UDC 519.3

MATHEMATICS

Academician of the Academy of Sciences of the Azerbaijan SSR Z. I. KHALILOV, B. N. PANAYOTI

ON THE MINIMIZATION OF A QUADRATIC FUNCTIONAL GENERATED BY A LINEAR OPERATOR IN NORMED SPACES

Some problems in the theory of optimal control lead to finding the minimum of a functional of the form

\[ \Phi(f)=\|Af-g\|^2,\quad f\in D(A), \tag{1} \]

where \(f\) is an element of a Banach space \(B\); \(g\) is a given element of a Hilbert space \(H\); \(A\) is a linear operator acting from \(B\) into \(H\); \(D(A)\) is the domain of definition of the operator \(A\).

Let us denote by \(R(A)\) the range of values of the operator \(A\). For example, the functional

\[ \Phi(f)=\|A_1f-g_1\|^2+\|A_2f-g_2\|^2, \tag{2} \]

where \(f\) is an element of a Banach space \(B\); \(g_1\) is a given element of a Hilbert space \(H_1\); \(g_2\) is a given element of a Hilbert space \(H_2\); \(A_1\) is a linear operator acting from \(B\) into \(H_1\), with domain of definition \(D(A_1)\); \(A_2\) is a linear operator acting from \(B\) into \(H_2\), with domain of definition \(D(A_2)\), is reduced to (1).

Indeed, consider the Hilbert space

\[ H=H_1\times H_2=\{\varphi,\ \varphi=[\varphi_1,\varphi_2],\ \varphi_1\in H_1,\ \varphi_2\in H_2, \]

\[ \langle\varphi',\varphi''\rangle=(\varphi_1',\varphi_1'')+(\varphi_2',\varphi_2'')\}. \]

Introduce a linear operator \(A\), acting from \(B\) into \(H\), and defined as follows:

\[ Af=[A_1f,A_2f],\quad \text{where } f\in D=D(A_1)\cap D(A_2). \]

Then the functional (2) is reduced to the form (1).

To solve the problem of minimizing the functional (1), consider the special case when \(A\) is a monomorphism, i.e., maps \(B\) into \(H\) in a one-to-one manner. Then the following obvious equality holds:

\[ \min_{f\in D(A)}\|Af-g\|^2=\min_{h\in R(A)}\|h-g\|^2. \tag{3} \]

The problem of minimizing the simplest functional \(\Phi(f)=\|f-g\|^2\) in a Hilbert space has been investigated in the following formulation: find an element \(f_0\) from some manifold \(G\) for which

\[ \min_{f\in G}\Phi(f)=\|f_0-g\|^2, \tag{4} \]

where, generally speaking, \(g\) does not belong to \(G\) (obviously, if \(g\in G\), then \(f_0=g\) and \(\min \Phi(f)=0\)).

Theorem 1. If \(G\) is a linear manifold, then the solution of the problem is unique; if, in addition, \(G\) is a closed manifold (i.e., \(G\) is a subspace of the space \(H\)), then there also exists a solution, and if \(f_0\) is a solution of the problem, then the element

\(f_0-g \perp G\), i.e., the equality holds

\[ (f_0-g,f)=0 \quad \text{for all } f\in G. \tag{5} \]

Theorem 2. If in the expression of functional (2) the operator \(A_2\) has a bounded inverse, and \(A_1\) is a closed operator, then the minimization problem for functional (2) has a solution, and moreover a unique one.

Proof. For the proof it suffices to establish that in the corresponding functional (1) the operator \(A\) is a monomorphism and \(R(A)\) is a subspace of the space \(H\).

First let us prove that the operator \(A\) is a monomorphism. For this it is necessary to prove that from \(Af=0\) it follows that \(f=0\). Let \(Af=0\). By definition, then \(A_1 f=0\) and \(A_2 f=0\). Since \(A_2\) is a monomorphism, \(f=0\), as was required to prove.

Further, from the linearity of the operator \(A\) follows the linearity of the manifold \(R(A)\). It remains to show that \(R(A)\) is a closed manifold. For this purpose take an arbitrary sequence \(h_\nu\in R(A)\) converging to some element \(h\). It is necessary to prove that \(h\in R(A)\). This is proved as follows. In view of the monomorphism, to each \(h_\nu\) there corresponds a unique \(f_\nu\) such that \(h_\nu=Af_\nu\), i.e. if \(h_\nu=(h_\nu',h_\nu'')\), then \(h_\nu'=A_1 f_\nu,\ h_\nu''=A_2 f_\nu\). Then \(h_\nu'\) converge to \(h'\) and \(h_\nu''\to h''\), respectively. Further we have the sequence \(f_\nu=A_2^{-1}h_\nu''\), which converges to the element \(f=A_2^{-1}h''\). Then \(h'=A_1 f\) by virtue of the closedness of \(A_1\), since \(f_\nu\to f\) and \(A_1 f_\nu\to h'\), so \(h'=A_1 f\). From the latter and from \(h''=A_2 f\) it follows that \(A\) is closed and \(h=Af\), as was required to prove.

Then, by the classical theorem, there exists \(f_0\), \(Af_0=h_0\in R(A)\), and it is unique, minimizing functional (2). This completes the proof of the theorem.

Quadratic functionals considered, for example, in (1) by Bellman et al. in a finite-dimensional Hilbert space, and by Z. Khalilov and E. Aslanov \((^2,^3)\) in a Hilbert space, are particular cases of functional (2). The theorem proved above is general in the theory of partial equations considered in works \((^1–^3)\). Its applications are especially effective in the investigation of optimal-control problems for partial differential equations.

For finding the existing unique minimizing element \(f_0\), under additional conditions one can construct a certain functional equation. Indeed, if we apply (5), then

\[ (Af_0-g,Af)=0 \]

for all \(f\in D(A)\).

Let \(D(A)\) be dense in \(B\). Then from \((A^*(Af_0-g),f)=0\) we have

\[ A^*Af_0=A^*g. \tag{6} \]

It can be shown that (6) is equivalent to the equation:

\[ A_1^*A_1 f_0+A_2^*A_2 f_0=A_1^*g_1+A_2^*g_2. \tag{7} \]

Let us now consider another characteristic functional:

\[ \Phi(f_1,f_2)=\|A_1'f_1+A_1''f_2-g_1\|^2+\|A_2'f_1+A_2''f_2-g_2\|_2^2, \tag{8} \]

where \(f_1\in B_1,\ f_2\in B_2,\ g_1\in H_1,\ g_2\in H_2,\ A_1':B_1\to H_1,\ A_1'':B_2\to H_1,\ A_2':B_1\to H_2,\)

\[ A_2'':B_2\to H_2. \]

It is not difficult to see that (8) can also be reduced to (2). Indeed, introduce the Banach space \(B=B_1\times B_2\) with elements \(f=(f_1,f_2)\) with norm \(\|f\|^2=\|f_1\|^2+\|f_2\|^2\) and operators \(A_1 f=A_1'f_1+A_1''f_2,\ A_2 f=A_2'f_1+A_2''f_2,\ Af=[A_1 f,A_2 f]\). Then (8) takes the form (2), which in turn reduces to (1).

Thus, the main theorem can be reformulated also for (8). If \(A_1'\) is a bounded linear operator, \(A_1''\) is a closed li-

linear operator and \(A_2\) is a linear operator having a bounded inverse, then, according to the theorem, the minimization problem for the functional (8) has a solution, and moreover a unique one.

To obtain equation (7) and other functional equations, the following formulas are used:

\[ \text{if } Af=[A_1f,A_2f], \quad \text{then } A^*\varphi=A_1^*\varphi_1+A_2^*\varphi_2; \tag{9} \]

\[ \text{if } Af=A_1f_1+A_2f_2, \quad \text{then } A^*\varphi=[A_1^*\varphi,A_2^*\varphi]. \tag{10} \]

Then, if we are dealing with the functional (8), the functional equation of type (7) for the functional (8), on the basis of (9) and (10), has the form (see (6))

\[ A_1^{\prime *}\bigl(A_1'f_1^0+A_1''f_2^0\bigr)+ A_2^{\prime *}\bigl(A_2'f_1^0+A_2''f_2^0\bigr) = A_1^{\prime *}g_1+A_2^{\prime *}g_2, \]

\[ A_1^{\prime\prime *}\bigl(A_1'f_1^0+A_1''f_2^0\bigr)+ A_2^{\prime\prime *}\bigl(A_2'f_1^0+A_2''f_2^0\bigr) = A_1^{\prime\prime *}g_1+A_2^{\prime\prime *}g_2. \]

Let us note that, in an analogous way, one can consider a functional of a more general form:

\[ \Phi(f_1,f_2,\ldots,f_m)= \sum_{i=1}^{n}\left\lVert \sum_{j=1}^{m} A_{ij}f_j-g_i \right\rVert_i^2, \]

where \(f_j\in D(A_{ij})\subset B_j\), \(g_i\in H_i\) for any \(i\), which, by the corresponding grouping, is reduced to a functional of the form (2). It is of interest to consider the corresponding problem for operators mapping into a Banach space.

Institute of Mathematics and Mechanics
Academy of Sciences of the Azerbaijan SSR

Received
28 X 1968

References

\(^{1}\) R. Bellman et al., Some Problems in the Mathematical Theory of Control Processes, IL, 1962.
\(^{2}\) Z. I. Khalilov, E. J. Aslanov, DAN, 169, No. 5 (1966).
\(^{3}\) Z. I. Khalilov, E. J. Aslanov, DAN, 182, No. 4 (1968).