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UDC 517.94
UNIQUENESS OF THE SOLUTION OF THE CORRECTION PROBLEM FOR SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS
B. V. KVEDARAS
1. Introduction. In the note [1], some necessary and sufficient conditions were indicated for the existence and uniqueness of a solution of the correction problem for a nonlinear differential equation of the second order.
In the present article, some conditions are found for the uniqueness of the solution of the correction problem for systems of linear differential equations; namely, a class of systems is indicated for which the solution of the correction problem, if it exists, is unique. The question of the existence of a solution is not considered.
2. Statement of the problem. Let a vector function
\(a(t) = (a_1(t), a_2(t), \ldots, a_n(t))\) with bounded and measurable positive components \(a_i(t)\) be given on the interval \([0,T]\).
We shall call the vector-function \(f(t) = (f_1(t), \ldots, f_n(t))\) a correction function if \(f_i(t) \equiv 0\) outside some interval \([t_i,\tau_i]\) contained in \([0,T]\), and \(f_i(t) = a_i(t)\) on \([t_i,\tau_i]\) \((i = 1, 2, \ldots, n)\). If \(t_i = \tau_i\), then we shall assume \(f_i(t) \equiv 0\) on \([0,T]\).
By a solution of the correction problem we shall mean such a correction function \(f(t)\) for which there exists a solution \(x(t)\) of the system of differential equations
\[ \frac{d^2 x}{dt^2} - A(t)x = f(t), \tag{1} \]
satisfying the prescribed conditions
\[ x(0)=x(0), \qquad x(T)=x_T, \]
\[ \frac{dx(0)}{dt}=\dot{x}_0, \qquad \frac{dx(T)}{dt}=\dot{x}_T. \]
Here \(x\) is an \(n\)-dimensional vector; \(A(t)\) is a square matrix of order \(n\) with elements \(a_{ij}(t)\) \((i,j=1,2,\ldots,n)\), functions defined, bounded, and measurable on \([0,T]\).
Suppose that there exist two correction functions \(f^{(1)}(t)\) and \(f^{(2)}(t)\), such that the solutions \(x(t,f^{(1)})\) and \(x(t,f^{(2)})\) of equation (1) satisfy the same boundary conditions. Then the function \(x(t) = x(t,f^{(1)}) - x(t,f^{(2)})\) will be a solution of equation (1) with right-hand side \(f(t)=f^{(1)}(t)-f^{(2)}(t)\), satisfying the conditions
\[ x(0)=x(T)=0, \qquad \dot{x}(0)=\dot{x}(T)=0. \tag{2} \]
Thus, the question of the uniqueness of the solution of the correction problem is equivalent to the question of the unsolvability of problem (1), (2) for all functions \(f(t)\) that are differences of two correction ...
functions. We shall denote the set of such functions by \(K\). We describe the structure of the functions in \(K\) as follows.
Let
\[ \alpha^k(t)=(0,\ldots,\alpha_k^k(t),\ldots,0), \]
\[ \beta^k(t)=(0,\ldots,\beta_k^k(t),\ldots,0), \]
where
\[ \alpha_k^k(t)= \begin{cases} \pm a_k(t), & \text{if } t\in (a_k,b_k),\\ 0, & \text{if } t\notin (a_k,b_k); \end{cases} \]
\[ \beta_k^k(t)= \begin{cases} \pm a_k(t), & \text{if } t\in (c_k,d_k),\\ 0, & \text{if } t\notin (c_k,d_k). \end{cases} \]
The intervals \((a_k,b_k)\) and \((c_k,d_k)\), for each \(k=1,2,\ldots,n\), are assumed not to intersect. Then an arbitrary function \(f(t)\in K\) is representable in the form
\[ f(t)=\sum_{k=1}^{n}[\alpha^k(t)-\beta^k(t)]. \]
Here the function \(\alpha_k^k(t)\) is negative on \((a_k,b_k)\) if the support of the function \(f^{(1)}(t)\) is contained in the support of \(f^{(2)}(t)\), and \(\beta_k^k(t)\) is negative on \((c_k,d_k)\) under the opposite inclusion.
3. Reduction of the problem. Let \(H\) denote the Hilbert space of vector functions \(g(t)=(g_1(t),\ldots,g_n(t))\), whose components \(g_i(t)\) are defined and square-summable on the interval \([0,T]\). We define the scalar product in \(H\) by the formula
\[ (x,y)=\int_{0}^{T}\sum_{i=1}^{n} x_i(t)\overline{y_i(t)}\,dt, \]
and the norm by \(\|x\|^2=(x,x)\).
Let the set of functions \(x(t)\in H\) that are absolutely continuous, have absolutely continuous first derivatives, and have second derivatives belonging to the space \(H\) be denoted by \(H_0\). Obviously, \(H_0\) is everywhere dense in \(H\).
Define the operator \(L_0\) by the formula
\[ L_0x=\frac{d^2x}{dt^2}-A(t)x \]
on the domain \(D\), which consists of all functions \(x(t)\in H_0\) satisfying conditions (2).
Thus, the question of the uniqueness of the solution of the correction problem is equivalent to the unsolvability of the operator equation
\[ L_0x=f \tag{3} \]
for all \(f(t)\in K\).
Repeating known arguments (see [2]), one can show that the operator adjoint to \(L_0\) is the operator
\[ L_0^*y=\frac{d^2y}{dt^2}-A^*(t)y \]
with domain \(D(L_0^*)=H_0\). Here \(A^*(t)\) is the matrix with elements \(a_{ij}^*(t)=\overline{a_{ji}(t)}\). A well-known theorem on the zeros of the adjoint operator leads to the following assertion.
Theorem 1. In order that the operator equation \(L_0 x = f\) be solvable, it is necessary and sufficient that \(f(t)\) be orthogonal to every solution of the equation
\[ \frac{d^2 y}{dt^2} - A^*(t)y = 0 . \tag{4} \]
Let the vector functions \(\varphi^1(t), \varphi^2(t), \ldots, \varphi^n(t), \psi^1(t), \psi^2(t), \ldots, \psi^n(t)\) be a fundamental system of solutions of equation (4).
Let \(f(t) \in K\). Then
\[ (f,\varphi^i)=\sum_{j=1}^{n}\int_{0}^{T} f_j(t)\overline{\varphi_j^i(t)}\,dt = \]
\[ =\sum_{j=1}^{n}\int_{0}^{T}\left[\alpha_j^i(t)-\beta_j^i(t)\right]\overline{\varphi_j^i(t)}\,dt =\sum_{j=1}^{n}(\alpha^j,\varphi^i)-\sum_{j=1}^{n}(\beta^j,\varphi^i); \]
\[ (f,\psi^i)=\sum_{j=1}^{n}(\alpha^j,\psi^i)-\sum_{j=1}^{n}(\beta^j,\psi^i) \quad (i=1,2,\ldots,n). \]
Theorem 2. If the solution of the correction problem is not unique, then at least one nontrivial solution \(y(t)\) of equation (4) satisfies the conditions
\[ (y,\alpha^i)=0,\qquad (y,\beta^i)=0 \quad (i=1,2,\ldots,n). \tag{5} \]
Proof. If the correction problem has more than one solution, then there exists a function \(f(t)\in K\), orthogonal (by Theorem 1) to all solutions of equation (4). Consequently,
\[ \sum_{j=1}^{n}(\alpha^j,\varphi^i)-\sum_{j=1}^{n}(\beta^j,\varphi^i)=0, \]
\[ \sum_{j=1}^{n}(\alpha^j,\psi^i)-\sum_{j=1}^{n}(\beta^j,\psi^i)=0 \quad (i=1,2,\ldots,n). \tag{6} \]
Denote
\[ b_{ij}=(\alpha^j,\varphi^i),\qquad b_{i\,n+j}=-(\beta^j,\varphi^i), \]
\[ b_{n+i\,j}=(\alpha^j,\psi^i),\qquad b_{n+i\,n+j}=-(\beta^j,\psi^i) \]
and consider the system of \(2n\) linear algebraic equations
\[ \sum_{j=1}^{2n} b_{ij}u_j=\gamma_i \quad (i=1,2,\ldots,2n), \]
or, in vector form, \(Bu=\gamma\), where \(B=(b_{ij})\) is a \(2n\)-dimensional square matrix with elements \(b_{ij}\).
It follows from (6) that the homogeneous system \(Bu=0\) has the nontrivial solution \(u^0=(1,1,\ldots,1)\).
If \(m \geqslant 0\) functions among \(\alpha^i(t)\) and \(\beta^i(t)\) \((i=1,2,\ldots,n)\) are identically equal to zero on \([0,T]\), then the system \(Bu=0\) will have another \(m\) nontrivial solutions of the form \((0,\ldots,1,\ldots,0)\), where the unit stands in that place
\(j=1,2,\ldots,2n\), for which column \(j\) in the matrix \(B\) is equal to zero (i.e., the corresponding \(\alpha^j(t)\) or \(\beta^{j-n}(t)\equiv 0\)).
It follows from this that the adjoint system \(B^{*}v=0\) also has \(m+1\) linearly independent nonzero solutions.
Let \(v=(c_1,c_2,\ldots,c_n,c'_1,\ldots,c'_n)\) be one of these solutions. Then, for \(i\leq n\), we have
\[ \sum_{j=1}^{2n}\overline{b}_{ji}v_j = \sum_{j=1}^{n}(\overline{\alpha^i,\varphi^j})c_j + \sum_{j=1}^{n}(\overline{\alpha^i,\psi^j})c'_j = \]
\[ = \left(\sum_{j=1}^{n}[c_j\varphi^j+c'_j\psi^j],\,\alpha^i\right) = (y,\alpha^i)=0, \]
where
\[ y(t)=\sum_{j=1}^{n}[c_j\varphi^j(t)+c'_j\psi^j(t)] \tag{7} \]
is a nonzero solution of equation (4).
In the same way, for \(n\leq i<2n\), we find that
\[
\sum_{j=1}^{2n}\overline{b}_{ji}v_j=(y,\beta^{i-n})=0.
\]
The theorem is proved.
From the fact that the system \(B^{*}v=0\) has at least \(m+1\) solutions, it follows that equation (3) also has at least \(m+1\) linearly independent solutions of the form (7), satisfying conditions (5).
Define the operator \(L\) by the equality
\[ Ly=\frac{d^2y}{dt^2}-A^{*}(t)y \tag{8} \]
with domain of definition \(D(L)\), consisting of vector-functions \(y(t)\in H_0\) satisfying conditions (5). Obviously, \(D(L)\) is not dense in \(H\). From the proof of Theorem 2 it follows that
Corollary. If \(m>0\) of the functions among \(\alpha^i(t)\) and \(\beta^i(t)\) \((i=1,2,\ldots,n)\) are identically equal to zero and the operator equation \(Ly=0\) has only \(m\) linearly independent solutions, then equation (3) is not solvable.
4. Uniqueness theorem. Before formulating the main result, we shall prove several auxiliary propositions. We shall consider the scalar equation
\[ -\frac{d^2x}{dt^2}-\lambda(t)x=0, \tag{9} \]
where \(\lambda(t)\geq 0\) is a measurable bounded function defined on \([0,T]\).
Lemma 1. Let \(\alpha(t)\), \(\beta(t)\) be sign-constant measurable and bounded scalar functions on \([0,T]\), not identically equal to zero. If \(\alpha(t)-\beta(t)\in K\), then equation (9) has only the trivial solution satisfying the conditions
\[ \int_{0}^{T}x(t)\alpha(t)\,dt=0,\qquad \int_{0}^{T}x(t)\beta(t)\,dt=0. \tag{10} \]
Proof. From the first equality it follows that the solution \(x(t)\) vanishes inside the interval \([a,b]\), and from the second—inside the interval \([c,d]\), i.e., the solution of problem (9)—(10) vanishes at two different points of the interval \((0,T)\). But, by virtue of the conditions on \(\lambda(t)\), this is possible only when \(x(t)\equiv 0\) (see, for example, [3]). The lemma is proved.
Lemma 2. If in the conditions of Lemma 1 \(\beta(t)\equiv 0\) \((\alpha(t)\equiv 0)\), then, by adding to the nontrivial condition from (10) the condition \(x(T)=0\) (or \(x(0)=0\)), we obtain that only \(x(t)\equiv 0\) satisfies equation (9) and the “augmented” conditions (10).
Proof. The condition
\[ \int_0^T x(t)\alpha(t)\,dt=0 \]
ensures that the function \(x(t)\) vanishes in \((0,T)\). Together with the additional condition this means that \(x(t)\) has two zeros on \([0,T]\), which is impossible. The contradiction obtained proves the lemma.
Theorem 3. If \(A(t)\) is a triangular matrix with nonnegative diagonal elements, then the solution of the correction problem is unique.
Proof. In view of what was said above, the theorem will be proved if we show that for every \(f(t)\in K\) the corresponding operator equation \(Ly=0\) has as many linearly independent solutions as there are functions among \(\alpha^i(t)\) and \(\beta^i(t)\) \((i=1,2,\ldots,n)\) that vanish identically on \([0,T]\).
Take an arbitrary function \(f(t)\in K\). From it we find \(\alpha^i(t)\) and \(\beta^i(t)\), and let \(m\) be the number of functions among \(\alpha^i(t)\) and \(\beta^i(t)\) that are identically zero. The nonzero \(\alpha^i(t)\) and \(\beta^i(t)\) generate \(2n-m\) conditions (5), which define the domain of definition \(D(L)\) of the operator \(L\). To these \(2n-m\) conditions of the form (5) we add another \(m\) conditions \(y_i(0)=0\) or \(y_i(T)=0\) as follows: for the number \(i\) for which \(\alpha^i(t)\equiv 0\), we add the condition \(y_i(0)=0\), and where \(\beta^i(t)\equiv 0\)—the condition \(y_i(T)=0\); if for some \(i\), \(\alpha^i(t)\equiv \beta^i(t)\equiv 0\), then we add two conditions: \(y_i(0)=y_i(T)=0\).
Denote by \(D(\widetilde L)\) the set of functions \(y(t)\) from \(D(L)\) satisfying the added conditions. Denote by \(\widetilde L\) the restriction of the operator \(L\) to \(D(\widetilde L)\). Obviously, \(D(L)\) and \(D(\widetilde L)\) differ by \(m\) dimensions.
We shall show that \(\widetilde L\) is invertible. Since \(A(t)\) is a triangular matrix, then (taking, for definiteness, the upper triangular one) the equation \(\widetilde L y=0\) can be written in expanded form as follows:
\[ \frac{d^2y}{dt^2}-\sum_{j=1}^{i} a_{ji}(t)y_j=0, \]
\[ (y,\alpha^i)=(y,\beta^i)=0\quad (i=1,2,\ldots,n), \tag{11} \]
\[ y^{i'}(0)=0,\qquad y^{i''}(T)=0, \]
where \(i'\) is that number \(i\) for which \(\alpha^i(t)\equiv 0\), and \(i''\)—if \(\beta^i(t)\equiv 0\). We shall solve system (11) successively. For \(i=1\) we have
\[ -\frac{d^2y_1}{dt^2}-a_{11}(t)y_1=0,\quad \int_0^T y_1(t)\alpha_1^1(t)\,dt=\int_0^T y_1(t)\beta_1^1(t)\,dt=0. \]
If \(\alpha_1^1(t)\) and \(\beta_1^1(t)\) are not zero, then by Lemma 1 we have \(y_1(t)\equiv 0\). If, however, one of these functions vanishes identically, then
then a boundary condition appears, which by virtue of Lemma 2 entails \(y_1(t) \equiv 0\). If, however, \(\alpha^1(t) \equiv \beta^1(t) \equiv 0\), then one obtains an ordinary boundary-value problem and, as is known (see, for example, [3]), \(y_1(t) \equiv 0\).
Thus, in all cases \(y_1(t) \equiv 0\). Consequently, in the system one may discard the first equation and the first column. There remains a system of order \(n-1\), which has the same property as the original one. Carrying out analogous reasoning, we obtain also \(y_2(t) \equiv 0\). Continuing this process by induction, we obtain that all \(y_i(t) \equiv 0\) \((i=1,2,\ldots,n)\), i.e. \(y(t) \equiv 0\). Thus, from \(\tilde L y=0\) it follows that \(y(t) \equiv 0\), i.e. the operator \(\tilde L\) is invertible. Now, from the fact that \(D(L)\) differs from \(D(\tilde L)\) by \(m\) measurements, it follows that the equation \(Ly=0\) has no more than \(m\) linearly independent solutions, as was required to prove.
In conclusion, the author expresses gratitude to S. G. Krein for posing the problem and for constant assistance in the work.
References
- Kvedaras B. V. Proceedings of the Seminar on Functional Analysis. Voronezh State Univ., issue 7, 1963, pp. 45–51.
- Naimark M. A. Linear Differential Operators. GITTL, 1954.
- Tricomi F. Differential Equations. IL, 1962.
Received by the editors
July 9, 1965
Institute of Physics and Mathematics
of the Lithuanian SSR