UDC 517.934

ON A PURSUIT PROBLEM IN THE CASE OF RESTRICTIONS ON THE IMPULSES OF CONTROL FORCES

N. N. Krasovskii, V. E. Tret’yakov

The problem of the pursuit of one controlled object by another under restrictions on the impulses of the control forces encounters serious difficulties. Difficulties arise already in the very formulation of the problem. Some of these difficulties are briefly touched upon in [1]. In the present article, in greater detail than in [1], a possible approach to the problem of effecting an encounter under the indicated restrictions is discussed, and ways of overcoming the difficulties are proposed. The approach to the problem presented here is illustrated by the solution of a simple example.

§ 1. Consider a pursuit problem for two objects whose motion is described by the differential equations

\[ \dot y = Ay + Bu, \tag{1.1} \]

\[ \dot z = Gz + Mv. \tag{1.2} \]

Here \(y(t)=\{y_1(t), \ldots, y_n(t)\}\) is the vector of phase coordinates of the first, pursuing object, \(z(t)=\{z_1(t), \ldots, z_n(t)\}\) is the vector of phase coordinates of the second, pursued object, \(u=\{u_1,\ldots,u_r\}\) and \(v=\{v_1,\ldots,v_l\}\) are the vectors of control actions, and \(A, G, B, M\) are constant matrices of the corresponding dimensions. Suppose that for each current instant of time \(\tau\) restrictions are given,

\[ \int_{\tau}^{\infty} \left[\sum_{j=1}^{r} u_j^2(t)\right]^{1/2} dt \leq \mu(\tau), \tag{1.3} \]

\[ \int_{\tau}^{\infty} \left[\sum_{s=1}^{l} v_s^2(t)\right]^{1/2} dt \leq \nu(\tau), \tag{1.4} \]

constraining the actions \(u(t)\) and \(v(t)\) for \(t\geq \tau\). In particular, if \(u\) and \(v\) are scalars, then we have

\[ \int_{\tau}^{\infty} |u(t)|\,dt \leq \mu(\tau), \qquad \int_{\tau}^{\infty} |v(t)|\,dt \leq \nu(\tau). \tag{1.5} \]

At the initial instant of time, for \(\tau=0\), the quantities \(\mu(0)\) and \(\nu(0)\) are given by the conditions of the problem. For \(\tau>0\), the values \(\mu(\tau)\) and \(\nu(\tau)\) are determined by the actually expended control resources.

\[ \mu(\tau)=\mu(0)-\int_0^\tau\left[\sum_{j=1}^r u_j^2(t)\right]^{1/2}\,dt,\qquad \nu(\tau)=\nu(0)-\int_0^\tau\left[\sum_{s=1}^l v_s^2(t)\right]^{1/2}\,dt. \tag{1.6} \]

The restrictions (1.3), (1.4), and (1.5) may be interpreted as restrictions on the impulses of the controlling forces.

Suppose that certain coordinates \(y_{i_k}, z_{i_k}\) \((k=1,\ldots,m)\) are singled out, whose coincidence is the aim of the pursuit. The sets of coordinates \(\{y_{i_k}\}=y_{[m]}\) and \(\{z_{i_k}\}=z_{[m]}\) will be regarded as vectors \(\{x_k\}=x_{[m]}\) in the \(m\)-dimensional space \(X\). We shall call the instant \(\vartheta\) the instant of meeting if, at this instant, \(y_{i_k}(\vartheta)=z_{i_k}(\vartheta)\) \((k=1,\ldots,m)\). Suppose that at each instant \(t=\tau\) both partners, the pursuer and the pursued, know the realized values \(y(\tau), z(\tau), \mu(\tau)\), and \(\nu(\tau)\). If one considers the pursuit process under the conditions (1.3), (1.4) as a game of two persons with opposing interests, in which the payoff is the time \(T=\vartheta-\tau\) until the meeting of the motions, and seeks optimal controls \(u(t)\) and \(v(t)\) according to the feedback principle in the form \(u[y(\tau), z(\tau), \mu(\tau), \nu(\tau)]\), \(v[y(\tau), z(\tau), \mu(\tau), \nu(\tau)]\), then the formulation of such a pursuit problem encounters difficulties [1]. The point is that the restrictions (1.3) and (1.4) admit impulsive \(\delta\)-controls \(u(t)\) and \(v(t)\), under which the phase motions \(y(t)\) and \(z(t)\) become discontinuous. (For example, the velocity of a material point changes by a jump at the moment when an instantaneous finite impulse is applied.) This circumstance makes the meeting extremely difficult if the pursuer at each instant \(t=\tau\) has only information about the quantities \(y(\tau), z(\tau), \mu(\tau), \nu(\tau)\). Therefore we introduce an additional condition that expands, for the pursuer, the information about the behavior of the pursued system. If such conditions are not introduced, then even in very simple cases the point \(z(t)\) may slip away from the point \(y(t)\). We shall suppose that at each instant \(t=\tau\) the pursuing partner already knows the choice of control \(v(\tau)\) made by the second partner at this same instant \(t=\tau\). This condition places the pursued object in unequal informational conditions in comparison with the pursuer; therefore we shall call it the condition of discrimination of the pursued object. From the point of view of real conditions, this assumption means that, in idealizing the problem, we ignore the inevitable delay of information about a change in the motion \(z(t)\) (1.2) reaching the control organ for the motion \(y(t)\) (1.1), i.e., we admit an instantaneous reaction of the pursuer to the actions of the pursued. Discrimination consists in the fact that information to the pursued about the pursuer, on the contrary, is admitted by us only with a (infinitely small) delay. This idealization is expedient for giving the problem a form convenient from the mathematical point of view.

In mathematical form, the discrimination condition may be given various forms. One may, for example, assume that the motion \(y(t)\) is continuous from the left, while \(z(t)\) is continuous from the right [1]. In the present article, however, another condition is adopted. Namely, we shall require that both motions \(y(t)\) and \(z(t)\) be continuous from the left, and we assume that at the instant \(t=\tau\), the organ producing the control \(u(\tau)\) already knows, about the motion \(z(t)\), the values \(z(\tau+0)\), \(\dot z(\tau+0)\), and \(v(\tau+0)\), while the organ producing the control \(v(\tau)\) knows about the motion \(y(t)\) only the values \(y(\tau)\) and \(\dot y(\tau)\).

Moreover, let us agree on the following. We shall regard realizations of the controls \(u(t)\) and \(v(t)\) of the form

\[ u(t)=u_R(t)+\sum_{s=1}^N p_s\cdot \delta(t-\tau_s),\qquad v(t)=v_R(t)+\sum_{s=1}^N q_s\cdot \delta(t-\tau_s). \tag{1.7} \]

Here the regular components \(u_R(t)\) and \(v_R(t)\) are piecewise-continuous functions, which we agree to consider continuous from the right, while the symbol \(\delta(t-\tau_s)\) denotes the \(\delta\)-function. In this case impulses of the form \(\delta(t-\tau)\) are included in the integrals (1.3) and (1.4), which estimate the remaining control resources, and are not included in the integrals (1.6), which estimate the expended control resources. In equations (1.1) and (1.2) the differentiation symbol denotes the right derivative. It is now expedient to choose the meeting condition in the form \(y_{i_k}(\vartheta+0)=z_{i_k}(\vartheta+0)\) \((k=1,\ldots,m)\).

In addition to the difficulties connected in the present case with the discontinuous character of the motions, the problem of the maximum time \(T\) until the meeting of \(y(t)\) and \(z(t)\) also has other difficulties,

generally characteristic of such differential games [1–3]. To avoid these difficulties, we shall discuss here not a game for the maximin \(T\), where the existence of a saddle point is required, but rather a different meeting problem which, however, has common features with the problem of \(\min_u \max_v T\). In doing so, we shall approach the problem mainly from the pursuer’s point of view. The problem may be formulated as follows.

Problem 1. Let the pursuit have begun at the time \(t=t_0=0\). We assume that, for the pursued partner, at each instant \(t=\tau \geq 0\), the coordinates \(y_i(\tau)\), \(z_i(\tau)\) and estimates of the control resources \(\mu(\tau)\) and \(\nu(\tau)\) remaining for the future are known, while for the pursuer the quantities \(y_i(\tau)\), \(z_i(\tau+0)\), \(\mu(\tau)\), \(\nu(\tau+0)\), \(\vartheta(\tau+0)\) are known. However, the future motion of the adversary is unknown to each of the partners. It is required to determine a regulation law
\[ u(\tau)=\xi^0[y(\tau),\, z(\tau+0),\, \mu(\tau),\, \nu(\tau+0),\, \vartheta(\tau+0)] \]
\((\tau\geq t_0=0)\), consistent with constraint (1.3) and ensuring the meeting of the motions (1.1) and (1.2) in the chosen coordinates, no matter how the control
\[ v(\tau)=\eta[y(\tau),\, z(\tau),\, \mu(\tau),\, \nu(\tau)] \]
or \(v=v(t)\), consistent with constraint (1.4), is chosen.

In this case we shall say that the control \(u=\xi^0\), together with some control \(v=\eta^0\), constitutes a pair of optimal controls if

1) for \(u=\xi^0\) and \(v=\eta^0\) the meeting of \(y(t)\) and \(z(t)\) is necessarily realized at some instant \(t=\vartheta^0\);

2) for deviations of \(v\) from \(\eta^0\), the meeting occurs no later than at the instant \(t=\vartheta^0\).

Moreover, one should also require that the number \(\vartheta^0\) be the smallest among all numbers \(\vartheta\) that can satisfy the two conditions listed.

Remark. Controls that are optimal in the sense defined here are not, generally speaking, best from the point of view of the saddle-point conditions of the corresponding game for the time \(T\) until the meeting; but in any case they usually give a satisfactory solution of the problem. Conditions (1) and (2), taking into account the requirement of minimality of \(\vartheta^0\), may be interpreted as a requirement to ensure \(\min_u \max_v T\), which, however, as is known, need not coincide with \(\max_v \min_u T\).

§ 2. Let us discuss the possibility of solving Problem 1 by methods of the theory of dynamic programming [4, 1]. Suppose that in some domain \(G\) of the space \(\{y,z,\mu,\nu\}\), where \(\mu>0\), \(\nu>0\), a function \(T^0[y,z,\mu,\nu]\) has been constructed, continuous inside \(G\), and such that in this domain, when \(y_{i_k}=z_{i_k}\) \((k=1,\ldots,m)\), the equality \(T^0=0\) holds, while when \(y_{i_k}\ne z_{i_k}\) for at least one \(k\), we have \(T^0>0\). We also assume that on the boundary of the domain \(G\) the inequality \(T^0\geq \beta>0\) is satisfied, if
\[ \sum_{k=1}^{m}(y_{i_k}-z_{i_k})^2>0. \]

Suppose further that, when the controls \(u(t)\) and \(v(t)\) are chosen in the form of functions
\[ u(t)=\xi[y(t),\, z(t+0),\, \mu(t),\, \nu(t+0),\, v(t+0)],\quad v(t)=\eta[y(t),\, z(t),\, \mu(t),\, \nu(t)] \]
or \(v=\eta(t)\) from some set \(Q\{\xi,\eta\}\) of possible controls, the systems (1.1), (1.2) realize piecewise-continuous motions \(y(t)\), \(z(t)\) and controls \(u(t)\), \(v(t)\) satisfying the conditions specified above for (1.7).

Let, for some pair \(\xi=\xi^0,\ \eta=\eta^0\), the relation
\[ \max_{\eta}\min_{\xi}\left(\frac{dT^0}{dt}\right)_{\eta,\xi} = \left(\frac{dT^0}{dt}\right)_{\eta^0,\xi^0} =-1, \tag{2.1} \]
hold everywhere in \(G\), where \(dT^0/dt\) is understood in the sense of the right derivative. Then, whatever the initial conditions \(y(\tau)\), \(z(\tau)\), \(\mu(\tau)\), \(\nu(\tau)\) for which \(T^0<\beta\), under \(u=\xi^0\) the meeting of \(y(t)\) and \(z(t)\) in the chosen coordinates will occur no later than at the instant \(t=\tau+T^0\). Moreover, if in addition \(v=\eta^0\), then the meeting will take place precisely at the instant \(t=\tau+T^0\).

Indeed, it follows from (2.1) that

\[ \left(\frac{dT^0}{dt}\right)_{\eta,\xi^0}\leq -1, \tag{2.2} \]

where for \(\eta=\eta^0\) the equality sign will hold. Since the function
\(T^0[y(t),z(t),\mu(t),\nu(t)]\), by virtue of the conditions on \(y(t)\), \(z(t)\), \(\mu(t)\), \(\nu(t)\), is continuous in \(t\) from the left, the inequality (2.2) can be integrated. Therefore, integrating relation (2.2) from \(\tau\) to \(t\), we obtain the inequality

\[ T^0[y(t),z(t),\mu(t),\nu(t)]\leq T^0[y(\tau),z(\tau),\mu(\tau),\nu(\tau)]-(t-\tau), \tag{2.3} \]

which is valid at least for those \(t\) for which \(y(t)\), \(z(t)\), \(\mu(t)\), \(\nu(t)\) do not leave the domain \(G\). Consequently, for these \(t\) the quantity
\(T^0[y(t),z(t),\mu(t),\nu(t)]\) will be a monotonically decreasing function of time, and hence it cannot exceed the number \(\beta\) for those \(t\geq \tau\) as long as the point \(\{y(t),z(t),\mu(t),\nu(t)\}\) remains in the domain \(G\), if at \(t=\tau\) the inequality \(T^0<\beta\) was valid. It follows that throughout the time up to \(t=\tau+T^0\) this point does not leave \(G\). Thus, inequality (2.3) proves to be valid for all \(\tau\leq t<\tau+T^0\), and it follows directly from it that, for \(u=\xi^0\), \(v=\eta\), encounter will occur no later than at \(t=\tau+T^0\). If, however, \(u=\xi^0\) and \(v=\eta^0\), then, having equality in (2.3), we obtain encounter exactly at the moment \(t=\tau+T^0\).

Thus, we have verified that if relation (2.1) is satisfied, then the pair of controls \(\xi^0\) and \(\eta^0\) satisfies conditions (1) and (2) listed for optimal controls in the formulation of problem 1. At the same time it turns out that precisely
\(\vartheta^0=t_0+T^0[y(t_0),z(t_0),\mu(t_0),\nu(t_0)]\). It would now be desirable to verify that this pair of controls \(\xi^0\) and \(\eta^0\) constitutes a saddle point for the game corresponding to the given pursuit problem, i.e., in addition to the inequality \(T_{\eta\xi^0}\leq T_{\eta^0\xi^0}=T^0\), one would like to derive also the inequality \(T_{\eta^0\xi}\geq T_{\eta^0\xi^0}\). If this cannot be done, then one should at least verify that the number \(\vartheta^0=t_0+T^0\) is the smallest among all numbers \(\vartheta\) satisfying conditions (1) and (2). Unfortunately, the proof of the inequality \(T_{\eta^0\xi}\geq T_{\eta^0\xi^0}\), if it is carried out by analogy with the proof of the inequality \(T_{\eta\xi^0}\leq T_{\eta^0\xi^0}\), encounters the following obstacle. We can again integrate the inequality \([dT^0/dt]_{\eta^0,\xi}\geq -1\) and obtain \(T^0(t)\geq T^0(t_0)-(t-t_0)\). However, from this it still cannot be concluded that encounter of the motions \(y(t)\) and \(z(t)\) cannot occur earlier than at the moment \(t=\vartheta^0=t_0+T^0\), since from our assumptions on the properties of the function \(T^0[y,z,\mu,\nu]\) it does not follow that, when \(y_{[m]}(t)\to z_{[m]}(t)\), one must necessarily have \(\lim T^0=0\). This occurs because, for points where \(y_{[m]}=z_{[m]}\), we allow the possibility of being on the boundary of the domain \(G\), inside which alone continuity of the function \(T^0\) is assumed. It is, however, inexpedient to give up this possibility, since in the examples below one obtains functions \(T^0\) for which the set of points where \(y_{[m]}=z_{[m]}\) lies precisely on the boundary of the domain \(G\). A similar situation is analyzed in detail in example 7.1 in article [3], and we shall not dwell further on this question here. The proof of the minimality of \(\vartheta^0\) also presents a difficulty, but of a different kind. Here the difficulty is connected with the condition, introduced by us, for discriminating the motion \(z(t)\). This condition is in good agreement with the condition (2.1) considered here, but it introduces a certain artificiality into the reasoning which could have been carried out in the general case in order to justify the fact that
\(\vartheta^0=\min_{\xi}\max_{\eta}\vartheta\). The authors are not yet able to indicate a satisfactory path that would universally resolve the difficulties noted in the general case. Therefore, here we confine ourselves for the controls \(\xi^0,\eta^0\) only to proving the above-mentioned properties (1) and (2).

In conclusion of this paragraph we give the simplest example of a function \(T^0\) satisfying the criterion formulated. To this end, consider on the line \(\zeta\) two points \(m_1\) and \(m_2\) with coordinates \(\zeta_1\) and \(\zeta_2\), whose motions are described by the equations: \(\dot{\zeta}_1=u\), \(\dot{\zeta}_2=v\). Here restrictions of the form (1.5) are prescribed. Let us write the equations of motion in normal form

\[ \dot y_1=y_2,\qquad \dot y_2=u;\qquad \dot z_1=z_2,\qquad \dot z_2=v \tag{2.4} \]

and consider, for the system (2.4), the pursuit problem under the condition that at the moment of encounter only the coordinates \((y_1=z_1)\) of the moving points must coincide. It is not difficult to verify that for this problem all the conditions of the indicated criterion are satisfied by the function

\[ T^0=-\frac{x_1}{x_2+(\mu-\nu)}, \qquad x_1=y_1-z_1, \qquad x_2=y_2-z_2, \]

which is definite, continuous, and positive in every case when \(x_1<0\), \(x_2+\mu-\nu>0\). Condition (2.1) here means that at the very first instant \(\tau\) of the process the maximum possible impulse \(\mu(\tau)\) must be applied to the pursuing point \(m_1\) in the direction of the point \(m_2\), and to the point \(m_2\) the maximum impulse \(\nu(\tau)\), directed away from the point \(m_1\). We have given this example in order to show that even in the simplest case, where the answer is known in advance, working with the function \(T^0\) is not entirely obvious. Incidentally, here the level surfaces \(T^0=\mathrm{const}\), which are planes in the space \(\{x_1,x_2,\mu-\nu\}\), intersect at the point \(x_1=x_2=\mu-\nu=0\), which therefore lies on the boundary of the region \(G\).

§ 3. In the paper [1] a solution of pursuit problems is described that is based on the notion of regions of attainability for the processes (1.1) and (1.2). In this section we shall discuss the possibilities of this approach in the case of the constraints (1.3) and (1.4) considered here. Suppose that at some instant \(t=\tau\) the quantities \(y_i(\tau)\), \(z_i(\tau)\), \(\mu(\tau)\), \(\nu(\tau)\), \(\vartheta(\tau)\) have been realized. Analogously to what was done in [1], one can construct for the first and second objects the regions of attainability \(H^{(1)}[y(\tau),\mu(\tau),\vartheta]\) and \(H^{(2)}[z(\tau),\nu(\tau),\vartheta]\), i.e., such regions in the \(m\)-dimensional space \(X\) that for each point \(y'_{[m]}\) and \(z'_{[m]}\) in them one can construct controls \(u'(t)\) and \(v'(t)\) constrained by

\[ \int_{\tau}^{\vartheta+0} \left(\sum_{j=1}^{r}[u'_j(t)]^2\right)^{1/2}dt \leq \mu(\tau), \qquad \int_{\tau}^{\vartheta+0} \left(\sum_{s=1}^{l}[v'_s(t)]^2\right)^{1/2}dt \leq \nu(\tau) \]

and transferring the objects in the time \(\vartheta-\tau\) from the positions \(y(\tau)\) and \(z(\tau)\) to the positions \(y_{[m]}(\vartheta+0)=y'_{[m]}\) and \(z_{[m]}(\vartheta+0)=z'_{[m]}\), respectively.

Besides the regions of attainability \(H^{(1)}[y(\tau),\mu(\tau),\vartheta]\) and \(H^{(2)}[z(\tau),\nu(\tau),\vartheta]\), we shall also consider here the regions \(H^{(1)}[y(\tau+0),\mu(\tau+0),\vartheta]\) and \(H^{(2)}[z(\tau+0),\nu(\tau+0),\vartheta]\). These are such regions into each point \(y'_{[m]}\) and \(z'_{[m]}\) of which one can arrive in the time \(\vartheta-\tau\) from the position \(y(\tau+0)\), \(z(\tau+0)\) by means of controls \(u'(t)\) and \(v'(t)\) constrained by

\[ \int_{\tau+0}^{\vartheta+0} \left(\sum_{j=1}^{r}[u'_j(t)]^2\right)^{1/2}dt \leq \mu(\tau+0), \qquad \int_{\tau+0}^{\vartheta+0} \left(\sum_{s=1}^{l}[v'_s(t)]^2\right)^{1/2}dt \leq \nu(\tau+0). \]

The choice of optimal controls \(\xi^0\) and \(\eta^0\) at each instant of time \(t=\tau\geq 0\) in problem 1 may be connected with the indicated regions of attainability \(H^{(1)}[y(\tau),\mu(\tau),\vartheta^0]\), \(H^{(1)}[y(\tau+0),\mu(\tau+0),\vartheta^0]\) and \(H^{(2)}[z(\tau),\nu(\tau),\vartheta^0]\), \(H^{(2)}[z(\tau+0),\nu(\tau+0),\vartheta^0]\), where \(\vartheta^0\) is the moment of absorption of the process (1.2) by the process (1.1), i.e., the instant \(\vartheta\) at which the region \(H^{(2)}[z(\tau+0),\nu(\tau+0),\vartheta]\) first falls entirely inside the region \(H^{(1)}[y(\tau),\mu(\tau),\vartheta]\). In the case of constraints on the “energy” of the control action ([1], p. 7), the control \(\xi^0\) was chosen so that the region \(H^{(2)}[z(\tau+d\tau),\nu(\tau+d\tau),\vartheta^0]\) remained inside the region \(H^{(1)}[y(\tau+dt),\mu(\tau+dt),\vartheta^0]\), and this was achieved by aiming the motion \(y(t)\) at the point \(y_{[m]}(\vartheta^0)=x^*_{[m]}(\tau)\), where the boundaries of the regions \(H^{(1)}\) and \(H^{(2)}\) touch. This rule was workable there at least if there did not arise a situation in which the boundaries of the regions \(H^{(1)}\) and \(H^{(2)}\) had

more than one common point (including coincident points in cases of higher-order contact). Such situations could be regarded as exceptional. In the problem under consideration the situation is more complicated. Let us note first of all that, in the case of the constraints (1.3), (1.4), the choice of the control \(u=\xi^0\) by aiming the motion \(y(t)\) at a point of contact of the boundaries of the domains \(H^{(1)}\) and \(H^{(2)}\) does not, generally speaking, ensure the condition

\[ H^{(2)}[z(\tau+d\tau),\nu(\tau+d\tau),\vartheta^0]\subset H^{(1)}[y(\tau+d\tau),\mu(\tau+d\tau),\vartheta^0]. \tag{3.1} \]

This is connected, in particular, with the fact that, under constraints of the form (1.3), (1.4), the control resources \(\mu(\tau)\) and \(\nu(\tau)\) may change discontinuously. The instantaneous change of the resources leads, in turn, to discontinuous deformations of the domains \(H^{(1)}\) and \(H^{(2)}\). Therefore the choice, at a given instant \(t=\tau\), of the control \(u(\tau)\) on the basis of aiming at the point \(y_{[m]}(\vartheta^0)=x^*_{[m]}(\tau)\) may lead to such an instantaneous deformation of the domain \(H^{(1)}\) over the time \(d\tau\) as a result of which the domain \(H^{(2)}\) (or a part of it) will lie outside the domain \(H^{(1)}\), and for this reason meeting the point \(z(t)\) may become impossible. This can be verified on the simple example considered below in § 4. If, however, we want successfully to complete the pursuit process, relying on \(H^{(1)}\) and \(H^{(2)}\), then at the instant \(t=\tau\) it is desirable to choose the control \(u(\tau)\) so that, after the resource \(\mu(\tau)-\mu(\tau+d\tau)\) has been expended, there still remains the possibility of choosing the control \(u(t)\) for \(t>\tau+d\tau\), with the aid of which one could reach any point of the domain
\(H^{(2)}[z(\tau+d\tau),\nu(\tau+d\tau),\vartheta^0]\), without violating the constraint (1.3). Thus, the choice of the control is connected with the fulfillment of relation (3.1). Unfortunately, under the constraints (1.3), (1.4), in a broad class of cases it is not possible to indicate a general rule for choosing the control which would not violate, during the entire pursuit time, condition (3.1), if at the initial time \(\tau=0\) the domain \(H^{(2)}\) lies inside the domain \(H^{(1)}\). A simple example in which no control \(u(\tau)\) can keep the domain \(H^{(2)}\) inside the domain \(H^{(1)}\) is furnished by the problem of pursuit of the point \(m_2\) by the point \(m_1\) on the line \(\zeta\), when the motions of these points are described by the equations:

\[ \ddot{\zeta}_1=u,\qquad \ddot{\zeta}_2=-\zeta_2+v. \]

Here \(\zeta_1\) is the coordinate of the point \(m_1\); \(\zeta_2\) is the coordinate of the point \(m_2\).

Write the equations of motion in normal form

\[ \dot y_1=y_2,\qquad \dot y_2=u;\qquad \dot z_1=z_2,\qquad \dot z_2=-z_1+v \]

and, assuming that the constraints (1.5) are imposed on \(u\) and \(v\), require that at the instant of meeting both the coordinates \((y_1=z_1)\) and the velocities \((y_2=z_2)\) of the moving points coincide. Then it is not difficult to choose such initial conditions
\(z_1(0)=z_2(0)=0,\ \nu(0)=\nu_0,\)
\(y_1(0)=-\lambda,\ y_2(0)=0,\ \mu(0)=\mu_0\)
and a number \(\vartheta>2\pi\), for which, for the domains
\(H^{(1)}[y(0),\mu(0),\vartheta]\) and
\(H^{(2)}[z(0),\nu(0),\vartheta]\), we obtain the picture shown in Fig. 1.

Fig. 1

The only control \(u(+0)\) that keeps the point \(x^*\) in the domain
\(H^{(1)}[y(+0),\mu(+0),\vartheta]\) is the control \(u(+0)=\rho_1\delta(t)\), since \(\rho_1+\rho_2=\mu_0\). But this control instantaneously deforms the domain
\(H^{(1)}[y(0),\mu(0),\vartheta]\) (the parallelogram \(ABCD\)) into the domain
\(H^{(1)}[y(+0),\mu(+0),\vartheta]\) (the parallelogram \(MBx^*N\)). If, in addition, \(v(\tau)\equiv0\) up to the time \(t=\vartheta-2\pi\), then the domain
\(H^{(2)}[z(\tau),\nu(\tau),\vartheta]\) for \(\tau\le \vartheta-2\pi\) will remain an unchanged circle of radius \(\nu_0\). Thus, in attempting to keep the point \(x^*\) in \(H^{(1)}\), aiming at this point, we immediately throw out of the domain \(H^{(1)}\) an essential part of the domain \(H^{(2)}\) (this lost part of the domain \(H^{(2)}\) in Fig. 1 is sha-

shaded). In the example given, a considerable complication in the pursuit problem is introduced by the fact that the objects are not of the same type. For objects of the same type one can specify a rule for choosing a control that gives a satisfactory solution of Problem 1; however, here too a general study of the question of the principle for choosing the control \(\xi^0\), not violating the relation (3.1), requires a detailed mathematical analysis which lies beyond the scope of the present article. In this article we shall only examine the situation that arises in a simple example, where, however, many basic regularities already become apparent.

§ 4. Let two free material points move in the plane \(\{\zeta,\varkappa\}\): the pursuing point \(m_1\{\zeta_1,\varkappa_1\}\) and the pursued point \(m_2\{\zeta_2,\varkappa_2\}\), and let the motion of these points be described by the differential equations

\[ \ddot{\zeta}_1=u_1,\qquad \ddot{\varkappa}_1=u_2, \tag{4.1} \]

\[ \ddot{\zeta}_2=v_1,\qquad \ddot{\varkappa}_2=v_2. \tag{4.2} \]

Denoting \(\zeta_1=y_1,\ \dot{\zeta}_1=y_2,\ \varkappa_1=y_3,\ \dot{\varkappa}_1=y_4,\ \zeta_2=z_1,\ \dot{\zeta}_2=z_2,\ \varkappa_2=z_3,\ \dot{\varkappa}_2=z_4\), we write equations (4.1) and (4.2) in normal form

\[ \dot y_1=y_2,\qquad \dot y_2=u_1,\qquad \dot y_3=y_4,\qquad \dot y_4=u_2, \tag{4.3} \]

\[ \dot z_1=z_2,\qquad \dot z_2=v_1,\qquad \dot z_3=z_4,\qquad \dot z_4=v_2 \tag{4.4} \]

and assume that the controlling forces \(u=\{u_1,u_2\}\) and \(v=\{v_1,v_2\}\) are constrained by impulse restrictions

\[ \int_{\tau}^{\infty} [u_1^2(t)+u_2^2(t)]^{1/2}\,dt\leq \mu(\tau), \tag{4.5} \]

\[ \int_{\tau}^{\infty} [v_1^2(t)+v_2^2(t)]^{1/2}\,dt\leq \nu(\tau). \tag{4.6} \]

We shall require the encounter of the motions (4.3) and (4.4) only in the coordinates \((y_1=z_1,\ y_3=z_3)\), but not necessarily in the velocities \(y_2,y_4\) and \(z_2,z_4\). Then Problem 1 for the objects (4.3) and (4.4) will consist in determining a control law

\[ u(\tau)=\xi^0[y(\tau),z(\tau+0),\mu(\tau),\nu(\tau+0),v(\tau+0)], \]

consistent with the restriction (4.5) and ensuring an encounter of the motions (4.3) and (4.4) in the coordinates, no matter how the control \(v(\tau)\) consistent with the restriction (4.6) is chosen. The attainability regions \(H^{(1)}[y(\tau),\mu(\tau),\vartheta]\) and \(H^{(2)}[z(\tau),\nu(\tau),\vartheta]\) in the plane \(\{\zeta,\varkappa\}\) under the conditions (4.5), (4.6) are closed disks (see Fig. 2)

\[ \{y_1(\vartheta)-[y_1(\tau)+y_2(\tau)(\vartheta-\tau)]\}^2+ \]

\[ +\{y_3(\vartheta)-[y_3(\tau)+y_4(\tau)(\vartheta-\tau)]\}^2\leq \mu^2(\tau)(\vartheta-\tau), \tag{4.7} \]

\[ \{z_1(\vartheta)-[z_1(\tau)+z_2(\tau)(\vartheta-\tau)]\}^2+ \]

\[ +\{z_3(\vartheta)-[z_3(\tau)+z_4(\tau)(\vartheta-\tau)]\}^2\leq \nu^2(\tau)(\vartheta-\tau), \tag{4.8} \]

while the regions \(H^{(1)}[y(\tau+0),\mu(\tau+0),\vartheta]\) and \(H^{(2)}[z(\tau+0),\nu(\tau+0),\vartheta]\) are open disks. In Fig. 2 the regions \(H^{(1)}\) and \(H^{(2)}\) are shown precisely for the absorption moment \(\vartheta^0\) of process (4.4) by process (4.3). If the pursuit is conducted

...from the calculation of bringing the motion (4.3) to the point of tangency \(x^*\) of the boundaries of the disks \(H^{(1)}\) and \(H^{(2)}\), then in the present case this will mean that the pursuer must immediately use his entire control resource, and the region \(H^{(1)}\) will then instantaneously deform into a point. And if, in doing so, the pursued does not use the full stock of control, then condition (3.1) will necessarily be violated. However, according to the arguments in the preceding paragraph, it is desirable to choose the control \(u=\xi^0\) so that condition (3.1) be satisfied, i.e., so that the region \(H^{(2)}[z(\tau+d\tau), \nu(\tau+d\tau), \vartheta^0]\) remains inside the region \(H^{(1)}[y(\tau+d\tau), \mu(\tau+d\tau), \vartheta^0]\).

Fig. 2

Hence one may derive the following conclusion. If at the instant \(t=\tau\) the disks \(H^{(1)}[y(\tau), \mu(\tau), \vartheta^0]\) and \(H^{(2)}[z(\tau+0), \nu(\tau+0), \vartheta^0]\) do not coincide in their interiors, and if we wish that the capture time \(\vartheta^0\) not increase in the course of pursuit, then the control \(u=\xi^0\) is determined at this instant uniquely in the form of a \(\delta\)-function

\[ u=\xi^0=p\cdot\delta(t-\tau). \tag{4.9} \]

The vector \(p\) is computed in the following way:

\[ p_\xi=-\left[x_2+\frac{1}{\vartheta^0-\tau}x_1\right],\qquad p_\chi=-\left[x_4+\frac{1}{\vartheta^0-\tau}x_3\right], \tag{4.10} \]

\[ \vartheta^0-\tau= \frac{(x_1x_2+x_3x_4)}{(L^2-x_2^2-x_4^2)} + \]

\[ +\frac{\left[(x_1x_2+x_3x_4)^2+(x_1^2+x_3^2)(L^2-x_2^2-x_4^2)\right]^{1/2}} {(L^2-x_2^2-x_4^2)}, \tag{4.11} \]

where \(x_i(\tau)=y_i(\tau)-z_i(\tau+0)\), \(L(\tau)=\mu(\tau)-\nu(\tau+0)\). In this case, as is not difficult to verify, \([p_\xi^2+p_\chi^2]^{1/2}=L\), so that the equality \(\mu(\tau+0)=\nu(\tau+0)\) must hold. Then the region \(H^{(1)}[y(\tau+0), \mu(\tau+0), \vartheta^0]\) proves to coincide with the region \(H^{(2)}[z(\tau+0), \nu(\tau+0), \vartheta^0]\), and in the subsequent motion these regions must coincide all the time. Only in this case will condition (3.1) be satisfied at all times with a nonincreasing capture time \(\vartheta^0(\tau)\). It follows from this that the choice by the pursued partner of any control \(\nu(\tau)\) consistent with the restriction (4.6) uniquely determines the choice of the control \(u(\tau)\). Let us explain the meaning of the formulated pursuit rule. For this purpose we consider a coordinate system \(\{\xi',\chi'\}\) moving translationally together with the pursued point (see Fig. 3).

Let \(l\) be the straight line perpendicular to the direction \(AB\) from the point \(m_1\) to the point \(m_2\). At the initial instant \(t=t_0\) the pursuing point \(m_1\) receives the impulse (4.9), which in the system \(\{\xi',\chi'\}\) directs the point \(m_1\) toward the point \(m_2\). In this case, for the future (for \(t>t_0\)) the control resource is preserved as \(\mu(t_0+0)=\nu(t_0+0)\). If the point \(m_2\) continues to move along the

inertia (i.e., \(v(t)\equiv 0\)), then the point \(m_1\) must also move by inertia (i.e., one should put \(u(t)\equiv 0\)), and the meeting will occur at the instant \(\vartheta^0=t_0+T^0\), \(T^0=AB/|w|\), where \(w\) is the velocity of motion of the point \(m_1\) toward the point \(m_2\) in the relative motion. If a control force \(v(\tau)\) is applied to the point \(m_2\) which displaces it relative to the straight line \(l\) in the direction opposite to that in which \(m_1\) is located, then a control action \(u(\tau)\), equal to the action \(v(\tau)\), should be applied to the point \(m_1\), as a result of which the point \(m_1\) will copy the motion of the point \(m_2\)

Fig. 3

Fig. 4

in the system \(\{\xi'',\chi''\}\), moving relative to the system \(\{\xi',\chi'\}\) with velocity \(w\). When this condition is fulfilled for all \(\tau>0\), the meeting will again take place at the instant \(t=\vartheta^0\). If, however, a control action \(v(\tau)\) is applied to the point \(m_2\) which displaces it relative to the straight line \(l\) toward the side where the point \(m_1\) is located, then a control \(u(\tau)\) which is the mirror reflection of the control \(v(\tau)\) with respect to the straight line \(l\) must be applied to the point \(m_1\). In this case the approach of the points will accelerate, and the meeting will occur earlier than for \(t=\vartheta^0\) (see Fig. 4).

The behavior of the fleeing point should be regarded as reasonable if it does not choose a control action that displaces it relative to the straight line \(l\) toward the side where the point \(m_1\) is located; if, in this case, the pursuer copies its behavior, then the controls \(u=\xi^0\), \(v=\eta^0\) constitute a pair of optimal controls. The described law for choosing the control \(u(\tau)=\xi^0\) was implemented on an analog computing device for the system (4.3), (4.4)\(^1\).

Fig. 5

Fig. 6

In Figs. 5, 6, the trajectories of the pursuing object \((A)\) and the pursued object \((B)\) are shown in the plane \(\{\xi,\chi\}\); the meeting point is denoted by the letter \(C\). In Fig. 5 the trajectories of the objects are shown in the case of their optimal behavior; in Fig. 6 the tra—

\(^1\) In addition, the presence of constant terms on the right-hand sides of equations (4.3), (4.4) was assumed, which slightly complicates the problem and does not change the essence of the matter at all.

trajectories of the pursued and pursuing objects in the case when the pursued one chooses an uncontrolled method, and then the encounter occurs earlier than at \(t=\vartheta^0\). Thus, in the example considered, the control \(u(\tau)=\xi^0\) should be chosen according to the principle of copying and mirror reflection described above. Such a choice of \(u(\tau)\) in our example does not violate condition (3.1) at any time and satisfies all the conditions of Problem 1. Moreover, in this way we obtain here a solution of the problem on
\(\min_u \max_v T\).

However, it should be borne in mind that the indicated principle can be implemented only under the condition of discriminating the pursued object, and therefore in reality it cannot be applied. But the approach to the problem formulated above in § 3 is useful for the following reason: it determines the maximum of what the pursuer can achieve under constraints on the impulses of the control actions. We shall now show in what way one can get rid of the discrimination condition, and how the above pursuit principle can be approximated by a control law having a more realistic meaning.

§ 5. Let \(H^{(1)}[y(\tau),\mu(\tau),\vartheta^0]\) and \(H^{(2)}[z(\tau),\nu(\tau),\vartheta^0]\), as before, be the attainability domains constructed for the absorption instant \(\vartheta^0\). We can now do one of two things: either 1) slightly increase \(\mu(\tau)\) to a value \(\mu^*(\tau)>\mu(\tau)\), or 2) slightly increase the number \(\vartheta^0\) to a number \(\vartheta^*>\vartheta^0\). If, in doing so, we discard certain degenerate situations, which we shall leave aside, then in both cases it will turn out that the domains \(H^{(2)}[z(\tau),\nu(\tau),\vartheta^0]\) or \(H^{(2)}[z(\tau),\nu(\tau),\vartheta^*]\) lie strictly inside the domains \(H^{(1)}[y(\tau),\mu^*(\tau),\vartheta^0]\) or \(H^{(1)}[y(\tau),\mu(\tau),\vartheta^*]\), respectively. Let, then, \(\varepsilon(\tau)\) be the least distance between the boundaries of the corresponding domains \(H^{(1)}\) and \(H^{(2)}\). We shall now say that \(\vartheta^0_\varepsilon\) is an instant of \(\varepsilon\)-absorption of process (1.2) by process (1.1) if, for \(t=\vartheta^0_\varepsilon\), the distance between the boundaries of
\(H^{(1)}[y(\tau),\mu(\tau),\vartheta^0_\varepsilon]\) and
\(H^{(2)}[z(\tau),\nu(\tau),\vartheta^0_\varepsilon]\) first becomes equal to \(\varepsilon\) (under the condition that \(H^{(2)}\subset H^{(1)}\)). As already noted, the transition from \(\vartheta^0\) to \(\vartheta^0_\varepsilon\) can be carried out either by means of an additional reserve of impulse \(\mu\) of the control \(u\), or by means of some increase of \(\vartheta\). In the regular cases, to which we have agreed to restrict ourselves, \(\lim \vartheta^0_\varepsilon=\vartheta^0\) as \(\varepsilon\to0\).

Thus, let us set some small number \(\varepsilon(t_0)=\varepsilon>0\) and find the instant \(\vartheta^0_\varepsilon\). Suppose that we have succeeded in making such a choice of the control \(u\) that, during the pursuit, the quantities \(\varepsilon(\tau)\) and \(\vartheta^0_{\varepsilon(\tau)}\) do not increase with changing \(\tau\), and the function \(\varepsilon(\tau)\) remains positive in any case until the difference \(\vartheta^0_{\varepsilon(\tau)}-\tau\) becomes smaller than some quantity \(\sigma>0\). Obviously, then, from the pursuer’s point of view, the result of the pursuit may be considered satisfactory; and if the quantities \(\varepsilon\) and \(\sigma\) are chosen sufficiently small, the result of the pursuit may in general turn out to be sufficiently close to the optimal one. Such a choice of the control \(u\) is indeed possible, at least in the cases of objects of the same type. Moreover, such a choice of the control \(u\) is possible in the form of an algorithm relying only on the values \(y(\tau), z(\tau), \mu(\tau), \nu(\tau)\), which frees us from the restrictive condition of discriminating the motion \(z(t)\). At the same time, by choosing \(\varepsilon>0\) sufficiently small, one can obtain results close to those given by the use of the above-mentioned discrimination condition. Without discussing this method of constructing \(u\) in the general case of objects of the same type, and without giving proofs (this investigation will be the subject of a separate work), we shall demonstrate it here on the problem considered in the preceding paragraph.

Let, therefore, the motions of the objects be described by equations (4.3) and (4.4), and let conditions (4.5) and (4.6), restricting the resources, be given.

control of the objects. As before, we shall require the meeting of the motions (4.3) and (4.4) only in the coordinates. Let us construct the regions \(H^{(1)}[y(\tau),\mu(\tau),\vartheta_\varepsilon^0]\) and \(H^{(2)}[z(\tau),\nu(\tau),\vartheta_\varepsilon^0]\) for our example. These regions will be disks (see Fig. 7), again determined by relations (4.7) and (4.8), in which only \(\vartheta_\varepsilon^0\) is substituted for \(\vartheta\).

Let \(\vartheta_\varepsilon^0\) be the instant of \(\varepsilon\)-absorption of process (4.4) by process (4.3). The following relation holds:

\[ \vartheta_\varepsilon^0-\tau = \frac{x_1x_2+x_3x_4+\varepsilon L}{L^2-x_2^2-x_4^2} + \frac{\left[(x_1x_2+x_3x_4+\varepsilon L)^2+(x_1^2+x_3^2-\varepsilon^2)(L^2-x_2^2-x_4^2)\right]^{1/2}} {L^2-x_2^2-x_4^2}, \tag{5.1} \]

where \(x_i(\tau)=y_i(\tau)-z_i(\tau)\), \(L(\tau)=\mu(\tau)-\nu(\tau)\). We shall now treat the continuous pursuit process as the limiting case of a multistep discrete process under the condition that the step \(\Delta t\to0\). In doing so, only the motion \(y(t)\) need be specified, while the motion \(z(t)\) may be regarded from the very beginning as described by the corresponding differential equations. Then the algorithm for constructing \(u(t)\) is as follows. Suppose that at the instant \(t=t_k\) the quantities \(\varepsilon(t_k)>0\) and \(\vartheta_\varepsilon^0(t_k)=\vartheta_\varepsilon^0(t_k)\) have been realized. The control \(u(t)\) on the interval \(t_k<t\le t_k+\Delta t\) is chosen constant and equal to

\[ u(t)=\frac{p}{\|p\|}\,\gamma(t_k), \tag{5.2} \]

where the vector \(p\) is directed toward the point \(\alpha\) (see Fig. 7),

\[ p_\xi=-\left[x_2+\frac{1}{\vartheta_\varepsilon^0(t_k)-t_k}\,x_1\right],\qquad p_\chi=-\left[x_4+\frac{1}{\vartheta_\varepsilon^0(t_k)-t_k}\,x_3\right], \]

\[ x_i(t_k)=y_i(t_k)-z_i(t_k), \]

and the quantity \(\gamma(t_k)\) is defined by the equality

\[ \gamma(t_k)=\frac{[\mu(t_k)-\nu(t_k)][\vartheta_\varepsilon^0(t_k)-t_k]}{\varepsilon(t_k)}-1. \tag{5.3} \]

Consequently, on the indicated interval,

\[ u_1(t)=\frac{p_\xi}{\|p\|}\,\gamma(t_k),\qquad u_2(t)=\frac{p_\chi}{\|p\|}\,\gamma(t_k),\qquad \|p\|=[p_\xi^2+p_\chi^2]^{1/2}. \tag{5.4} \]

After carrying out the motions \(y(t)\) and \(z(t)\) on the interval \(t_k<t<t_k+\Delta t=t_{k+1}\), one can, preserving the old value \(\vartheta=\vartheta_\varepsilon^0(t_k)\), find the smallest distance \(\varepsilon^*(t_{k+1})\) between the boundaries of the regions \(H^{(1)}[y(t_{k+1}),\)

\(\mu(t_{k+1})\), \(\vartheta_\varepsilon^0(t_k)]\) and \(H^{(2)}[z(t_{k+1}), v(t_{k+1}), \vartheta_\varepsilon^0(t_k)]\). If it turns out that \(\varepsilon^*(t_{k+1}) \geqslant \varepsilon(t_k)\), then one must find a new instant of \(\varepsilon\)-absorption \(\vartheta_\varepsilon^0=\vartheta_\varepsilon^0(t_{k+1})\), putting \(\varepsilon=\varepsilon(t_k)\), and then continue the computation for the interval \(t_{k+1}<t<t_{k+1}+\Delta t\), proceeding from \(\varepsilon(t_{k+1})=\varepsilon(t_k)\) and from the new \(\vartheta_\varepsilon^0(t_{k+1})\). If, however, it turns out that \(\varepsilon^*(t_{k+1})<\varepsilon(t_k)\), then one must seek a new instant of \(\varepsilon\)-absorption \(\vartheta_\varepsilon^0=\vartheta_\varepsilon^0(t_{k+1})\), now putting \(\varepsilon=\varepsilon^*(t_{k+1})=\varepsilon(t_{k+1})\), and then continue the computation proceeding from the new values \(\varepsilon(t_{k+1})=\varepsilon^*(t_{k+1})\), \(\vartheta_\varepsilon^0(t_{k+1})\).

Fig. 8

It remains only to explain the meaning of the equalities (5.2), (5.3), which determine here the control \(u\). This meaning is as follows: the control \(u(t)\) (5.2), (5.3) is close to that \(\delta\)-impulse control which, at the instant \(t=t_k\), aims the motion \(y(t)\) at the point \(\alpha\) (see Fig. 7), where the boundary of the region \(H^{(1)}\) touches the boundary of the region \(H^{(2)}\), expanded homothetically until contact with the boundary of the region \(H^{(1)}\). Here, however, this \(\delta\)-impulse control is spread out into a certain ordinary function, constant on an interval of positive measure. Thus, in this form it is possible here to preserve the rule of aiming at the point \(\alpha\) of tangency of the boundaries of the regions \(H^{(1)}\) and \(H^{(2)}\), formulated in papers [1, 3].

Figure 8 shows several implementations of the process of pursuit of object (4.4) by object (4.3), modeled on an electronic digital computer. It is assumed that the pursuer chooses the control \(u(t)\) (5.2), while the pursued moves according to the law

\[ z_1=0.4t,\quad z_2=0.4,\quad z_3=0.2,\quad z_4=0,\quad 0\leqslant t<0.25; \]

\[ z_1=0.3t+0.025,\quad z_2=0.3,\quad z_3=0.3t+0.125,\quad z_4=0.3,\quad 0.25\leqslant t. \]

The initial state of the objects was chosen as follows:

\[ y_1(0)=y_2(0)=y_3(0)=y_4(0)=0,\quad \mu_0=0.4\sqrt{2}+\sqrt{0.1}, \]

\[ z_1(0)=0,\quad z_2(0)=0.4,\quad z_3(0)=0.2,\quad z_4(0)=0,\quad \nu_0=\sqrt{0.1}. \]

The evading object (dashed line) moves by inertia on the interval \(0\leqslant t<0.25\); at the instant \(t=0.25\), in a prescribed direction, it expends the entire reserve of control \(\nu_0\) by an impulse, and thereafter the motion again proceeds by inertia. The solid line in Fig. 8 denotes the motion of the pursuing object if it is guided by the best possible strategy described in the preceding paragraph, under the condition of discrimination of the motion \(z(t)\). The pursuit time is then determined by formula (4.11), and it will be equal to \(T^0=0.5\). Figure 8 also shows the trajectories of the pursuing object (lines with small circles) if it is controlled according to the law \(u(t)\) (5.2), for different initial values \(\varepsilon(t_0)=0.1;\ 0.05;\ 0.025\). The pursuit time \(T_\varepsilon^0=\vartheta_\varepsilon^0-t_0\) is then determined by formula (5.1), and it will be respectively equal to \(T_\varepsilon^0=0.91;\ 0.69;\ 0.59\).

In Fig. 9, a and b show the control actions \(u_1(t)\) and \(u_2(t)\) (5.4), corresponding to \(\varepsilon(t_0)=0.025\). We see that the control \(u(t)\) (5.2) indeed approximates the \(\delta\)-impulse control \(u=\xi^0\) (4.9), which is symbolically shown in Fig. 9 in the form of \(\delta\)-circles. This approximation will, of course, be the better the smaller \(\varepsilon(t_0)\) is. Finally, Fig. 10, a presents the graphs of the variation of the control resources, and Fig. 10, b the change in time of the function \(\varepsilon(t)\), which indeed turns out to be positive throughout the entire time of pursuit.

Fig. 9

Fig. 10

It should be noted that the behavior of the evader in the numerical example given is reasonable in the sense indicated in the preceding paragraph; therefore \(\vartheta_\varepsilon^0(\tau)\) remains a constant quantity, and encounter occurs at the moment \(t=\vartheta_\varepsilon^0(t_0)\).

References

1. N. N. Krasovskii, Yu. M. Repin, V. E. Tret’yakov, Izv. AN SSSR, Tekhnicheskaya kibernetika, No. 4, 1965.
2. L. S. Pontryagin, DAN SSSR, 156, No. 4, 1964.
3. N. N. Krasovskii, PMM, 30, issue 2, 1966.
4. R. Bellman, O. Glicksberg, J. Gross, Some Questions in the Mathematical Theory of Control Processes. Moscow: IL, 1962.

Received by the editors
February 2, 1966.

Ural State University
named after A. M. Gorky