Search for Targets with Conditionally Deterministic Motion

Abstract

Optimal search for targets with conditionally deterministic motion is investigated. The target motion takes place in $\mathcal{Y}$, a copy of Euclidean n-space, and depends on a stochastic parameter $\xi $ which takes values in $\mathcal{X}$ , another copy of Euclidean n-space. The target motion is deterministic given knowledge of $\xi $. That is, there is a function $Y:T \times \mathcal{X} \to \mathcal{Y}$, where T is a time interval, such that $Y( \cdot ,x)$ gives the target motion conditioned on $\xi = x$. Search plans are specified by functions $\mu :T \times \mathcal{Y} \to [ 0,\infty )$. A functional P is defined so that $P_t [ \mu ]$ gives the probability of detecting the target by time t using plan $\mu $. Let $J(t,x)$ be the absolute value of the Jacobian of $Y(t, \cdot )$ evaluated at x. If there exist functions $m:T \to (0,\infty )$ and $j:\mathcal{X} \to (0,\infty )$ such that $J(t,x) = m(t)j(x)$ for $(t,x) \in T \times \mathcal{X}$ , the target motion is called factorable Let $\varphi _2 :T \to [ 0,\infty )$. If the target motion is factorable, Theorems 4.1 and 4.2 give a method for finding a plan $\mu ^ * $ such that $\int_\mathcal{Y} \mu ^ * (t,y)dy\leqq \varphi _2 (t) $for $t \in T$ and $P_t [ \mu ^ * ]\geqq P_t [ \mu ]$, $t \in T$, for all search plans $\mu $ satisfying $\int_{\mathcal{y}} \mu (t,y)dy\leqq \varphi _2 (t) $ for $t \in T$. Let k and l be positive numbers. Theorem 5.1 gives sufficient conditions for finding a plan $\mu ^ * $ such that $\mu ^ * \leqq k$, $\int_T \int_\mathcal{Y}\mu ^ * (t,y)dydt \leqq l $ and $\lim _{t \to \infty } P_t [ {\mu ^ * } ]\leqq \lim _{t \to \infty } P_t [ \mu ]$ for any search plan $\mu $ satisfying $\mu \leqq k$ and $\int_T \int_\mathcal{Y} \mu (t,y)dydt\leqq l$. Examples of optimal search plans are computed to illustrate the use of the above theorems.