Full Text
UDC 517.934+62.50
MATHEMATICS
Corresponding Member of the Academy of Sciences of the USSR N. N. Krasovskii
ON THE GAME ENCOUNTER OF MOTIONS
Consider the problem \((^{1-5})\) of the game encounter of the pursuing \((y[t])\) and pursued \((z[t])\) motions
\[ \dot y = Ay + Bu, \qquad \dot z = Az + Bv, \tag{1} \]
where the realizations \(u[t]\) and \(v[t]\) of the controls \(u\) and \(v\) are constrained by
\[ \|u\| \le \mu, \qquad \|v\| \le \nu \qquad (\mu > \nu), \tag{2} \]
the symbol \(\|q\|\) denoting the Euclidean norm of the vector \(q\). Here \(y, z\) are \(n\)-vectors; \(u, v\) are \(r\)-vectors. The vectors under consideration are treated as column vectors; the superscript \('\) will denote transposition, and the symbol \(\{q\}_m\) will denote the vector composed of the first \(m\) components of the vector \(q\). The process begins at a prescribed time \(t=t_0\). The encounter of the motions \(y\) and \(z\) at the instant \(t^* = t_0 + T\) is defined as a situation satisfying the condition
\[ \{y[t_0+T] - z[t_0+T]\}_m = 0, \tag{3} \]
where the number \(m\) is given.
The solution of the minimax problem of the time \(T\) until the encounter of the motions \(y\) and \(z\) is determined by the theorems from works \((^2,^3,^6,^7)\), if, among the arguments of the function \(u=u^0\) specifying the control for the pursuer, the current values of the control \(v[t]\) are allowed. Otherwise, the solution of this problem can be obtained in limiting form from a certain approximating scheme \((^8)\). In this case, again, among the arguments of the function \(u^0\), along with the phase vectors \(y[t]\) and \(z[t]\), there appears an additional variable \(\vartheta(t)\). The limiting motions \(y[t], z[t], \vartheta[t]\) generated in this way are included within the framework of generalized solutions \((^9)\) of the resulting differential equations with discontinuous right-hand sides. This is achieved in the following way.
An extremal control \(u=u^0\) is constructed, defining equations (1) as equations in contingencies. For this purpose, in the \((2n+1)\)-dimensional space \(\{y,z,\vartheta\}\) \((\vartheta>0)\), two subsets \(W_0\) and \(W^e\) are distinguished, which are specified by the following conditions: let \(\vartheta^0(y,z)\) be the time until the moment of absorption \((^5,^10)\), \(t^0=t_*+\vartheta^0\), of the process \(z[t]\) by the process \(y[t]\) (from the state \(y[t_*]=y,\ z[t_*]=z\)); then the set \(W_0\) is the totality of all states \(\{y,z,\vartheta\}\) satisfying the condition \(\vartheta \ge \vartheta^0(y,z)\); the complementary set \(W^e\), on the contrary, is the totality of states \(\{y,z,\vartheta\}\) that satisfy the condition \(\vartheta < \vartheta^0(y,z)\). In the domain \(W_0\), the function \(u^0(y,z,\vartheta)\) is chosen nonuniquely and may take any values satisfying the condition
\[ \|u^0\| \le \mu. \tag{4} \]
In the domain \(W^e\), the function \(u^0(y,z,\vartheta)\) is defined by the rule of extremal aiming \((^5,^8,^10)\). Let \(\varepsilon^0(y,z,\vartheta)\) be the smallest value of \(\varepsilon\) for which the closed \(\varepsilon\)-neighborhood \(G^{(1)}[y[t],\vartheta;\varepsilon]\) of the reachability domain \(G^{(1)}[y[t],\vartheta]\) of the process \(y\) (from the state \(y[t]\) to the moment \(t_\vartheta=t+\vartheta\)) contains the reachability domain \(G^{(2)}[z[t],\vartheta]\) of the process \(z\) (from the state \(z[t]\) to the moment \(t_\vartheta\)). Further, let \(u[t]=u_*(y[t],z[t],\vartheta[t])\) be that control which, at the instant \(t\), aims the motion
\(y[t]\) to a point \(q^0 \in G^{(1)}[y[t], \vartheta]\) nearest to that point \(p^0\) at which the boundaries of the domains \(G^{(1)}[y[t], \vartheta; \varepsilon^0]\) and \(G^{(2)}[z[t], \vartheta]\) touch. If at the point \(\{y,z,\vartheta\}\in W^\varepsilon\) the function \(u_*(y,z,\vartheta)\) is continuous, then \(u^0(y,z,\vartheta)=u_*(y,z,\vartheta)\). In the opposite case \(u^0(y,z,\vartheta)\) is again a multivalued function, constrained only by condition (4). The additional variable \(\vartheta[t]\) is constrained by the regulating relation
\[ (d\vartheta[t]/dt)^+ \leq -1, \tag{5} \]
which is adjoined to the original equations of motion (1) (the symbol \((d\vartheta/dt)^+\) denotes the upper derivative).
By a generalized solution (motion) \(\{y[t],z[t],\vartheta[t]\}\) \((t\geq t_0)\) of the system (1), (5) for \(u=u^0(y,z,\vartheta)\) we shall mean any vector function \(\{y[t],z[t],\vartheta[t]\}\) that satisfies the conditions:
\(1^\circ\). The functions \(y[t]\) and \(z[t]\) are absolutely continuous and for almost all values of \(t\) satisfy equations (1), where \(u=u^0\) and arbitrary piecewise-continuous realizations \(v[t]\) are admissible, constrained by condition (2).
\(2^\circ\). The variable \(\vartheta[t]\) is right-continuous for all values of \(t\) and satisfies condition (5). In the domain \(W^\varepsilon\) the function \(\vartheta[t]\) is continuous and satisfies the equation \(d\vartheta/dt=-1\), while in the domain \(W_0\) the possible values of \(\vartheta[t]\) are constrained by the condition \(\varepsilon^0(y[t],z[t],\vartheta[t])=0\).
The following assertion is valid.
I. Suppose that for the initial state \(y[t_0]\), \(z[t_0]\) of system (1) the condition \(\vartheta^0(y[t_0],z[t_0])<\infty\) is fulfilled. Then, for \(u=u^0\), every generalized solution \(\{y[t],z[t],\vartheta[t]\}\) \((t\geq t_0)\) of the system (1), (5) with the initial condition \(y[t_0]\), \(z[t_0]\), \(\vartheta[t_0]=\vartheta^0(y[t_0],z[t_0])\) has the property that \(\{y[t],z[t],\vartheta(t)\}\in W_0\) for all \(t\geq t_0\), until the encounter occurs. This inclusion ensures the encounter of the motions \(y[t]\) and \(z[t]\) no later than at the moment \(t^0=t_0+\vartheta^0(y[t_0],z[t_0])\).
Assertion (I) means that
\[ \min_u \max_v T_{u,v}=T_{u^0(y,z,\vartheta),\,v^0}=\vartheta^0(y[t_0],z[t_0]) \tag{6} \]
among the controls \(u=u(y,z,\vartheta)\), \(v=v[t]\) \(\bigl(v^0[t]=v u^0(y[t],z(t),\vartheta[t])/\mu\bigr)\), and the given minimax is realized by the control \(u=u^0\), which, when \(v\ne v^0\), generally forces sliding regimes \(y[t]\).
Remark 1. Among the motions \(\{y[t],z[t],\vartheta[t]\}\) under consideration, for the pursuer perhaps the most interesting are those for which, for \(t\geq t_0\), the equality \(\vartheta[t]=\vartheta^0(y[t],z[t])\) holds throughout up to the encounter. Indeed, on these motions, for \(t\geq \tau\geq t_0\), the encounter for every \(\tau\) takes place no later than at the moment \(t^0[\tau]=\tau+\vartheta^0(y[\tau],z[\tau])\). From example (7), considered in article (8), it follows that the imposition of the regulating relation (5) is essential for ensuring the encounter of the motions \(y[t]\) and \(z[t]\).
Thus, the extremal control \(u^0(y,z,\vartheta)\) ensures the minimax (6) for the time \(T_{u,v}\) until encounter. On the other hand, one can verify by an example that the extremal control \(v^0(y,z)\) (or \(v^0(y,z,\vartheta)\)), selected at each instant of time \(t\) from the condition that the motion \(z[t]\) be aimed at the moment \(t^0=t+\vartheta^0(y[t],z[t])\) at the point \(q^0\), where the boundaries of the domains \(G^{(1)}[y[t],\vartheta]\) and \(G^{(2)}[z[t],\vartheta]\) touch, does not ensure the maximin
\[ \max_v \min_u T_{u,v}=\vartheta^0(y[t_0],z[t_0]). \tag{7} \]
An example of this kind may be furnished by the problem of the game encounter only in the coordinates of two controllable material points \(m^{(1)}\) and \(m^{(2)}\), described by the equations
\[ m^{(1)}\ddot{\xi}^{(1)}=u,\qquad m^{(2)}\ddot{\xi}^{(2)}=v, \tag{8} \]
where \(\xi^{(1)},\xi^{(2)}\) are three-dimensional vectors, and the controls \(u\) and \(v\) are constrained by conditions (2). However, the following assertion is valid.
II. For any number \(\delta>0\) one can construct a control
\(v_\delta[t]=v_\delta(y[t],z[t],\vartheta_\delta[t])\)
\((d\vartheta_\delta/dt=-1,\ \vartheta_\delta[t_0]=\vartheta^0(y[t_0],z[t_0])-\delta\), which preserves the motion \(z[t]\) from meeting the motion \(y[t]\) for
\(t_0\le t<t_0+\vartheta^0(y[t_0],z[t_0])-\delta\), whatever the realization of the control \(u[t]\) constrained by condition (2).
The control \(v_\delta\) again determines the motion \(z[t]\) as a generalized solution of the corresponding equation (1). Here the control \(v_\delta[t]\) is constructed from the condition of nonincrease of the Lyapunov function
\[ \chi(y[t],z[t],\vartheta_\delta[t])= \int_t^{t+\vartheta_\delta} \varepsilon^0(y[t],z[t],\tau-t)^{-1}\,d\tau \]
with respect to time \(t\). It is then verified that the function
\(\chi[t]=\chi(y[t],z[t],\vartheta_\delta[t])\) does not increase if the control \(v_\delta[t]\) is chosen from the minimum condition
\[ \int_t^{t+\vartheta_\delta} \varepsilon^0(y[t],z[t],\tau-t)^{-2}\psi'_\tau(t)B \left( \frac{vB'\psi_\tau(t)}{\|B'\psi_\tau(t)\|}-v_\delta[t] \right)d\tau = \min_{\|v\|\le v}, \tag{9} \]
where \(\psi_\tau(t)\) is the vector-function appearing in the conditions of the maximum principle (11) for the problem of determining the quantity
\(\varepsilon^0(y[t],z[t],\tau-t)\).
Remark 2. The quantity \(\vartheta^0(y[t_0],z[t_0])\), generally speaking, is greater than the optimal time \(\vartheta_0\) for the problem of program maximin pursuit considered in paper (12). Thus, the use of the realizing values \(y[t]\) and \(z[t]\) in the law determining the control \(v[t]\) by the feedback principle ensures, for the pursued motion, a better result than the program control \(v(t)\), which is based only on the initial data \(y[t_0]\) and \(z[t_0]\). At the same time, the control \(v_\delta\) described by us, constructed according to the feedback principle, ensures for the pursued object a result \(T_\delta=\inf_u T_{u,v}\) arbitrarily close to the best guaranteed result for the pursuer,
\[ T^0=\min_u\max_v T_{u,v}. \]
Above, the problem of exact meeting of the motions \(y\) and \(z\) in \(m\) coordinates was considered. Results I and II, however, are transformed in an understandable way to the case of the problem of \(\gamma\)-meeting, which is defined by the condition
\[ \|\{y[t_0+T]-z[t_0+T]\}_m\|\le \gamma, \tag{10} \]
where only the time \(\vartheta^0(y,z)\) until absorption is replaced by the time
\(\vartheta^{(\gamma)}(y,z)\) until \(\gamma\)-absorption. It should also be noted that results I and II carry over without substantial changes to the case of arbitrary convex and similar constraints \(u\in U\) and \(v\in V\) on the controls \(u\) and \(v\).
Ural State University
named after A. M. Gorky
Received
19 IV 1968
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