Control of the Search for Observation Objects from a Spatio-Temporal Poisson Flow in a Multi-Channel Search System

Abstract

The article considers the problem of searching for objects of observation for the case, when the sequence of their appearance satisfies the laws of spatial and temporal Poisson flow. Its solution is obtained without taking into account the limitations associated with the significant excess of the search effort intensity over the intensity of the flow of observation objects. As a mathematical model, used for optimization of the search, the system of differential equations describing dynamics of changing of mathematical expectation of number of objects present in subdomains of the search system’s field of view, but not yet detected. A procedure for optimizing the distribution of search effort intensity in search system channels for of dynamic and steady state search modes. Presented are examples.

Appendices

APPENDIX A

1.1 Sequence of Transformation of Differential Kolmogorov Equations (2.5)

Let us multiply the kth equation of each ith (i = $\overline {1,I}$) of the system (2.5) by k (k = 1, 2, …) and sum them by k. As a result, we obtain

$$\sum\limits_{k = 1}^\infty {{{{\dot {P}}}_{{ik}}}k} = {{\lambda }_{i}}F + {{\xi }_{i}}G,$$

(A.1)

where $\sum\nolimits_{k = 1}^\infty {{{{\dot {P}}}_{{ik}}}k}$ = ${{\dot {\mu }}_{i}}$ is the rate of change of the mathematical expectation of the number of undetected OOs in X_i, i = $\overline {1,I}$:

$$G = - \sum\limits_{k = 1}^\infty {{{P}_{{ik}}}k} + \sum\limits_{k = 1}^\infty {{{P}_{{ik - 1}}}k} ,$$

(A.2)

$${{F}_{i}} = - \sum\limits_{k = 1}^\infty {{{P}_{{ik}}}{{k}^{2}}} + \sum\limits_{k = 1}^\infty {{{P}_{{ik + 1}}}{{k}^{2}}} + \sum\limits_{k = 1}^\infty {{{P}_{{ik + 1}}}k.} $$

(A.3)

By revealing the sums in (A.2), it can be shown that

$$G = \sum\limits_{k = 0}^\infty {{{P}_{{ik}}}} = 1.$$

(A.4)

Carrying out a similar operation for (A.3), we obtain

$${{F}_{i}} = - \sum\limits_{k = 1}^\infty {{{P}_{{ik}}}k} = - {{\mu }_{i}}.$$

(A.5)

From (A.1), (A.4), (A.5) it follows that

$${{\dot {\mu }}_{i}} = - {{\lambda }_{i}}(t){{\mu }_{i}} + {{\xi }_{i}}(t),\quad i = \overline {1,I} .$$

(A.6)

The description of the system of differential Eqs. (A.6) corresponds to vector notation (2.6).

APPENDIX B

1.1 Rationale for Inequality (5.11)

From (5.11) it follows that

$$I\sum\limits_{i = 1}^I {{{\xi }_{i}}} \geqslant \sum\limits_{i = 1}^I {{{\xi }_{i}}} + 2\sum\limits_{j = 1}^{I - 1} {\sum\limits_{i = j + 1}^I {\sqrt {{{\xi }_{j}}{{\xi }_{i}}} } } $$

(B.1)

or

$$(I - 1)\sum\limits_{i = 1}^I {{{\xi }_{i}}} \geqslant 2\sum\limits_{j = 1}^{I - 1} {\sum\limits_{i = j + 1}^I {\sqrt {{{\xi }_{j}}{{\xi }_{i}}} .} } $$

(B.2)

Let us write a system of inequalities

$$\begin{gathered} {{\xi }_{1}} + {{\xi }_{2}} \geqslant 2\sqrt {{{\xi }_{1}}{{\xi }_{2}}} , \hfill \\ {{\xi }_{1}} + {{\xi }_{3}} \geqslant 2\sqrt {{{\xi }_{1}}{{\xi }_{3}}} , \hfill \\ \ldots \ldots \ldots \ldots \ldots \ldots \ldots \hfill \\ {{\xi }_{{I - 1}}} + {{\xi }_{I}} \geqslant 2\sqrt {{{\xi }_{{I - 1}}}{{\xi }_{I}}} . \hfill \\ \end{gathered} $$

(B.3)

Summing up the inequalities in (B.3), we obtain (B.2) and, respectively (B.1). Equation (5.11) is valid.