Models and methods for three external Ballistics inverse problems Текст научной статьи по специальности «Математика»

MSC 93A30, 93B40, 90C56, 90C90 DOI: 10.14529/mmp170408

MODELS AND METHODS FOR THREE EXTERNAL BALLISTICS INVERSE PROBLEMS

N.K. Arutyunova, A.M. Dulliev, V.I. Zabotin

Kazan National Research Technical University named after A.N. Tupolev - KAI, Kazan, Russian Federation

E-mail: [email protected], [email protected], [email protected]

We consider three problems of selecting optimal gun barrel direction (or those of selecting optimal semi-axis position) when firing an unguided artillery projectile on the assumption that the gun barrel semi-axis can move in a connected nonconvex cone having a non-smooth lateral surface and modelling visibility zone restrictions. In the first problem, the target is in the true horizon plane of the gun, the second and the third problems deal with some region of 3D space. A distinctive feature of the models is that the objective functions are e-Lipschitz ones. We have constructed a unified numerical method to solve these problems based on the algorithm of projecting a point onto e-Lipschitz level function set. A computer programme has been based on it. A series of numerical experiments on each problem has been carried out.

Keywords: mathematical modelling; inverse problem of external ballistics; optimization;

e

Introduction

Let us consider a problem of firing an unguided projectile. It is required to minimize the distance from the projectile drop point to the target. This problem is a problem of the external ballistic theory. It has been studied fairly well (see [1]) if the following conditions hold: air resistance is not taken into account, movement of the gun barrel is restricted by the true horizon plane of the gun, and the Earth surface is spherical.

However, in reality, a gun barrel direction can be arbitrarily selected, as a rule, only within some connected nonconvex cone that has non-smooth lateral surface and arises in conditions narrowing the selecting gun barrel direction due to some obstacles.

To restrictions on the selecting gun barrel direction, it is often necessary to add conditions such that the target can lie outside of the true horizon plane of the gun or, possibly, on the surface defining the relief of a landscape. In the latter case, the problem of selecting optimal gun barrel direction becomes much more complicated. The foremost reason for this difficulty is that the minimal Euclidean distance from the target to the projectile drop point does not always correspond to the optimal shot or even close to the optimal one. For example, this lack of correspondence can be observed in case the target and the projectile drop point are separated from each other by some substantial obstacle.

A mathematical model can be described as follows. Suppose dist(,) is the Euclidean distance, O is the point in which located the gun, l is a ray with origin O, N is the

projectile drop point, N = N(l), M is the target, K is a cone with vertex O, D' is a set in space R3, and M G D. It is required

mindist (N(l),M) , subject to l G K, N(l) G D'.

Naturally, there may be other possible mathematical models with other objective functions. In particular, below we consider a problem in which the objective function is

M

We stress that, in this paper, air resistance is not taken into account. At the same time, first, our models that are described and investigated below can be applied for firing an unguided heavy projectile. Secondly, the solutions obtained in the models can be served as the base for further finding for a more precise solution and can be refined by appropriate methods.

Below, we formulate and investigate three problems for which a unified algorithm for their solving is constructed. Also, we present numerical results for several test examples; these results were obtained with a computer programme implementing the algorithm.

1. Formulation of the Problems

Throughout, the gun is modelled by a point T, the direction of the gun barrel is modelled by a ray starting from this point, the oblateness of the Earth is not taken into account.

Let us choose a Cartesian coordinate system Oxyz with the origin O coinciding with T. Suppose the plane Oxy is the horizontal plane at O, the axis Oz is directed vertically up, and the gravitational acceleration g, vertically down. The directions of the axes Ox and Oy will be given below. Throughout, we use the following notation: || • ||p is the p-norm on Rra, n ^ 1 (the subscript p will be omitted for p = 2); int and fr are the operators of taking interior and boundary in Rn, respectively; Pr A is the projection of a set A on the z-th axis (here, the axes x, y, and z are denoted by 1, 2, and 3, respectively); given X C Rra and Y C R, we denote by Lip(X; Y) the set of all Lipschitz continuous on X functions f: X — Y with Lipschitz constant lzp(f).

We will assume that the direction (i.e. ray position) of the gun barrel can be freely selected within a closed cone K with the apex O and the cone K contains no vertical rays O

velocity vector v has the initial point at O and has a constant length v0. In other words, v G S fl K, where S = {v | ||v|| = vo} C R3 is the sphere of radius vo centered at O. By v denote the fraction v0j\J||g||.

Throughout, we assume that the selecting direction of the gun barrel is equivalent to the selecting the vector v = (vx,vy,vz) with a given fixed v0. This vector will be described by means of a spherical coordinate system in the form

vx = v0 cos 0 cos p, vy = v0 cos 0 sin p, vz = v0 sin 0, (p,0) G E C 8 := {(p, 0) | p G [0, 2n),0 G (-nj2, nj2)}, ^ '

where the closed set E corresponds to the set S f K, which is given by coordinates (p,0).

We assume that the air resistance is negligible. It follows that the trajectory travelled by a projectile is a parabolic trajectory on the gun plane.

In this paper we will consider three problems of selecting optimal gun barrel direction when firing an unguided projectile to the given target. We will suppose that the target M is the point with y-coordinate equal to zero and M belongs to the convex set bounded by the paraboloid reached by the projectile (see [1]), and this paraboloid is obtained by the union of all the projectile trajectories on the condition of the constraints absence on gun barrel directions. (The case, where the target can lie outside of the paraboloid, and the problem of the joint selecting optimal gun barrel direction (p, V) and magnitude v0 of the initial projectile velocity are not considered in this paper.)

Among the points lying on the projectile trajectories, we will consider only the points with ^-coordinate such that x ^ k, where k = const and k > 0. In practical terms, the constant k, for example, may correspond to the blast radius of a projectile.

Other constraints on x, y, and z-coordinates are different and depend on the problems that will be considered below.

Problem I. Let the point M be the target, lie on the plane xOy, and have coordinates (a, 0) a > k. Suppose the projectile drop point N also lies on the plane xOy and has coordinates (x, y). For all the trajectories, there are no barriers determined by v G S fl K. Out of all those trajectories, it is required to choose one for which the Euclidean distance MN

Following [1], the distance r from O to N(x,y) is calculated by

Determined by (3) the map (p,V) ^ (x,y) from ©to R2 is denoted by hI.

Since v is fixed, we obtain the upper bound pI = v2 for the distance travelled by a projectile. Let us introduce notation

Obviously, the feasible set of the projectile drop points is given by the set hl (E). Thus, the gun barrel direction determined by pair (p, p) and the corresponding optimal projectile trajectory can be found by solving the problem

r = \Jx2 + y2 = v2 sin Using (1) and (2), we see that the coordinates x and y are determined from x = r cos p = v2 sin 2p cos p, y = r sin p = v2 sin 2p sin ifi.

r

(2)

(3)

(4)

\\M - N||2 ^ min, s.t. N G hi(E) f Wi.

(5)

E

Let [9\,92] C [0, 2^) and the functions g\(p) and g2(p) satisfy the condition -n/2 ^ gi(p) ^ g2(p) ^ n/2, p E [tfiA]-

We put

E = {(p,V) G © | p G [91,92],gi(p) ^ V < 92(p)} ,

or, in other words

E = {(p,V) G © | g(p,V) < 0} ,

where

д(ф,0) := max {вг — ф,ф — 02,д1(ф) - 0,0 — д2(ф)} . (6)

Note that Problem I admits a generalization to the case when the target is the set M = {M1, M2,..., Mn} of the poi nts Mi = (ai,bi), i = 1,... ,n, n > 1, on the plane xOy. Indeed, the Chebyshev с enter of M can be choosen as the target at a single shot, for example.

Problem II. Let the point M(a, 0, c) E R3 be the target. Suppose the target and all the possible projectile drop points lie not below the plane z = zmin (zmin < 0). As in Problem I, assume that a > к. We will distinguish two subproblems, which depend on

M

Problem II.b, respectively.

MM trajectory selection is limited only the set E. Among all admissible trajectories, it is

M

M

Il.b. Let the point M belong to the boundary fr D of a set D determined by inequality H(x, y, z) ^ 0, where H is a continuous function over R3 and H(0, 0, 0) ^ 0. This function defines the relief of a landscape. (It means that the target can be located on a surface of ground/water or some fixed ground object, and the gun, on a surface of ground/water or in the air). We assume that the point N(x, y, z) belongs to fr D, the section of the trajectory from O to N has no common points with int D, and z ^ zmin implies H(x,y,z) ^ 0. Among all admissible trajectories, it is required to choose the one for which the segment MN int D

there is no such one. This requirement means that the projectile drop point must be within the line-of-sight of the target and be located maximally close to the target.

Note that the target can lie above (c > 0) or below (c < 0) relative to the plane xOy and the optimal value of angle can be less than zero when c < 0.

Before giving analytical formulations of Problem II.a and Problem Il.b, let us make some important remarks.

Unlike Problem I, for Problems II.a and Il.b, it is either impossible or very difficult to specify a single-valued map from в to R3 that analogous to hi. A reason for this difficulty

R3

Speaking in more detail, in Problem II.a, to each point (ф,0) E E there corresponds not one point but a subset of the set of projectile trajectory points; in Problems Il.b an analytical determination of coordinates of the point N(x,y,z) for given (ф,0) E E may

D

в

R3. This map, denoted be low by hii: в x R+ ^ R3, takes each (ф,0,г) E в x R+ to the

(x, y, z) E R3 (x, y, z)

(ф, 0) and the distance from O to the projection of the point (x, y, z) on the plane xOy is equal to r.

hii

^(ф,0,г) = {(x,y,z) | x = r cos ф,у = r sin ф, z = r tan0 — (1 + tan2 0)(2v2)-1r2} .

(7)

Despite single-valuedness of the maps hl and hll, their inverse maps h"^d h—1 are multivalued: the drop point can be reached by selecting one of the two appropriate values

of the angle p E [—n/2,n/2] for the direction of the gun barrel. At the same time, the

p

for Problem I and II.a,b, the maps h-1 and h~1 play an important role. A reason, on which this constructing is possible, is that the maps h-1 and h~1 can be parameterized by single-valued maps. Indeed, it follows from (3), (7) that there exist the single-valued maps h~\ h—j, j E {1, 2} for which

h-1 = h—1 U h-1, dom(h j = dom (h—1), k E {I, II}, j E {1, 2};

Pr2 Im(h—1) = (0, n/4], Pr2 Im(h—1) = [n/4,n/2), (8)

Pr2 Im(h—1) = (-n/2,n/2), Pr2 Im(h—21) = (0,n/2).

The explicit expressions for h— will be given below.

Since all the projectile drop points lie not below the plane z = zmin and using (7), the r

Pll = V\/v2 - 2Zmin •

Taking into account this bound and the envelope of the family of the possible projectile trajectories, we define the set

Wll = j(x,y,z) \y/x2 + y2 ^ Pll,x Z K,Zmin ^ Z ^ 1 ^V2 - ^^ } , (9)

which is the set of all the points reached by a projectile.

In view of the above remarks, we can now give an analytical formulation of Problem II.a in the form

\\M - NII2 ^ min, n n

s.t. N E hu(E) n W„. V ;

For Problem II.b we have such an analytical formulation

\\M - N\\2 ^ min,

s.t. N E hll(E) n Wll n fr D, {XN + (1 - X)M \ X E [0,1]} n int D = 0, (11) 3j E {1, 2} : hu (Pr1 h—KN), Pr2 K](N), [0, Pra h—}(N)]) n int D = 0.

Obviously, the last two equations in (11) mean that the section of the trajectory from O to N and the segment MN does not intersect int D, respectively.

By N* denote a point that belongs to the solution set of problem (11). Unfortunately, problem (11) admits the existence of common points, in addition to N*, between the segment MN or the section of the trajectory from O to N* and fr D. Of course, adding conditions to the constrains of problem (11), we can eliminate this situation. However, as it is easy to show, a formalization of these conditions does not allow us correctly use any optimization method because the feasible set is not closed, moreover, an attempt to take its closure again leads to problem (11). To solve this problem we

propose the following procedure. It is necessary to solve problem (11) and if its solution

N*

find another solution of problem (11) in the union of sufficiently small closed "ring like" neighborhoods of the points (Pr1 h—1 (N*), Pr2 h—11(N*^) and (Pr1 h —21(N*), Pr2 h —2(N*)), respectively.

2. Minimizing the Objective Functions

Problems (5), (10), and (11) are complex in general case. Despite the fact that the

(x, y, z)

the functions describing the feasible set can be non-differentiable. In particular, this applies to the functions g1 and g2. Similarly, we can assert that these statements hold for problems (5) and (10) with respect to the variables p, 0 and the variables p, 0, r, respectively. Moreover, problem (11) with respect to the variables p, 0, r may have both objective function and feasible set functions that are non-differentiable.

Surely, in each problem (5), (10), and (11), we could try smoothing all its functions and then solve the corresponding smoothed problem by invoking some optimization method, for example, linearization method or sequential quadratic programming method. At the same time, a complicated form of the feasible set functions, especially in Problems II.a, II.b, (see below), indicates that using smoothing procedures is not useful, because the ones do not give substantial simplification of the original problem. Moreover, after this

E

the gun barrel or may be far from the optimal solution.

It is obvious that we can minimize the objective function in problem (5) either with

(p, 0) (x, y)

(p, 0, r) (x, y, z)

case, the found optimal solution allows us to find right away the required optimal gun

(p, 0)

a point closest to the target and lying on the optimal trajectory.

As above, in Problem II.b, we should not rely on an analytical determination of coordinates of the point N(x, y, z) by using the variables p, 0. Therefore, it is reasonable

xyz

which confirms that problems (5) and (10) should be also solved with respect to the x y z

Thanks to smoothness of the objective functions for problems (5) and (10) with respect

p0r

algorithm. However if the functions gi or g2 are non-differentiable or the set E is non-convex, then the use of this algorithm is difficult, because the conditions for its convergence may be not satisfied or because solving the auxiliary problems, for example, as finding a E

gi g2

8

E

of the photography results.

gi g2

about the variables in problems (5), (10), and (11), we will solve all problems with respect xyz

In addition, since the maps h-1 and h~1 are multivalued, it follows that the feasible sets in each problems should be presented as the union of two subsets that correspond to h-11 and h—1, k G {I, II} (see (8)). This implies that instead of one problem either (5), or (10), or (11), we have two corresponding subproblems, which may be solved independently of each other.

Clearly, each problem (5), (10), or (11) (and, therefore, its subproblems) is a problem of finding a projection the point M onto the feasible set. We claim that the feasible set for each problem can be defined by the corresponding functional inequality in the form

Fij(x,y,z) ^ 0, i G {I, II.a, II.b}, j G {1, 2},

where j is the subproblem number and the left-hand side of inequality satisfies so-called the e-Lipschitz condition. This condition will allow us to use the methods proposed in [2, 3]. For problems (5) and (10), the functions Fij and Fii^j can be defined by

Fj (N) := max {j g (Pn hj (N), Pr2 h-1(N)) , ^ (k - Pri N)} , (12)

F^(N) := max{WlLaji g (Pri h-(N), Pr2 h-(N)) ,

^iij (k - Pri N) , wILaj3 (zm;n - Pra N ) }. Here and in what follows, by Uj we denote the weighting coefficients whose values depend

Fij

for solving Problems I, II.a, II.b, we will propose a technique for choosing uij\. Since the constraints for problem (11) can be formalized by

' g (Pri hjN), Pr2 h~j(N)) ^ 0, H (N) = 0, Pri N ^ k, min {H (AN + (1 - A)M)} ^ 0,

ag [0, i]

mmf H(hii (Pri h—j(N), Pr2 h—(N),» Pra h^N)))} ^ 0,zmin - Pra N ^ 0,

(14)

we can define the function Fiihj by the formula

Fn.bj(N) := max|un.bji g (Pri h-j(N), Pr2 h-](N)) , u„.b j2 (k - Pri N) , Uii^jalH(N)|, -un^j4 mon {H(AN + (1 - A)M)} ,

j5 m* {H (hii (Pri h-](N), Pr2 h-j(NPra h-j(N)))}

(15)

The upper bounds for the variables x, ^d z in the definition of the sets Wi (see (4), (9)) are implicitly contained in (12) - (15), since they follow from projectile motion equations and from the condition N G h.(E). But the lower bounds for the variables x and

z

practical considerations.

Thus, each Problem i, i G {I, II.a, II.b}, for each j G {1, 2} has the form

\\M - N||2 ^ min, s.t. Fij (N) ^ 0. 1 j

Obviously, we should not rely on an analytical methods for solving any problem of the form in (16). Among numerical methods, the choice is small due to complicated properties Fij

At the same time, the functions Fj, i G {I, II.a, II.b}, j G {1, 2} satisfy the so-called the e-Lipschitz condition (see [4]) in the corresponding set Wi. Now we give the corresponding definition.

Let f be a function from a subset A contained in a normed space X to a normed space Y. The function f is railed e-Lipschitz continuous on A, if for any number 0 < e < e0 there exists some number L(e) > 0 such that for all x,y E A, the following inequality holds

\\f(x) - f(y)\\ ^ L(e)\\x - y\\ + e. (17)

By e-Lip(A; Y) denote the set of e-Lipschitz continuous on A functions; for a fixed function f and fixed e, by lip(f; e) denote the smallest function (i.e., the infimum) of all L(e) satisfying (17). (The properties for lip(f; e) can be found in [2], where it is denoted by l(e))).

To present the assertions that each function Fj, i E {I, II.a, II.b}, j E {1, 2} is e-Lipschitz continuous on a certain set in Rra (n equal to 2 or 3), we shall give the explicit formulas for Pn h—1(N), Pr2 h—1(N), Wk, k E {I, II} j E {1, 2}.

Using (3) and (7), we get

p(N) := Pn h—1(N) = arctan(y/x), k E {I, II}, j E {1, 2};

n . ( j (n 1 . \Л2 + У2 4+(_1Г 4 - 2аГС81П-^

^j(N) := Pr2 h~\N) = - + (—1)j - — - arcsln Jy ,j E {1, 2};

фиj(N): = Pr2 h-i(N)

= arctan ((x2 + y2)-1/2 (v2 + (-1)Vv4 - (x2 + y2 + 2v2z))) ,j E {1, 2}.

(18)

Note there exist some values of the variables x, ^^d z such that the expression under the last square root in (18) can be negative. But this fact indicates that, for the given magnitude vo of the initial projectile velocity, the coordinates of the projectile drop point can not be equal to those x, y, and z for any values of p.

The e-Lipschitz continuity of the functions Fj, i E {I, II.a, II.b}, j E {1,2}, is formulated below by Propositions 1-3. We also give several lemmas in which some functions that are used in the definitions of the functions Fj are e-Lipschitz continuous or Lipschitz continuous. For the subsequent estimates of lip(-) and lip(s e), we use the norm \\ • \\1; we also assume that gug2 E Lip([01,^]; R), H E Lip(R3; R), 2e E (0,n/2 - 1).

Lemma 1. For the function g(p, p) defined by (6), we have g E Lip(6; R) and

lip(g) ^ max{lip(g1 ),lip(g2), 1} . (19)

The proofs of this lemma and the following ones can be found in [6]. Lemma 2. For each k E {I, II} it is true that p(x,y) E Lip(Wk; [0, 2n)) and

lip(p) (20) Lemma 3. For each j E {1, 2} it is true that j(x,y) E ^Lip(Wl; (0,n/2)) and

lip(hj; e) ^ (V2vV 1 - t2(2e)Y', (21)

where т(2e) is the unique root of the equation (n/2 — 2e — arcsln т)л/1 — т2 = 1 — т in the interval [0,1).

Proposition 1. For each j G {1, 2} it is true that Fi j(N) G e-Lip(Wi; R) and

lip(Fij; e) ^ max{Ujlip(g) (lip(p) + lip 0j; e(uijilip(g))_i)) ,Uij^ , (22)

where lip(g), lip(p), and lip (0i j; e(uijilip(g))~l) determined by (19), (20), and (21), respectively.

Lemma 4. For each j G {1, 2} it is true that 0ii j(N) G e-Lip(Wn; (-n/2,n/2)) and

lip (0n j; e) ^ (pn(2e)-i + pV^j k-2, (23)

where 3 = v2 + Vv4 - k2 - 2v2zmin.

Proposition 2. For each j G {1, 2} it is true that Fihaj(N) G e-Lip(Wn; R) and

lip(Fn.aj; e) ^ max{u„.ajilip(g) (lip(p) + lip 0j; e(u„.ajilip(g))~i)) ,uUMj2,uUMj^ ,

(24)

where lip(g), lip(p), and lip (0ii j; e(uiiMjilip(g)) i) determined by (19), (20), and (23), respectively.

Proposition 3. For each j G {1, 2} it is true that Fihbj(N) G e-Lip(Wii; R) and

lip(Fu.bj; e) ^ max {un.bjilip(g) (lip(p) + lip 0j; e(un^jilip(g))"^) ,un^j2,

uu.bj3lip(H),uu.bj4lip(H),uu.bj5lip(H) (pulip(H)/(8e) + 1)} ,

where lip(g), lip(p), and lip (0ii j; e(uiLbjilip(g))~i) determined by (19), (20), and (23), respectively.

According to our assumptions, we have M G W^ i G {I, II}, whence Fij(M) ^ 0, j G {1, 2}

projection can be solved by using the algorithms proposed in [2, 3]. Each algorithm either generates an infinite or a finite sequence Qm. In the first case, the sequence Qm converges

Fij M satisfies inequality Fj (Qm) > 0 for each m =1, 2,.... In the second case, the last element of the sequence is the required point N*, in which Fj(N*) = 0. In this paper we use one of these algorithms. Before describing it, as we said above, we turn to the weighting coefficients Uj used in constructing the functions Fj because choosing Uj affects the accuracy of the solution obtained by the algorithm.

Since the algorithm finds the next approximation to the optimal solution by using

Fij

Fij

components, we must take into account how these function-components are related to each other. If these relations are ignored, then the approximations generated by the algorithm may not reach the stopping criterion Fj (Qm) < e* within a reasonable time (see also

Fij

may be too high, whence the number of algorithm iterations to reach accuracy is too large. It is known (see [5]) that an upper bound of the number of iterations for the gradient-free algorithms, including our algorithm, is determined by using a possible change in the value of the objective function; therefore, if we know these changes for all function-components

of Fij, then we can take uijl to provide the number of iterations adequate to the overall accuracy e*. In other words, having taken a value of the overall accuracy e* for the values of Fj, we can specify accuracies for every function-component of Fij by using the values of the corresponding weighting coefficients: accuracy for l-th function-component will be equal to e* /uijl.

In this article, we suppose uij1 = 1, i E {I, II.a, II.b}, j E {1,2}, that is, the e* Fij

The accuracies uijl for other function-components are taken according to remark above.

Let us present the algorithm.

Step 0. Choose: an initial value e0 > 0 of the e-Lipschitz parameter, the initial projectile velocity magnitude determined by v or v0, a lower bound k of x, an initial point N0 = M E Wij, parameters % X E (0,1). Assign the visibility zone constraints determined by g(p, p) satisfying (6). Set Q0 := №, k := 0, and m := 0.

Step 1. Depending on Problem I, II.a, or II.b, calculate Fij(Nk), i E {I, II.a, II.b}, j E {1, 2} using (12), (13), or (15), respectively. If Fij(Nk) < ek(1 + %), then sequentially set Nk+1 := Nk, Qm+1 := N^^d m := m +1 and pass to Step 2. Otherwise, go to Step 3.

Step 2. If Fj(Nk) ^ ek, then set ek+1 := XFj(Nk). Otherwise, set ek+1 := Xek. In either case, go to Step 4.

Step 3. Find Nk+1 by the following scheme. Depending on Problem I, II.a, or II.b, calculate lip(Fij,ek), i E {I, II.a, II.b}, j E {1,2} by formulas (22), (24), or (25), respectively, using (19)—(21), and (23). Solve the problem

Nk+1 = argmin {Fij(X) | X E fr Kk n Wij} , i E {I, II.a, II.b},j E {1, 2}, (26)

where

Kk H X E R | ||X - M|| O jA ,rk

|x E I ||X - MII ^ ,

Fij (Nk) - £k

y/nlip(Fij; £k)'

n = 2 for ftoblem ^d n = 3 for Problems II.a and II.b. Next, set £k+i := £k and go to Step 4.

Step 4- If Fij(Nk+1) = 0 or Fij(Qm) = 0 then Nk+1 or Qm is regarded as a solution of problem (16), respectively. Otherwise, set k := k + 1 and pass to Step 1.

Remark 1. According to the main statement about the algorithm convergence (see

Nk

Fj(Qm) -> 0. Therefore, the condition of the form Fj(Qm) < £*, £* E (0,1), should be

included in the stopping criterion. We can also include the conditions setting accuracies for some function-components of Fij or the condition of the form IQm — Qm+1|| < £Q, £q E (0,1) in the stopping criterion.

Remark 2. It is clear that the main computational costs are produced by Step 3, where the optimization subproblem for finding the point Nk+1 is solved. We will give two pieces of advice about solving this subproblem. First, since each function Fj, j E {1, 2}, is minimized either on a circle (Problem i = I) or on a 2-sphere (Problems i = II.a, II.b), it follows that the dimension of the subproblem can be reduced by one by applying transformation to polar coordinates or spherical ones, respectively. Secondly, it is not necessary to solve this subproblem precisely. Indeed, since the value Fij (Nk) is compared

with e^d ek (1 + YX instead of the point Nk+i determined by (26), one can find a point NVk+i such that Fj(Nk+i) ^ Fj(NVk+i) < Fj(Nk+i) + 5k, where 8k < ekY-

Remark 3. For computing values of the 4-th function-component of FiLb in problem II.b, we must solve the one-dimensional optimization problem of minimization of the function f (A) = H(AN + (1 - A)M). This problem can be solved by some corresponding numerical method. A similar remark also concerns the 5-th function-component of Fn.b.

3. Numerical Examples

M Wi

i G {I, II}, and the overall accuracy e* that taken equal to an initial value e0 of the e-Lipschitz parameter were varied. The optimization subproblem for finding the point Nk+i (see Step 3) was solved by the uniform search method.

For all examples, the following parameter values were set: v0 = 180m/s; k = 100m; Y = A = 0, 5. The values of the variable parameters are presented in Tables 1-3. Hereinafter and in Tables 1-3, we use the following notation: Mi = (110,0, 20) M2 = (2700, 0,-10) are the target M (the third coordinate of M for Problem I was equal to 0); Ei = {(p, 0) | p G [0,2n],0 G [7n/36,8n/36]}, E2 = {(p,0) | p G [0,2n], |4 + sin p|n/36 ^ 0 ^ |1 + sin p^/9} (xN, yN ,zN) is the solution of Problem I, II.a, or II.b; (p, 0) is the angles for the direction of the gun barrel, which ones corresponding to (xN,yN,zN); ktot is the total number of algorithm iterations; t is the total computational time in seconds. The value zmin in Problems II.a and II.b was equal to -10, and the set D in Problems II.b was determined by function

H(x,y,z) = min{max{90 - x,x - 130,-10 - y,y - 30, z - 20}, z + 10}.

Table 1

Results of test examples for Problem I

M E £ xN, m Vn ,m v,0 0, 0 ktot t, s

Mx Ei 0,1 2818,6 0,0 0,0 29,3 300 0,146

0,05 2976,8 0,0 0,0 32,1 353 0,170

0,01 3081,4 0,0 0,0 34,4 575 0,306

E2 0,1 1523,4 -152,1 -5,7 13,8 301 0,419

0,05 1834,6 -91,4 -2,9 16,9 442 0,822

0,01 2068,1 -20,6 -0,6 19,4 891 3,514

M2 Ei 0,1 2820,3 0,0 0,0 29,3 40 0,034

0,05 2977,1 0,0 0,0 32,2 144 0,094

0,01 3081,4 0,0 0,0 34,4 331 0,213

E2 0,1 2606,4 48,4 1,1 26,0 36 0,031

0,05 2447,8 138,6 3,2 24,0 158 0,086

0,01 2316,7 217,0 5,4 22,4 456 0,244

The weighting coefficients in the functions Fi7 FiLaj, and Fii^j (see (12), (13), and (15)) were chosen as follows: I) uiji = 1, uij2 = 0, 01; II.a) uii^ji = 1, uiLaj2 = uiLaj3 = 0,01; II.b) Uii.bji = 1 Uii.bj2 = ° 01 Uii.bj3 = Uii.bj4 = Uii.bj5 = ^ 001-

Table 2

Results of test examples for Problem II.a

M E £ xN, m vN ,m zN, m Ф,0 ф,0 ktot t, s

M1 Ei 0,1 100,0 0,1 58,0 0,1 31,0 1889 1,249

0,05 100,0 1,8 65,4 1,0 34,1 4182 2,244

0,01 102,0 2,1 72,1 1,2 36,1 26297 5,227

E2 0,1 108,7 -0,5 25,9 -0,3 14,3 614 1,073

0,05 106,8 -1,0 31,4 -0,6 17,3 2444 1,495

0,01 104,8 -0,3 35,9 -0,2 19,8 17619 8,854

M2 Ei 0,1 2730,7 0,0 48,3 0,0 29,3 8478 4,064

0,05 2771,2 0,0 121,4 0,0 32,2 33069 12,750

0,01 2803,2 -0,1 173,5 0,0 34,4 213343 103,261

E2 0,1 2502,2 107,1 -8,6 2,5 24,4 27281 5,952

0,05 2336,3 826,1 -4,3 19,5 24,2 244288 131,227

0,01 2324,5 663,2 -2,8 15,9 23,4 313626 163,866

Table 3

Results of test examples for Problem II.b

M E £ xN, m vN ,m zn , m Ф,0 ф, 0 ktot t, s

M1 Ei 0,1 100,0 0,0 58,0 0,0 31,0 2748 9,753

0,05 100,2 16 62,6 9,1 32,6 7308 14,240

E2 0,1 108,7 -0,5 25,9 -0,3 14,3 1072 11,303

0,05 106,8 -1,0 31,4 -0,6 17,3 4258 11,179

M2 Ei 0,1 2730,7 0,0 48,3 0,0 29,3 12746 19,626

0,05 - - - - - - -

E2 0,1 2502,2 107,1 -8,6 2,5 24,4 47995 88,068

0,05 2437,0 145,7 -10,0 3,4 23,6 69230 121,723

The tests were performed on an Intel Core i3-4020 1,50 GHz personal computer. The

£

solution in Problem I faster than in other two Problems. Moreover, in Problem II.a and especially in Problem II.b, there is a large increase in the number of iterations with £o greater than 0, 05. (The dash in Table 3 means that the algorithm did not give a solution for 200s.) A reasonable explanation for this increase may be that, for the functions FILajb a decrease in £ implies an increase in L(e). However, for a finite number, number of iterations (when L(£) is not very high), it is still possible to cut off a set that does not contain zero of F^; this yields that the set Wi is narrowed down to its subset. It is readily seen that computational time can be reduced by parallelizing some parts of the algorithm. For example, one can be applied in subproblems, in which the uniform search method are used.

References

1. Konovalov A.A. Vneshnyaya ballistika [External Ballistics]. Moscow, Tsentral'nyy nauchno-issledovatel'skiy institut informacii, 1979. (in Russian)

2. Arutyunova N.K., Dulliev A.M., Zabotin V.I. Algorithms for Projecting a Point onto a Level Surface of a Continuous Function on a Compact Set. Computational Mathematics and Mathematical Physics, 2014, vol. 54, no. 9, pp. 1395-1401. DOI: 10.7868/S0044466914090038

3. Zabotin V.I., Arutyunova N.K. [Two Algorithms for Finding the Projection of a Point onto a Nonconvex Set in a Normed Space]. Computational Mathematics and Mathematical Physics, 2013, vol. 53, no. 3, pp. 344-349. DOI: 10.7868/S0044466913030162 (in Russian)

4. Vanderbei R.J. Extension of Piyavskii's Algorithm to Continuous Global Optimization. Journal of Global Optimization, 1999, vol. 14, pp. 205-216.

5. Nesterov Yu. Vvedenie v vypukluyu optimizaciyu [Introduction into a Convex Optimization]. Moscow, Moskovskiy tsentr nepreryvnogo matematicheskogo obrazovaniya, 2010. (in Russian)

6. Arutyunova N., Dulliev A., Zabotin V. Models and Methods for Three External Ballistics Inverse Problems. Available at: http://arxiv.org/abs/1610.02933 (accessed 16 October 2017).

Received January 20, 2017

УДК 519.863+519.853.4+519.853.6+531.554 DOI: 10.14529/mmpl70408

МОДЕЛИ И МЕТОДЫ ДЛЯ ТРЕХ ОБРАТНЫХ ЗАДАЧ ВНЕШНЕЙ БАЛЛИСТИКИ

Н.К. Арутюнова, A.M. Дуллиев, В.И. Заботин

Казанский национальный исследовательский технический университет им. А.Н. Туполева - КАИ, г. Казань, Российская Федерация

Рассматриваются три математические модели задачи выбора оптимального направления ствола орудия при стрельбе неуправляемым снарядом в предположении, что полуось ствола может перемещаться в связном невыпуклом конусе, имеющем негладкую боковую поверхность и моделирующем ограничения на зону видимости. В первой задаче цель расположена в плоскости истинного горизонта орудия, во второй и третьей - в некоторой области пространства. Отличительной особенностью моделей является е-липшицевость целевых функций. Построен единый численный метод решения поставленных задач, базирующийся на одном алгоритме проектирования точки на е

ЭВМ. По каждой из задач проведена серия вычислительных экспериментов.

Ключевые слова: математическое моделирование; обратная задача внешней бале

ближенное решение.

Литература

1. Коновалов, A.A. Внешняя баллистика / A.A. Коновалов. - М.: Центральный научно-исследовательский институт информации, 1979.

2. Арутюнова, Н.К. Алгоритмы проектирования точки на поверхность уровня непрерывной на компакте функции / Н.К. Арутюнова, A.M. Дуллиев, В.И. Заботин // Журнал вычислительной математики и математической физики. - 2014. - Т. 54, № 9. -С. 1448-1454.

3. Заботин, В.И. Два алгоритма отыскания проекции точки на невыпуклое множество в нормированном пространстве / В.И. Заботин, Н.К. Арутюнова // Журнал вычислительной математики и математической физики. - 2013. - Т. 53, № 3. - С. 344-349.

4. Vanderbei, R.J. Extension of Piyavskii's Algorithm to Continuous Global Optimization / R.J. Vanderbei // Journal of Global Optimization. - 1999. - V. 14. - P. 205-216.

5. Нестеров, Ю.Е. Введение в выпуклую оптимизацию / Ю.Е. Нестеров. - М.: МЦНМО, 2010.

6. Arutyunova, N. Models and Methods for Three External Ballistics Inverse Problems / N. Arutyunova, A. Dulliev, V. Zabotin. - URL: https://arxiv.org/abs/1610.02933.

Наталья Константиновна Арутюнова, КаНДИДаТ физико-математических наук, доцент, кафедра «Прикладная математика и информатика:», Казанский национальный исследовательский технический университет им. А.Н. Туполева - КАИ (г. Казань, Российская Федерация), [email protected].

Айдар Мансурович Дуллиев, кандидат физико-математических наук, доцент, кафедра «Прикладная математика и информатика», Казанский национальный исследовательский технический университет им. А.Н. Туполева - КАИ (г. Казань, Российская Федерация), [email protected].

Владислав Иванович Заботин, доктор технических наук, профессор, кафедра «Прикладная математика и информатика», Казанский национальный исследовательский технический университет им. А.Н. Туполева - КАИ (г. Казань, Российская Федерация), [email protected].

Поступила в редакцию 20 января 2011 г.