The article defines the main tasks of optimal search control for targets with the given coordinates distributions on routes with the specified distributions of the risk of the search units loss. The tasks have been investigated under maximized quality criteria: “the target detection probability”and “the difference between the probabilities of target detection and of a search unit's loss”. Calculation formulae for the results of non-optimized route scanning are obtained, as well as the tasks of optimal interruption of the search in real time search planning and control are formalized. Search tasks with the real time adjustment of the initial target coordinate distributions and risk distributions are discussed. Imitation experiments verifying the analytical results of the work are described.
L f (x) is the target coordinates distribution on the route, ∫ f (x)dx 0 1; Copyright ⃝c by the paper's authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
L r(x) is the RU coordinates distribution on the route, ∫ r(x)dx
0 Ji;i = 1; n, are control quality functionals. 1; 2
Verifying imitation experiments (see below) were carried out for the test case with linearly decreasing and linearly increasing distribution of the target and RU coordinates respectively. Figure 1 shows the route sampling used and the probabilities of the objects’ presence at the points, given to the random coordinate generator. J1 = r(s)ds)f (x)dx; (1) where the first multiplier of the sub-integral function describes the probability of the search unit reaching coordinate x.
When substituting in (1) explicit forms f (x); r(x) of the test case, J1 = 5=6 = 0:83(3).
In imitation experiments (see Figure 3, Figure 4) at 106 passes, J1 = 0:834. In this case, not elementary formula (1), obviously, was verified, but the accuracy of the test case and the imitation experiment. The discrepancy between the theoretical and experimental values seems to be mainly explained not by the small number of route passes (106), but by the roughness of its sampling (20 points).
Note to Figure 3, Figure 4: Abscissae of J1 values are assigned right search boundaries, and mathematical expectations of the SU loss coordinates in all cases are less than the corresponding abscissae.
In contrast to the single-criterion task, the maximized quality criterion in this group of tasks is the difference between the target detection probability and the SU loss probability.
J2 = 56 1 = 0:16(6): In imitation experiment at 106 passes J2 = 0:166: The above formalized strategy (2) generates the task of assigning the optimal final scanning coordinate l that maximizes the difference in probabilities of the target detection and the SU loss:
J2(l ) = l2m(0a;xL) J2(l); J2(l ) J2: Solving task (3) for explicit forms f (x); r(x) of the test case provides J2(l ) = 0:454: In imitation experiments (see Figure 7, Figure 8) at 106 passes J2(l ) = 0:434:
It should be noted that even at linear distribution of the objects’ coordinates the analytical solution of the simplest problem of search optimization (2), (3) turns out to be rather cumbersome. An imitation computer program [Nar19] was developed by the authors, allowing the values of the specified search quality criteria to be calculated and search parameters for arbitrary distributions f (x); r(x) to be optimized.
L (3) The route (0; L) scanning before the target detection or the SU loss. When the target is detected, unlike strategy (2), the scanning stops. The difference between the target detection probability and the SU loss probability in this control strategy is:
J3 =
Assigning the optimal end coordinate of the SU for two-criteria route scanning with a stop at the target detection Let us strengthen strategy (4) by specifying the corresponding task of assigning the optimal final scanning coordinate, maximizing the difference between the probabilities of the target detection and the SU loss: J3(l ) J3:
It should be clarified that at strategy (5) the scanning can be interrupted before reaching l , both when the SU is lost and when the target is detected 5
1. The bi-criteria search problems (2) – (5) formalized in general form are connected with the following system of inequalities:
J3(l ) = l2m(0a;xL) J3(l); { J2(l )
J3(l )
J2 J3
J2 (5) (6)
Formalizations (2) – (5), however, do not allow the full set of functions to be arranged in descending order, since pair order (J2(l ); J3) already depends on the explicit forms f (x); r(x).
2. Search strategies (3) and (5) can be enhanced by introducing real-time coordinates correction of the target and registering units f (x); r(x) (this does not mean revaluation of the initial hypotheses about target and RU coordinate distributions, but recalculation of these distributions for the unscanned route section).
Sections 1-3 of the work were supported by the program of fundamental scientific researches of the SB RAS N I.5.1., project N 0314-2019-0020. Sections 4, 5 of the work were supported by RFBR, projects N 18-08-01284, N 18-07-00526. [Hel80]
O. Hellman. Introduction to optimal search theory. Moscow: Nauka, 1980.