Abstract
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Reports of the Academy of Sciences of the USSR
- Volume 140, No. 1
MATHEMATICS
M. K. KERIMOV
ON THE THEORY OF THE SECOND VARIATION FOR DISCONTINUOUS VARIATIONAL PROBLEMS IN SPACE
(Presented by Academician A. A. Dorodnitsyn on 6 III 1961)
The present note is devoted to the theory of the second variation for discontinuous variational problems with movable ends in a space of many dimensions ((^1)). We have chosen the nonparametric case, since in this case the following essential difference between parametric and nonparametric problems becomes apparent: in the parametric case ((^2)), both primary and secondary extremals are polygonal lines, whereas in the nonparametric case the primary extremals are polygonal, while the secondary extremals have discontinuities of the first kind. Thus there arise discontinuous variational problems with discontinuous extremals. As in ((^1)), we shall confine ourselves to the case in which the integrand in the integral (J), for which the minimum is sought, has only one surface of discontinuity. However, our results are without difficulty carried over to the case of any finite number of discontinuity surfaces.
- Let (E_{102}) be a polygonal extremal realizing the minimum of the functional (J). To compute the second variation of the discontinuous functional (J), we generalize the method proposed by Morse ((^3)) for continuous variational problems. Let (\alpha_h(a), \beta_h(a), \gamma_h(a)) ((h=1,2,\ldots,n)) be functions of class (C^{(2)}) for (a) near (a=0). Substituting them into the primary end conditions ((^1)) and differentiating with respect to (a), at (a=0) we find
[
\eta_i^1=\eta_i(x^1)=C_{ih}^1\tau_h^1,\qquad
\eta_i^2=\eta_i(x^2)=C_{ih}^2\tau_h^2,
]
[
\eta_i^{0-}=\eta_i^{0-}(x^0)=C_{ih}^{0-}\tau_h^0,\qquad
\eta_i^{0+}=\eta_i^{0+}(x^0)=C_{ih}^{0+}\tau_h^0
\qquad (h,\ i=1,2,\ldots,n),
\tag{A}
]
where
[
\eta_i(x)=y_a(x,0),\qquad
\tau_h^1=\alpha_h'(0),\qquad
\tau_h^2=\gamma_h'(0),\qquad
\tau_h^0=\beta_h'(0),
]
[
C_{ih}^k(0)=y_{ih}^k(0)-y_{ix}(x^k)x_h^k(0),
]
[
C_{ih}^{0-}=y_{ih}^0(0)-y_{ix}^{-}(x^0)x_h^0(0),\qquad
C_{ih}^{0+}=y_{ih}^0(0)-y_{ix}^{+}(x^0)x_h^0(0)
]
[
(i,h=1,2,\ldots,n;\ k=1\ \text{or}\ 2).
]
We shall call conditions (A) the secondary end conditions. One can construct ((^1)) a one-parameter family of admissible curves
[
y_i=y_i(x,a)
\begin{cases}
y_i^{-}(x,a), & x^1(a)\leq x\leq x^0(a),\
y_i^{+}(x,a), & x^0(a)\leq x\leq x^2(a),
\end{cases}
\tag{1}
]
containing the given polygonal extremal (E_{102}). The functions (\eta_i(x)) and the constants (\tau_h^1,\tau_h^0,\tau_h^2) will be called the variations of the family (1) along (E_{102}).
Differentiating the function (J(a)) twice with respect to (a) and putting (a=0), taking into account the primary transversality and discontinuity conditions, we obtain the expression
[
J''(0)=J_2(\eta,\tau)
=b_{hl}^1\tau_h^1\tau_l^1+
\left(b_{hl}^{0-}+b_{hl}^{0+}\right)\tau_h^0\tau_l^0+
b_{hl}^2\tau_h^2\tau_l^2
]
[
+\int_{x^1}^{x^0}2\omega^1(x,\eta,\eta')\,dx+
\int_{x^0}^{x^2}2\omega^2(x,\eta,\eta')\,dx,
\tag{2}
]
which we shall call the second variation of the functional (J) along (E_{102}). In formula (2) (b_{hl}^{1}), (b_{hl}^{0-}), (b_{hl}^{0+}), (b_{hl}^{2}) are rather complicated expressions depending on (F^{1}), (F^{2}) and their derivatives at the points (1,0,2), and
[
2\omega^{k}(x,\eta,\eta')=
F^{k}{y_i y_j}\eta_i\eta_j
+2F^{k}{y_i y'j}\eta_i\eta'_j
+F^{k}\eta'_i\eta'_j
]
[
(i,j=1,2,\ldots,n;\ k=1\ \text{or}\ 2).
]
By an admissible set of variations (\eta_i(x)), (\tau_h^1), (\tau_h^0), (\tau_h^2) ((x^1\le x\le x^2)) we shall mean a set for which (\tau_h^1,\tau_h^0,\tau_h^2) are constants, and the (\eta_i(x)) belong to the class (D^{(1)}) on (x^1\le x\le x^2), with the exception of the point (x=x^0), where they have a discontinuity of the first kind subject to the condition
[
\eta_i^{0-}=C^{0-}{ih}\tau_h^0,\qquad
\eta_i^{0+}=C^{0+}\tau_h^0.
\tag{3}
]
2. We shall say that the broken extremal (E_{102}), intersecting the manifolds (M^1,M^2) at the points (1,2) with observance of the primary transversality conditions, satisfies the necessary condition of Jacobi in terms of the sign of the second variation (condition IV), if (J_2(\eta,\tau)) along (E_{102}) is nonnegative for every admissible set of variations satisfying conditions (A).
Theorem 1. In order that the broken extremal (E_{102}) realize a minimum of the functional (J), it is necessary that it satisfy condition IV.
The proof is based on the following lemma.
Lemma. For every admissible set of variations (\eta_i(x)), (\tau_h^1), (\tau_h^0), (\tau_h^2) along (E_{102}), satisfying conditions (A), there exists a one-parameter family of admissible curves (1), containing (E_{102}) for (a=0), such that (\eta_i(x)), (\tau_h^1), (\tau_h^0), (\tau_h^2) is the set corresponding to this family along (E_{102}); the functions (y_i^-(x,a)), (y_i^+(x,a)) possess all the continuity and differentiability properties required in computing the second variation of the functional (J) along (E_{102}).
3. Let us formulate a new (so-called secondary) variational problem for the functional (J_2(\eta,\tau)) in the class of admissible sets of variations (\eta_i(x)), (\tau_h^1,\tau_h^0,\tau_h^2), satisfying conditions (A). We have a problem of the same type as the primary variational problem (1), with the sole difference that now the functions (\eta_i(x)) of the class (D^{(1)}), having at (x=x^0) a discontinuity of the first kind, are taken as admissible functions, while in place of the manifolds (M^1), (M^0) and (M^2) there appear (n)-dimensional linear manifolds (N^1,N^0,N^2) with equations (A).
A set (\eta_i(x)), (\tau_h^1,\tau_h^0,\tau_h^2), realizing a minimum of the functional (J_2(\eta,\tau)), must satisfy the system of secondary Euler equations
[
\mathcal{J}i(\eta)=\frac{d}{dx}\omega{\eta'i}-\omega
\begin{cases}
\displaystyle \frac{d}{dx}\omega^1_{\eta'i}-\omega^1=0,
& x^1\le x\le x^0, \tag{4}\[6pt]
\displaystyle \frac{d}{dx}\omega^2_{\eta'i}-\omega^2=0,
& x^0\le x\le x^2,
\tag{5}
\end{cases}
]
[
(i=1,2,\ldots,n)
]
and the boundary conditions
[
C^1_{ih}\xi_i^1(x^1)=b^1_{hl}\tau_l^1,\qquad
C^2_{ih}\xi_i^2(x^2)=-\,b^2_{hl}\tau_l^2,
\tag{6}
]
[
C^{0-}{ih}\xi_i^1(x^0)-C^{0+}\xi_i^2(x^0)
=-\,(b^{0-}{hl}+b^{0+})\tau_l^0
\quad (i,l,h=1,2,\ldots,n),
\tag{7}
]
where the notation has been introduced
[
\xi_i^1(x)=\omega^1_{\eta'i}\,[x,\eta(x),\eta'(x)],\qquad
\xi_i^2(x)=\omega^2\,[x,\eta(x),\eta'(x)].
\tag{8}
]
We shall call (6) the secondary transversality conditions, and (7) the secondary corner condition.
A broken curve
[
\eta_i=\eta_i(x)
\begin{cases}
\eta_i^{-}(x), & x^1 \leqslant x \leqslant x^0,\
\eta_i^{+}(x), & x^0 \leqslant x \leqslant x^2
\end{cases}
\qquad (i=1,2,\ldots,n),
]
whose arcs (\eta_i^{-}(x)) and (\eta_i^{+}(x)) satisfy equations (4), (5), respectively, on (x^1 \leqslant x \leqslant x^0) and (x^0 \leqslant x \leqslant x^2), and condition (7) for (x=x^0), is called a secondary broken (or simply broken) extremal. Since at the points 1, 0, 2 of the polygonal extremal (E_{102}) the transversality condition is satisfied, it follows (cf. (3)) that in neighborhoods of these points the linear manifolds (N^1, N^0, N^2) will be regular.
Theorem 2. For any extremal (\eta_i^{-}(x),\ \tau_h^1) ((x^1 \leqslant x \leqslant x^0)) of equation (4), satisfying the first of conditions in (A) and (6) and intersecting the manifold (N^0), there exists a unique extremal (\eta_i^{+}(x),\ \tau_h^0) ((x^0 \leqslant x \leqslant x^2)) of equation (5), satisfying together with (\eta_i^{-}(x)) conditions (3) and (7).
This theorem makes it possible, for each family of extremals of equation (4), to construct a unique supplementary family, which together with the first constitutes a family of broken extremals of the secondary variational problem.
Introduce canonical variables (\eta_i,\ \xi_i^k) by means of the equations
[
\xi_i^k=\omega_{\eta_i'}^{\,k}(x,\eta,\eta'),
\tag{9}
]
from which one can find
[
\eta_i'=\Pi_i^k(x,\eta,\xi^k), \qquad i=1,2,\ldots,n
\tag{10}
]
((k=1) corresponds to the interval (x^1 \leqslant x \leqslant x^0), and (k=2) to the interval (x^0 \leqslant x \leqslant x^2)). Write the Hamiltonian functions
[
\mathfrak{H}^k(x,\eta,\xi^k)
=
[\eta_i'\omega_{\eta_i'}^{\,k}-\omega^k]_{\eta_i'=\Pi_i^k}
=
\Pi_i^k\xi_i^k-\omega^k(x,\eta,\Pi^k)
\tag{11}
]
((k=1) on (x^1 \leqslant x \leqslant x^0) and (k=2) on (x^0 \leqslant x \leqslant x^2)).
The canonical systems of equations, equivalent to the Euler equations (5), (6), have the form
[
d\eta_i/dx=\mathfrak{H}{\xi_i^k}^{\,k}, \qquad
d\xi_i^k/dx=-\mathfrak{H},}^{\,k
\qquad i=1,2,\ldots,n
\tag{12}
]
((k=1) on (x^1 \leqslant x \leqslant x^0) and (k=2) on (x^0 \leqslant x \leqslant x^2)).
Take system (12) for (k=1) with the initial conditions
[
\eta_i(x^1)=C_{il}\tau_l^1, \qquad
\xi_i^1(x^1)C_{ih}^1=b_{hl}^1\tau_l^1
\quad (i,h,l=1,2,\ldots,n).
\tag{13}
]
The system of (2n) linear equations (13) has a maximal system of (n) linearly independent solutions (\eta_{il}(x^1),\ \xi_{il}^1(x^1),\ \tau_{hl}^1) ((l=1,2,\ldots,n)), since the rank of the matrix (|C_{ih}|) is equal to (n). Consequently, system (12) for (k=1) has (n) linearly independent solutions (\eta_{il}(x),\ \xi_{il}^1(x)) ((x^1 \leqslant x \leqslant x^0,\ l=1,2,\ldots,n)), taking the initial values (\eta_{il}(x^1),\ \xi_{il}^1(x^1)) ((l=1,2,\ldots,n)).
Theorem 3. For any linearly independent system of solutions (\eta_{il},\ \xi_{il}^1,\ \tau_{hl}^1) of equations (12) for (k=1), satisfying the initial conditions (13), there exists a unique supplementary system of linearly independent solutions (\eta_{il},\ \xi_{il}^2,\ \tau_{hl}^0) of equations (12) for (k=2), and the resulting system of broken solutions of equations (12) is linearly independent.
In what follows we shall denote this system simply by (\eta_{il}(x),\ \xi_{il}(x),\ \tau_{hl}).
Let two systems of variables (x,\eta_i,\eta_i') and (x,u_i,u_i') be given. Introduce the notation
[
\bar{\xi}i^k(x)=\omega\ 2.}^k(x,u,u'),\qquad i=1,2,\ldots,n;\quad k=1\ \text{or
]
Then
[
\eta_i\omega_{\eta_i}^k(x,u,u')+\eta_i'\bar{\xi}i^k(x)
=
u_i\omega^k(x,\eta,\eta')+u_i'\xi_i^k(x),
]
[
\eta_i\mathcal{J}_i^k(u)-u_i\mathcal{J}_i^k(\eta)=\frac{d}{dx}Q^k(\eta;u),
]
where
[
Q^k(\eta;u)=\eta_i\bar{\xi}_i^k-u_i\xi_i^k,\quad k=1\ \text{or}\ 2.
]
If (\eta_i) and (u_i) are a pair of discontinuous solutions of the system (4)—(5), then (Q^1(\eta;u)=\mathrm{const}) or (Q^2(\eta;u)=\mathrm{const}). If these constants are equal to zero, then (\eta_i(x)), (u_i(x)) are called mutually adjoint.
Theorem 4. Any two solutions (\eta_i,\xi_i,\tau_h) and (\bar{\eta}_i,\bar{\xi}_i,\bar{\tau}_h) belonging to a system of discontinuous linearly independent solutions of equations (12) are mutually adjoint.
A system of pairwise mutually adjoint discontinuous linearly independent solutions (\eta_{il},\xi_{il},\tau_{hl}) will be called an adjoint basis, and the determinant
[
\Delta(x)
\begin{cases}
\Delta^-(x)=|\eta_{il}^-(x)|, & x^1\leq x\leq x^0,\[4pt]
\Delta^+(x)=|\eta_{il}^+(x)|, & x^0\leq x\leq x^2\quad (i,l=1,2,\ldots,n)
\end{cases}
\tag{14}
]
the basis determinant.
Every solution (\eta_i(x),\xi_i(x),\tau_h) of problem (12)—(13) is linearly expressible in terms of the solutions (\eta_{il},\xi_{il},\tau_{hl}) of the adjoint basis.
The totality of all solutions of the canonical system depending on an adjoint basis will be called an adjoint family.
Definition of a focal point. Corresponding to the value (x=x^3) ((x^3\neq x^1,\ x^3\neq x^0)), a point of a nonsingular broken extremal (E_{102}), transversal to the manifold (M^1) at point 1, is called a focal point of the manifold (M^1) if problem (4)—(5)—(13) has a discontinuous solution (u_i(x)) such that (u_i(x^3)=0), but not all (u_i) are identically equal to zero on the interval (x^1