Controlled Search for Targets Arriving According to a Spatio-Temporal Poisson Point Process by an Information Measurement System with Inhomogeneous Scope

Abstract

We consider the surveillance target search problem for the case in which the arrival of targets is modeled by a spatio-temporal Poisson point process and the information measurement system scope is inhomogeneous. A probabilistic description of the inhomogeneity subdomains is given, and patterns of taking them into account when forming the a posteriori intensity density of a random point process are revealed. A search control law synthesis procedure for the case of an inhomogeneous scope is defined. An example is given.

APPENDIX

Proof of the Theorem. Consider a subdomain of the scope,

$$ X_{\mathrm {Ev}} (x)\in X, $$

(A.1)

where $X_{\mathrm {Ev}} (x)$ is a small neighborhood of point $x$ with Lebesgue measure $\left |X_{\mathrm {Ev}} \right |\ll 1$, and for this neighborhood, let us define two hypotheses

$$ \begin {gathered} H_{0} :\;X_{\mathrm {Ev}} (x)\notin \bigcup _{k}D_{k} (t) ,\quad H_{1} :\; X_{\mathrm {Ev}} (x)\in \bigcup _{k}D_{k} (t) ,\\ t\in [t,t+\Delta t],\quad \Delta t\ll 1, \end {gathered} $$

(A.2)

for which $P\{ H_{0} \} =P_{0}$, $P\{ H_{1} \} =P_{1}$, $P_{0} +P_{1} =1$.

By $A$ we denote the event associated with the arrival of subsequent surveillance targets in $X_{\mathrm {Ev}} (x)$ from the point process on the time interval $[t,t+\Delta t]$. It is associated with the two conditional probabilities $P(A/H_{0} )$ and $P(A/H_{1} )$. Then we have [13]

$$ P(A)=P(A/H_{0} )P_{0} +\; P(A/H_{1} )P_{1} . $$

(A.3)

Under the assumption of impossibility of the arrival of the next surveillance target in the spatial domains occupied by tracked targets, it is assumed that $P(A/H_{1})=0$; then

$$ P(A)=P(A/H_{0} )(1-P_{1} ). $$

(A.4)

Let us determine the values of $P(A)$ and $P(A/H_{0} )$,

$$ P(A)=P\Big \{ \varphi _{q\mathrm {Ev}} (x,t+\Delta t)-\varphi _{q\mathrm {Ev}} (x,t)=1,2,3, \ldots \Big \} =1-\exp \left \{-\int \limits _{t}^{t+\Delta t}\tilde {\xi }_{q\mathrm {Ev}} (x,t)dt \right \}, $$

(A.5)

$$ P(A/H_{0} )=P\Big \{ \varphi _{\mathrm {Ev}} (x,t+\Delta t)-\varphi _{\mathrm {Ev}} (x,t)=1,2,3, \ldots \Big \} =1-\exp \left \{-\int \limits _{t}^{t+\Delta t}\xi _{\mathrm {Ev}} (x,t)dt \right \}, $$

(A.6)

where $\varphi _{q\mathrm {Ev}} (x,t)=\varphi _{q} (X_{\mathrm {Ev}}(x),t)$ and $\varphi _{\mathrm {Ev}} (x,t)=\varphi (X_{\mathrm {Ev}}(x),t)$; $\tilde {\xi }_{q\mathrm {Ev}} (x,t)$ and $\xi _{\mathrm {Ev}} (x,t)$ are the a posteriori and a priori intensity measures of the random point process in $X_{\mathrm {Ev}} (x)$ for $P\{H_{1} \} =P_{1}$ and $P\{ H_{0} \} =1$, respectively.

Taking into account the smallness of $\left |X_{\mathrm {Ev}} \right |$, we obtain

$$ \begin {aligned} \xi _{\mathrm {Ev}} (x,t)&=\int \limits _{X_{\mathrm {Ev}} (x)}\nu (x,t)dx= \nu (x,t)\left |X_{\mathrm {Ev}} \right |+o(\left |X_{\mathrm {Ev}} \right |), \\ \tilde {\xi }_{q\mathrm {Ev}} (x,t)&=\int \limits _{X_{\mathrm {Ev}} (x)}\tilde {\nu }_{q} (x,t)dx= \tilde {\nu }_{q} (x,t)\left |X_{\mathrm {Ev}} \right |+o(\left |X_{\mathrm {Ev}} \right |), \end {aligned}$$

(A.7)

where $o(\left |X_{\mathrm {Ev}} \right |)$ is a remainder of the order of smallness of at most $\left |X_{\mathrm {Ev}} \right |$.

In view of (A.5) and (A.6), relation (A.4) acquires the form

$$ 1 - \exp \left \{ - \int \limits _{t}^{t+\Delta t} \tilde {\xi }_{q\mathrm {Ev}} (x,t)dt \right \} = \left \{ 1 - \exp \left \{ - \int \limits _{t}^{t+\Delta t} \xi _{\mathrm {Ev}} (x,t)dt \right \} \right \}(1 - P_{1}).$$

(A.8)

Let us expand the left- and right-hand sides of (A.8) in a series in a neighborhood of $t$. Taking into account the condition $\Delta t\ll 1$, we will obtain

$$ \tilde {\xi }_{q\mathrm {Ev}} (x,t)\Delta t+o(\Delta t)=\big (\xi _{\mathrm {Ev}} (x,t)\Delta t+o(\Delta t)\big )(1-P_{1} ). $$

(A.9)

Divide the left- and right-hand sides of (A.9) by $\Delta t$ and pass to the limit as $\Delta t\to 0$. Since $\lim \limits _{\Delta t\to 0} \frac {o(\Delta t)}{\Delta t} =0$, it follows from (A.9) that

$$ \tilde {\xi }_{q\mathrm {Ev}} (x,t)=\xi _{\mathrm {Ev}} (x,t)(1-P_{1} ).$$

(A.10)

Substituting the values from (A.7) into (A.10), we obtain

$$ \tilde {\nu }_{q} (x,t)\left |X_{\mathrm {Ev}} \right |+o\big (\left |X_{\mathrm {Ev}} \right |\big )=\Big (\nu (x,t)\left |X_{\mathrm {Ev}} \right |+o\big (\left |X_{\mathrm {Ev}} \right |\big )\Big )(1-P_{1} ). $$

(A.11)

Dividing the left- and right-hand sides of (A.10) by $\left |X_{\mathrm {Ev}}\right |$, passing to the limit as $\left |X_{\mathrm {Ev}}\right |\to 0$, and taking into account the fact that $\lim \limits _{\left |X_{\mathrm {Ev}} \right |\to 0} \frac {o(\left |X_{\mathrm {Ev}}\right |)}{\left |X_{\mathrm {Ev}} \right |} =0$, we arrive at a relation between the a priori and a posteriori intensity densities of the spatio-temporal Poisson point process,

$$ \tilde {\nu }_{q} (x,t)=\nu (x,t)(1-P_{1} ). $$

(A.12)

It should be noted that relation (A.10) is similar in its structure to the relation determining the intensity of the Poisson point process obtained under random sparsification [14].

Let us determine the value of $P_{1}$. To this end, for the components of the vector variable $\gamma _{k}$, respectively, $\{ \gamma _{k1}\ldots \gamma _{kn} \}$ $0<n\le 3$, we introduce the discretization

$$ \begin {aligned} &{}\Big \{\gamma _{k1}^{-L_{k}}=-L_{k}\Delta \gamma _{k},\ldots ,\gamma _{k1}^{-l_{k1}} =-l_{k1}\Delta \gamma _{k},\ldots ,\gamma _{k1}^{-1} =-\Delta \gamma _{k},0, \\ &{}\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \gamma _{k1}^{1}=\Delta \gamma _{k},\ldots ,\gamma _{k1}^{l_{k1}} =l_{k1}\Delta \gamma _{k},\ldots ,\gamma _{k1}^{L_{k}} =L_{k}\Delta \gamma _{k}\Big \}, \\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ldots , \\ &\Big \{\gamma _{kn}^{-L_{k}} =-L_{k}\Delta \gamma _{k},\ldots ,\gamma _{kn}^{-l_{kn}} =-l_{kn}\Delta \gamma _{k},\ldots ,\gamma _{kn}^{-1} =-\Delta \gamma _{k},0, \\ &{}\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \gamma _{kn}^{1}=\Delta \gamma _{k},\ldots ,\gamma _{kn}^{l_{kn}} =l_{kn}\Delta \gamma _{k},\ldots ,\gamma _{kn}^{L_{k}} =L_{k}\Delta \gamma _{k}\Big \} \end {aligned} $$

with step $\Delta \gamma _{k}$, where ${[l_{k1} \ldots l_{kn} ]=\overline {l}_{k}^{T}}$ is the vector of indices, $l_{k1} =-\overline {L_{k} ,L}_{k}$, $\ldots$, and $l_{kn}=-\overline {L_{k} ,L}_{k}$.

The values of $L_{k}$ and $\Delta \gamma _{k}$ are taken the same for all $n$. The Lebesgue measure of the $n$-dimensional cube $\Delta _{\gamma k}$ with sides $\Delta \gamma _{k}$, defined as $\left |\Delta _{\gamma k} \right |=(\Delta \gamma _{k} )^{n}$, is substantially less than unity, $\left |\Delta _{\gamma k} \right |\ll 1$.

The values of $L_{k}$ and $\Delta \gamma _{k}$ are selected so that the domain $D_{k}$ corresponding to the surveillance target position at the origin falls completely into the $n$-dimensional cube with sides $[-L_{k} \Delta \gamma _{k} ,L_{k} \Delta \gamma _{k}]$.

To the $k$th tracked target at time $t\in \omega _{q}$ we assign the random variables generated by the vectors of indices $\overline {l}_{D_{k} }^{\thinspace T}=[l_{1D_{k} } \ldots l_{nD_{k} } ]$ such that $\overline {l}_{D_{k} } \Delta \gamma _{k} \in D_{k}$ and

$$ \varsigma _{k}^{\overline {l}_{k} } (t)=\eta _{k} (t)+\overline {l}_{D_{k} } \Delta \gamma _{k} .$$

(A.13)

The condition

$$ {\overline {l}_{D_{k} } \Delta \gamma _{k} =\left [\begin {array}{c} {l_{1D_{k} } \Delta \gamma _{k} } \\ {\vdots } \\ {l_{nD_{k} } \Delta \gamma _{k} } \end {array}\right ]\in D_{k}}$$

determines points belonging to the domain $D_{k}$ when the surveillance target center is at the origin. It is obvious that ${\rm M} (\overline {l}_{D_{k} } )\in {\rm M} (\overline {l}_{k} )$, where ${\rm M} (\overline {l}_{k} )$ is the set of points corresponding to the vectors of indices $\overline {l}_{k}$ and ${\rm M} (\overline {l}_{D_{k} } )$ is the set of points corresponding to the vectors of indices $\overline {l}_{D_{k} }$.

We represent the distribution densities of the random variables (A.13) in the form

$$ w_{k}^{\overline {l}_{k} } (x-\overline {l}_{k} \Delta \gamma _{k} ,t)=\left \{\begin {array}{{ll}} w_{k} (x-\overline {l}_{k} \Delta \gamma _{k} ,t), & \overline {l}_{k} \in {\rm M} (\overline {l}_{D_{k} } ) \\[.3em] 0, & \overline {l}_{k} \notin {\rm M} (\overline {l}_{D_{k} } ). \end {array}\right .$$

The probability of $\varsigma _{k}^{\overline {l}_{k} } (t)$ into $X_{\mathrm {Ev}} (x)$ at the time $t$ for some $\overline {l}_{k}$ is determined by the relation

$$ P\left \{ \varsigma _{k}^{\overline {l}_{k} } (t)\in X_{\mathrm {Ev}} (x)\right \} =P_{1k\overline {l}_{k} }^{\Delta } (x,t)=w_{k}^{\overline {l}_{k} } \left (x-\overline {l}_{k} \Delta \gamma _{k} ,t\right )\left |X_{\mathrm {Ev}} \right |+o\big (\left |X_{\mathrm {Ev}} \right |\big ).$$

(A.14)

Set

$$ \left |\Delta _{\gamma k} \right |=\left |X_{\mathrm {Ev}} \right |.$$

(A.15)

The events related to random variables from the collection (A.13) corresponding to the $k$th tracked target falling into $X_{\mathrm {Ev}} (x)$ are incompatible. Indeed, two distinct elements of one and the same tracked target from a discrete tuple of elements determined in accordance with (A.13) cannot be simultaneously in one elementary domain of the space $X_{\mathrm {Ev}} (x)$ by virtue of (A.15). For incompatible events one has the relation

$$ P_{1k}^{\Delta } (x,t) = \sum \limits _{l_{k1} =-L_{k} }^{L_{k} } \ldots \sum \limits _{l_{kn} =-L_{k} }^{L_{k} } w_{k}^{\overline {l}_{k} } (x - \overline {l}_{k} \Delta \gamma _{k} ,t)\left |X_{\mathrm {Ev}} \right | + o(\left |X_{\mathrm {Ev}} \right |),\quad 0 < n \le 3. $$

(A.16)

Taking into account (A.15) and passing to the limit in (A.16), we obtain the probability of being at a point $x$ for some of the elements of the $k$th tracked target,

$$ \begin {aligned} P_{1k} (x,t)&=\lim \limits _{\substack {\left |\Delta _{\gamma } \right |\to 0\\ L_{k} \to \infty }} P_{1k}^{\Delta } (x,t) \\[.3em] &{}=\lim \limits _{\substack {\left |\Delta _{\gamma k} \right |\to 0\\ L_{k} \to \infty }} \sum \limits _{l_{k1} =-L_{k} }^{L_{k} }\ldots \sum \limits _{l_{kn} =-L_{k} }^{L_{k} }w_{k}^{\overline {l}_{k} } \left (x-\overline {l}_{k} \Delta \gamma _{k} ,t\right )\left |\Delta _{\gamma k} \right | +o\big (\left |\Delta _{\gamma k} \right |\big ) \\ &{}=\int \limits _{D_{k} }w_{k} (x-\gamma _{k} ,t) d\gamma _{k} . \end {aligned}$$

(A.17)

If some element of the $k$th tracked target is at the point $x$, then no element of the other tracked targets may belong to it.

The probability of the event that no element of the $m$th tracked target ($m={1,\ldots ,K}_{q}$, $m\ne k$) belongs to $x$ can be determined in accordance with (A.17),

$$ 1-P_{1m} (x,t)=1-\int \limits _{D_{m} }w_{m} (x-\gamma _{m} ,t) d\gamma _{m} .$$

(A.18)

Consider the event $B$ that one random variable from the tuples $\{\varsigma _{k}^{\overline {l}_{k}}(t),\;\overline {l}_{k}\in {\rm M}(\overline {l}_{D_{k}}),\;k={1,\ldots ,K}_{q}\}$ corresponding to all tracked targets falls into an elementary domain of the space $X_{\mathrm {Ev}} (x)$ at time $t$. This event may occur in several ways; i.e., it splits into several incompatible versions,

• – At time $t$, $X_{\mathrm {Ev}} (x)$ captures one of the random variables of the first group ${\{ \varsigma _{1}^{\overline {l}_{1} } (t),\ \overline {l}_{1} \in {\rm M} (\overline {l}_{D_{1} } )\}}$ (event $B_{1}$), while the random variables of the other groups $\left\{ \varsigma_m^{\overline{l}_m}(t),\; \overline{l}_m\in\mathrm{M}(\overline{l}_{D_m}), \mathrm{Im}=\overline{2,K}_{q} \right\}$ do not fall into $X_{\mathrm {Ev}} (x)$ (events $\overline {B}_{2},\overline {B}_{3} ,\ldots ,\overline {B}_{K_{q} }$).

• – At time $t$, $X_{\mathrm {Ev}} (x)$ captures one of the random variables of the second group $\{ \varsigma _{2}^{\overline {l}_{2} } (t),\overline {l}_{2}\!\in \!{\rm M}(\overline {l}_{D_{2}})\}$ (event $B_{2}$), while the random variables of the other groups $\{ \varsigma _{m}^{\overline {l}_{m} }(t),\ \overline {l}_{m} \in {\rm M} (\overline {l}_{D_{m} } ),\;m=1,3,\ldots ,K_{q} \}$ do not fall into $X_{\mathrm {Ev}}$ (events $\overline {B}_{1},\overline {B}_{3},\ldots ,\overline {B}_{K_{q} }$); and so on.

Consequently,

$$ B=B_{1} \overline {B}_{2} \overline {B}_{3} \ldots \overline {B}_{K_{q} } +B_{2} \overline {B}_{1} \overline {B}_{3} \ldots \overline {B}_{K_{q} } +\ldots +B_{K_{q} } \overline {B}_{1} \overline {B}_{2} \ldots \overline {B}_{K_{q} -1} .$$

(A.19)

Applying theorems on the composition and multiplication of probabilities and using the property of complementary events, we obtain

$$ \begin {aligned} P_{1} &=P(B)=P(B_{1} )P(\overline {B}_{2} )\ldots P(\overline {B}_{K_{q} } ) \\ &\qquad {}+P(B_{2} )P(\overline {B}_{1} )P(\overline {B}_{3} )\ldots P(\overline {B}_{K_{q} } )+\ldots +P(B_{K_{q} } )P(\overline {B}_{1} )\ldots P(\overline {B}_{K_{q} -1} ). \end {aligned} $$

(A.20)

Taking into account (A.17) and (A.18), for (A.20) we obtain

$$ P_{1} (x,t)=\sum \limits _{k=1}^{K_{q} }\left [\thinspace \thinspace \int \limits _{D_{k} }w_{k} (x-\gamma _{k} ,t)d\gamma _{k} \prod \limits _{m\ne k}^{K_{q} }\left (1- \int \limits _{D_{m} }w_{m} (x-\gamma _{m} ,t) d\gamma _{m} \right )\right ],\quad x\in X.$$

(A.21)

The assertion in the Theorem follows from (A.12) and (A.21). $\quad \blacksquare$