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UDC 517.917
ON THE CONSTRUCTION OF A FAMILY OF INTEGRAL LINES SITUATED IN A PRESCRIBED REGION
A. D. MYSHKIS, A. A. SHARSHANOV
1. Statement of the problem. In a number of applications there arises a problem which, in general form, may be formulated as follows.
Let there be given a system of equations
\[ \frac{dx}{dt}=f(x,t)\qquad (0\le t<\infty;\ x=(x_1,\ldots,x_n);\ f=(f_1,\ldots,f_n)) \tag{1} \]
and some set \(K\subset E_{x,t}\) (i.e., the space of \(x,t\); the assumptions will be indicated below). It is required to construct the set \(L\subset E_x\) of initial values \(x(0)\) of all solutions of system (1) for which
\[ x(t)\in K\qquad (0\le t<\infty). \tag{2} \]
Suppose, for example, that a mechanical system is considered, on whose generalized coordinates \(q_1,\ldots,q_m\) constraints of the nature of inequalities are imposed, so that the representative point must, in the course of the evolution of the system, remain at all times in some prescribed region \(Q\subset E_q\). Then, in constructing possible solutions, we arrive at the general problem formulated above, putting \(K=Q\oplus E_{\dot q}\oplus(0\le t<\infty)\); this is the problem of the “passage” of a mechanical system through a prescribed aperture. Such a problem for a linear system with periodic coefficients for \(m=1\) and \(2\), arising in connection with the theory of accelerators, was considered in [1]; in this case the corresponding region \(L\) of the space of initial states of the system was called the “capture region” of the aperture \(Q\).
Various variants of the original general problem are possible. Thus, the right-hand side of system (1), as well as the set \(K\), may depend on some parameter \(\lambda\), and it is then required that condition (2) be satisfied for all values of \(\lambda\) under consideration. This variant is obtained, in particular, in the problem of the passage of a mechanical system through a prescribed aperture if the initial conditions are given at any \(t=-\lambda\ge0\): it suffices to consider the system
\[ \frac{dx}{dt}=f(x,t+\lambda)\qquad (0\le t<\infty) \]
with the correspondingly shifted \(K\), in order to arrive at the indicated variant.
2. A fundamental solution. Assume, for simplicity, that the function \(f(x,t)\) is continuous throughout \(E_x\oplus(0\le t<\infty)\) and ensures local uniqueness of the solution of the initial-value problem \(x(t_0)=x_0\) (\(t_0\ge0\)); denote this solution by \(x=\varphi(t;x_0,t_0)\). For any set \(S\subset E_x\), denote by \(\varphi(t_1;S,t_0)\) the totality of all values \(x(t_1)\) of those solutions, extendable from \(t_0\) to \(t_1\), for which \(x(t_0)\in S\). Finally, denote by
\(K_\tau \subset E_x\) \((\tau \geqslant 0)\) be the intersection of the set \(K\) with the plane \(t=\tau\). Then the solution of the initial general problem has the form
\[ L=\bigcap_{0\leqslant \tau<\infty}\varphi(0;\,K_\tau,\tau). \tag{3} \]
From this the following simple properties follow.
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If all the sets \(K_\tau\) \((0\leqslant \tau<\infty)\) are closed and all solutions of system (1) are continuable to the whole semiaxis \(0\leqslant t<\infty\), then the set \(L\) is closed.
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If for each \(\tau\geqslant 0\) the set \(K_\tau\) is closed, and the set
\[ \bigcup_{0\leqslant r\leqslant \tau} K_r \]
is bounded, then the set \(L\) is closed.
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If system (1) is linear (not necessarily homogeneous) and all the sets \(K_\tau\) \((0\leqslant \tau<\infty)\) are convex, then the set \(L\) is also convex.
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If \(f(-x,t)=-f(x,t)\) and all the sets \(K_\tau(0\leqslant \tau<\infty)\) are centrally symmetric (with respect to the origin), then the set \(L\) is also centrally symmetric.
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If for all \(x\in K_0\) one has \(\varphi(t;\,x,0)\in K_t(0\leqslant t<\infty)\), then \(L=K_0\).
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If for each \(\tau\geqslant 0\) the set \(K_\tau\) is nonempty and compact (in itself), and for every \(r\in[0,\tau]\), \(x\in K_\tau\) one has \(\varphi(r;\,x,\tau)\in K_r\), then the set \(L\) is nonempty and compact.
If the function \(f\) and the set \(K\) depend on a parameter \(\lambda\), then on the right-hand side of (3) one must put an additional intersection over all considered values of \(\lambda\). In this case properties 1–5 remain valid if the formulated conditions are imposed for each considered value of \(\lambda\).
3. Construction of the boundary of the sought domain. The problem of an exact or approximate construction of the set \(L\), as well as of estimating the measure of this set, is in the general case rather difficult. In [1] this problem was investigated for the system indicated in Sec. 1 and for a domain \(K\) of a special form. Here we shall give a general method for constructing the set \(L\), which is easily justified in concrete examples; however, the limits of applicability of this method are not yet completely clear, so that the arguments given below are valid only “generally speaking.”
Suppose that the function \(f\) is sufficiently smooth, all solutions of system (1) are continuable to the semiaxis \(0\leqslant t<\infty\), and the set \(K\) is described by the inequality \(F(x,t)\leqslant 0\), where the function \(F(x\in E_x,\ 0\leqslant t<\infty)\) is sufficiently smooth and \(\operatorname{grad} F\ne 0\) everywhere when \(F=0\). Then each set \(\varphi(0;\,K_\tau,\tau)\) is bounded by a sufficiently smooth surface \(D_\tau\), and therefore, by virtue of formula (3), the envelope of this family of surfaces has greater significance.
Since the equation of \(D_\tau\) has the form
\[ F(\xi,\tau)\big|_{\xi=\varphi(\tau;\,x,0)}=0, \tag{4} \]
the equation of the envelope is obtained by adjoining to (4) the result of differentiating (4) with respect to \(\tau\):
\[ \left[\operatorname{grad} F(\xi,\tau)\cdot f(\xi,\tau)+F'_{\tau}(\xi,\tau)\right]\big|_{\xi=\varphi(\tau;\,x,0)}=0. \tag{5} \]
If \(\tau\) is eliminated from equalities (4) and (5), one obtains the equation of the envelope of the family of surfaces \(D_\tau\). The envelope is constructed analogously if the boundary of the domain \(K\) is given in parametric form (cf. Sec. 5).
The set \(L\), generally speaking, is bounded both by pieces of the envelope and by pieces of individual surfaces \(D_\tau\). Suppose that \(n=2\), each of the sets \(\varphi(0;\,K_\tau,\tau)\) is a half-plane (in Fig. 1 one of these half-planes is shaded), and the envelope has the form shown
in Fig. 1 by the dashed line. Then the set \(L\) is the densely shaded part of the plane in Fig. 1. In concrete simple examples this construction is carried out without difficulty. It would be interesting to investigate such a construction in some general case, in particular to indicate conditions under which the boundary of \(L\) belongs entirely to the envelope.
If system (1) is not integrable in explicit form, then the envelope of the family \(D_\tau\) can be constructed by numerical integration. For this it is necessary, for each \(\tau \geqslant 0\), to solve system (4), (5) with respect to \(\xi\), which gives
\[ \xi=\psi(\tau;\alpha_1,\ldots,\alpha_{n-2}), \]
where \(\alpha_i\) are certain parameters; after this the desired envelope can be represented in parametric form
\[ x=\varphi(0;\psi(\tau;\alpha_1,\ldots,\alpha_{n-2}),\tau). \]
Fig. 1.
Having considered the position of the envelope relative to the surfaces \(D_\tau\), which can likewise be done by numerical integration, one can draw a conclusion about the form of the boundary of the set \(L\).
If the functions \(f\) and \(F\) depend on the parameter \(\lambda\), then the family of surfaces \(D=D_{\tau,\lambda}\) will be two-parameter, and in order to construct the boundary of the set \(L\) one must differentiate the equation of \(D_{\tau,\lambda}\) with respect to \(\tau\) and with respect to \(\lambda\), and then eliminate \(\tau\) and \(\lambda\) from the equalities obtained. As above, this envelope of a two-parameter family of surfaces in the general case gives only part of the boundary \(L\).
From the parametric equation of the envelope one can, by integration, obtain or estimate the measure of the parts of \(L\) adjacent to this envelope.
4. Simple examples. For simple autonomous problems the construction of the domain \(L\) presents no difficulty and is clear in principle from Fig. 2, where \(K_\tau\) does not depend on \(\tau\) and is a circle bounded by a bold line, while the domain \(L\) is shaded. For stable linear systems \((n=2)\)
\[ \begin{aligned} 1)\quad & \dot{x}=y,\qquad \dot{y}=-\omega^2 x \quad (\omega=\mathrm{const}>0),\\ 2)\quad & \dot{x}=y,\qquad \dot{y}=-\omega^2 x-2\eta y\\ & \qquad (\eta=\mathrm{const},\ \omega>\eta),\\ 3)\quad & \text{the same, } \omega<\eta \end{aligned} \]
in the case when \(K_\tau\) is a strip \(|x|\leqslant a\), the domain \(L\) is shown in Fig. 3. The same result is obtained by the method of § 3.
Fig. 2.
5. The envelope for linear systems. For simplicity we shall show the construction of the envelope of the family \(D_\tau\) for the second-order system
\[ \dot{x}=a(t)x+b(t)y,\qquad \dot{y}=c(t)x+d(t)y, \tag{6} \]
if \(K_\tau\), for each \(\tau \geqslant 0\), is bounded by the line with equation
\[ x=\psi(s;\tau),\qquad y=\chi(s;\tau). \tag{7} \]
(If there are several such lines, then for each of them the envelope is constructed independently of the others.)
Let \(x_1(t), y_1(t)\) and \(x_2(t), y_2(t)\) be solutions of system (6) for which
\[ x_1(0)=1,\quad y_1(0)=0,\quad x_2(0)=0,\quad y_2(0)=1. \]
Then the envelope of the lines \(D_\tau\) is determined by the equations
\[ x x_1(\tau)+y x_2(\tau)=\psi(s;\tau),\quad x y_1(\tau)+y y_2(\tau)=\chi(s;\tau), \]
\[ x x_1'(\tau)+y x_2'(\tau)=\psi_s'(s;\tau)s_\tau' + \psi_\tau'(s;\tau), \]
\[ x y_1'(\tau)+y y_2'(\tau)=\chi_s'(s;\tau)s_\tau' + \chi_\tau'(s;\tau), \]
Fig. 3
from which \(s,\ \tau\) and \(s_\tau'\) must be eliminated. The last two equations, in view of the first two and (6), can be rewritten in the form
\[ a(\tau)\psi(s;\tau)+b(\tau)\chi(s;\tau) = \psi_s'(s;\tau)s_\tau' + \psi_\tau'(s;\tau), \]
\[ c(\tau)\psi(s;\tau)+d(\tau)\chi(s;\tau) = \chi_s'(s;\tau)s_\tau' + \chi_\tau'(s;\tau). \]
Let, in particular, \(K_\tau\) for all \(\tau\ge 0\) be the straight line \(x=A\); then one may put \(\psi\equiv A,\ \chi\equiv s\), and we obtain the system of equations
\[ x x_1(\tau)+y x_2(\tau)=A,\quad x y_1(\tau)+y y_2(\tau)=s, \]
\[ a(\tau)A+b(\tau)s=0,\quad c(\tau)A+d(\tau)s=s_\tau', \]
whence we find the parametric equations of the envelope
\[ x=\frac{A}{W(\tau)} \left[ y_2(\tau)+\frac{a(\tau)}{b(\tau)}x_2(\tau) \right] = \frac{A x_2'(\tau)}{b(\tau)W(\tau)}, \]
\[ y=-\frac{A}{W(\tau)} \left[ \frac{a(\tau)}{b(\tau)}x_1(\tau)+y_1(\tau) \right] = -\frac{A x_1'(\tau)}{b(\tau)W(\tau)}, \tag{8} \]
where
\[ W(\tau)=x_1(\tau)y_2(\tau)-x_2(\tau)y_1(\tau). \]
For estimates of the region \(L\) in the case of periodic coefficients, see [1]. The construction of envelopes for a linear homogeneous or nonhomogeneous system of arbitrary order \(n\) in the general case, when instead of (7) there will be
\(x_i=\psi_i(s_1,s_2,\ldots,s_{n-1};\tau)\) \((i=1,2,\ldots,n)\), and also in the particular case of a boundary of the form \(x_1=A\), is carried out in a completely analogous way. In this case the envelope is determined by a system of \(n-1\) parameters, which may be taken to be \(\tau\) and the parameters \(s_1,\ldots,s_{n-1}\), connected by one relation.
References
- Sharshanov A. A. Differential Equations, 3, No. 4, 589—600, 1967.
Received by the editors
May 10, 1966
Physico-Technical Institute
of Low Temperatures, Academy of Sciences of the Ukrainian SSR,
Physico-Technical Institute
Academy of Sciences of the Ukrainian SSR