result by proposing a forced Fokker-Planck equation that captures the time evolution of PDF under the influence of search [see (2) below]. The derivation of (2) follows the approach outlined in Theorem 1 of Hellman [1970].

$$ \frac{\partial p}{\partial t} - \nabla\cdot\big(D\cdot\nabla p + p\, \nabla\cdot D - \mathrm{v}\,p\big) = p\, \bigg(\int_\Omega \phi\, p\,\mathrm{d}\mathrm{x} - \phi\bigg), \tag{2} $$

here $\phi(\mathrm{x}, t)$ is the instantaneous search density function [see Zhao and Bewley [2025]]. Finally, we conclude this section by highlighting the core assumption made regarding the target dynamics in Zhao and Bewley [2025].

Assumption In the absence of searchers, the PDF of target’s position is statistically stationary.

Problem formulation

This hybrid system is governed by:

  • A Fokker-Planck equation modeling the time evolution of the PDF $p(\mathrm{x}, t)$,
$$ \frac{\partial p}{\partial t} - \nabla\cdot\big(D\cdot\nabla p + p\, \nabla\cdot D - \mathrm{v}\,p\big) = 0. $$
  • Searchers’ dynamics $\dot{\mathrm{q}}_m(t) = \mathrm{g}_m(\mathrm{q}_m, \mathrm{u}_m)$, for $m=1,\ldots,M$.
  • A Bayesian rule updating the information collected by searchers at each observation time $t_k$,
$$ \phi({\bf x},\, t_k) = \prod_{m=1}^M\phi_m({\bf x},\, t_k), \quad \phi_m({\bf x},\, t_k) = 1 - \alpha_m\, \exp\big(-\beta_m\, \| {\bf x} - E_m\, {\bf q}_m(t_k)\|^2 \big). $$
  • An objective functional that maximizes the probability of finding the target while penalizing control efforts
$$ J = \left( \int_\Omega \bigl[ p^+({\bf x},\, t_f) \bigr]^2\, \mathrm{d}{\bf x} \right)^{\frac12} + \int_0^{t_f} \sum_{m=1}^M {\bf u}_m^T\, R_m\, {\bf u}_m\, \mathrm{d}t. $$
  1. B. L. Hanson, M. Zhao and T. R. Bewley, An extensible framework for the probabilistic search of stochastically-moving targets characterized by generalized Gaussian distributions or experimentally-defined regions of interest, Communications in Statistics-Theory and Methods (2025): 1-26.
  2. Jazwinski, Andrew H, Stochastic processes and filtering theory., Courier Corporation, 2013.