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UDC 517.221+517.216.2
MATHEMATICS
Academician of the Academy of Sciences of the Azerbaijan SSR Z. I. KHALILOV, E. J. ASLANOV
ON A VARIATIONAL PROBLEM IN HILBERT SPACE AND ITS APPLICATION TO PARTIAL DIFFERENTIAL EQUATIONS
1. Let \(H_1\) and \(H_2\) be real Hilbert spaces with scalar products \((x,y)_1\) and \((\xi,\eta)_2\); let \(K\) be a linear continuous operator acting from \(H_1\) into \(H_2\); and let \(E(x)\) be a quadratic functional in \(H_1\), defined by the formula
\[ E(x)=(Kx-\xi,\,Kx-\xi)_2+(x,x)_1 . \tag{1} \]
Then the following holds.
Theorem. For every \(\xi \in H_2\) there exists \(x_0 \in H_1\) minimizing the functional (1), which is determined uniquely from the equation
\[ x+K^*Kx=K^*\xi . \tag{2} \]
Proof. Obviously, equation (2) has, moreover, a unique solution \(x_0 \in H_1\). Therefore, for any \(y \in H_1\) we have
\[ \begin{aligned} E(x_0+y)&=E(x_0)+2(K^*Kx_0+x_0-K^*\xi,y)_1+(Ky,Ky)_2+(y,y)_1\\ &=E(x_0)+(Ky,Ky)_2+(y,y)_1 . \end{aligned} \tag{3} \]
It follows from (3) that \(E(x_0)<E(x_0+y)\) for every \(y\ne0\), i.e. \(x_0\in H_1\) is the unique element minimizing the functional.
2. Let \(H\) be a real Hilbert space with scalar product \((\varphi,\psi)\). Then the aggregate of vector functions with values in \(H\), defined on the interval \([0,T]\), with scalar product
\[ [f,g]=\int_0^T (f(s),g(s))\,ds, \]
will be a Hilbert space (1), which we shall denote by \(L_2=L_2(H,[0,T])\).
Take as \(H_1\) the set of pairs \(\langle f,\varphi\rangle\), \(f\in L_2\), \(\varphi\in H\), with scalar product of the elements \(x=\langle f,\varphi\rangle\) and \(y=\langle g,\psi\rangle\), defined by the formula
\[ (x,y)_1=a[f,g]+b(\varphi,\psi),\qquad a>0,\ b>0 . \]
We define the operator \(K\) by the formula
\[ Kx=Bf+U\varphi, \]
where \(B\) and \(U\) are linear continuous operators acting from \(L_2\) into \(H_2\) and from \(H\) into \(H_2\), respectively. The operator \(K^*\), acting from \(H_2\) into \(H_1\), will have the form
\[ K^*\xi=\langle a^{-1}B^*\xi,\ b^{-1}U^*\xi\rangle . \]
Hence
\[ K^*Kx=\langle a^{-1}B^*Bf+a^{-1}B^*U\varphi,\ b^{-1}U^*Bf+b^{-1}U^*U\varphi\rangle, \]
and equation (2) is written in the form of the system
\[ \begin{aligned} B^*Bf+af+B^*U\varphi&=B^*\xi,\\ U^*U\varphi+b\varphi+U^*Bf&=U^*\xi, \end{aligned} \tag{4} \]
whose solution minimizes the functional
\[ E(f,\varphi)=(Bf+U\varphi-\xi,Bf+U\varphi-\xi)_2 +a\int_0^T \|f(s)\|^2\,ds+b\|\varphi\|^2 . \tag{5} \]
3. The constructions given above can be applied to the solution of the following variational problems. Let
\[ u(t)=U(t,0)\varphi+\int_0^t U(t,s)f(s)\,ds \]
be a generalized solution of the Cauchy problem
\[ du(t)/dt+A(t)u(t)=f(t),\quad 0<t\leq T,\quad u(0)=\varphi, \tag{6} \]
in the space \(H\), where \(A(t)\) is, generally speaking, an unbounded operator in \(H\). Here \(U(t,s)\) is an operator-valued function, continuous in the aggregate of variables for \(0\leq s\leq t\leq T\), with values in the ring of linear continuous operators in \(H\) (for conditions under which \(A(t)\) can generate \(U(t,s)\), see \((^{2-4})\)).
Problem 1. Find a pair \(f\in L_2,\ \varphi\in H\), minimizing the functional
\[ E_1(f,\varphi)=\|u(T)-\psi\|^2 +a\int_0^T \|f(s)\|^2\,ds+b\|\varphi\|^2,\quad a>0,\ b>0, \tag{7} \]
where \(\psi\in H\) is a given element.
The functional \(E_1\) may be regarded as a special case of the functional (5), if one sets \(H_2=H\),
\[ Bf=\int_0^T U(T,s)f(s)\,ds,\quad U\varphi=U(T,0)\varphi. \]
Then \(U^*\varphi=U^*(T,0)\varphi,\ B^*\varphi=U^*(T,t)\varphi\).
System (4) in this case takes the form
\[ af(t)+\int_0^T U^*(T,t)U(T,s)f(s)\,ds +U^*(T,t)U(T,0)\varphi=U^*(T,t)\psi, \]
\[ b\varphi+U^*(T,0)U(T,0)\varphi +\int_0^T U^*(T,0)U(T,s)f(s)\,ds=U^*(T,0)\psi . \tag{8} \]
Putting \(t=0\) in the first equation and comparing with the second, we obtain \(b\varphi=af(0)\), where \(f(t)\) is determined from the loaded integral equation
\[ af(t)+\int_0^T U^*(T,t)U(T,s)f(s)\,ds +\frac{a}{b}U^*(T,t)U(T,0)f(0)=U^*(T,t)\psi, \tag{9} \]
Problem 2. Find a pair \(f\in L_2,\ \varphi\in H\), minimizing the functional
\[ E_2(f,\varphi)=\int_0^T \|U(s)-g(s)\|^2\,ds +a\int_0^T \|f(s)\|^2\,ds+b\|\varphi\|^2,\quad a>0,\ b>0, \tag{10} \]
where \(g\in L_2\) is a given function.
The functional \(E_2\) may be regarded as a special case of the functional (5), if one sets \(H_2=L_2,\ U\varphi=U(t,0)\varphi\),
\[ Bf=\int_0^t U(t,s)f(s)\,ds . \]
Then
\[ U^* f=\int_0^T U^*(s,0) f(s)\,ds,\qquad B^* f=\int_t^T U^*(s,t) f(s)\,ds . \]
System (4) in this case takes the form:
\[ \int_t^T\int_0^s U^*(s,t)U(s,\tau)f(\tau)\,d\tau\,ds +af(t)+\int_t^T U^*(s,t)U(s,0)\varphi\,ds = \]
\[ =\int_t^T U^*(s,t)g(s)\,ds, \]
\[ \int_0^T\int_0^s U^*(s,0)U(s,\tau)f(\tau)\,d\tau\,ds +b\varphi+\int_0^T U^*(s,0)U(s,0)\varphi\,ds = \]
\[ =\int_0^T U^*(s,0)g(s)\,ds . \]
Just as in problem 1, we obtain \(b\varphi=af(0)\), where \(f(t)\) is determined from the loaded integral equation
\[ af(t)+\int_t^T\int_0^s U^*(s,t)U(s,\tau)f(\tau)\,d\tau\,ds+ \]
\[ +\frac{a}{b}\int_t^T U^*(s,0)U(s,0)f(0)\,ds =\int_t^T U^*(s,t)g(s)\,ds . \tag{11} \]
The problems set forth, following Bellman \((^5)\), may be regarded as problems of optimal control. The first term in the functionals \(E_1\) and \(E_2\) represents the cost of deviation, and the other two terms the cost of control, with \(a\) and \(b\) determining the weight assigned to the controls \(f\) and \(\varphi\).
- Many nonstationary problems for partial differential equations can be written in the form (6). Therefore the solution of variational problems 1 and 2 can be applied to the investigation of analogous questions for partial differential equations. As an application, consider the Cauchy problem for the heat equation
\[ \partial u(t,x)/\partial t-\partial^2 u(t,x)/\partial x^2=f(t,x),\qquad -\infty<x<\infty,\quad t\geqslant 0, \]
\[ u(0,x)=\varphi(x). \]
Here \(H=L_2(-\infty,\infty)\), \(A(t)\) does not depend on \(t\) and is determined by the equality
\(Au=-\partial^2u/\partial x^2,\ D(A)=W_2^2(-\infty,\infty)\).
Then \(A\) is a self-adjoint positive operator in \(L_2(-\infty,\infty)\), and
\(U^*(t,s)=U(t,s)=\exp (-(t-s)A)\), where (see (6))
\[ \exp(-tA)\varphi=\frac{1}{2\sqrt{\pi t}}\int_{-\infty}^{\infty} \exp\left(-\frac{(x-\eta)^2}{4t}\right)\varphi(\eta)\,d\eta,\qquad t>0. \]
The functionals \(E_1\) and \(E_2\) take, for this case, the form
\[ E_1(f,\varphi)=\int_{-\infty}^{\infty}|u(T,x)-\psi(x)|^2\,dx +a\int_0^T\int_{-\infty}^{\infty}|f(t,x)|^2\,dx\,dt +b\int_{-\infty}^{\infty}|\varphi(x)|^2\,dx, \]
\[ E_2(f,\varphi)=\int_0^T\int_{-\infty}^{\infty}|u(t,x)-g(t,x)|^2\,dx\,dt +a\int_0^T\int_{-\infty}^{\infty}|f(t,x)|^2\,dx\,dt +b\int_{-\infty}^{\infty}|\varphi(x)|^2\,dx . \]
Problem 1 becomes the problem of the best approximation to a prescribed temperature \(\psi(x)\) over a definite time interval \(T\), and its solution is found from the nonhomogeneous integral equation
\[ a f(t,x)+\int_0^T \int_{-\infty}^{\infty} K(x-\eta,t,s) f(s,\eta)\,d\eta\,ds+ \]
\[ +\frac{a}{b}\int_{-\infty}^{\infty} K(x-\eta,t,0) f(0,\eta)\,d\eta=G(t,x), \tag{12} \]
\[ K(\xi,t,s)=\frac{1}{2\sqrt{\pi(2T-s-t)}}\exp\left(-\frac{\xi^2}{4(2T-s-t)}\right); \]
\[ G(t,x)=\int_{-\infty}^{\infty} \frac{\exp\left(-(x-\eta)^2/4(T-t)\right)}{2\sqrt{\pi(T-t)}}\, \psi(\eta)\,d\eta . \]
Problem 2 becomes the problem of the best approximation to a prescribed temperature regime \(g(t,x)\) over a definite time interval \(T\), and its solution is found from equation (12), where
\[ K(\xi,t,s)= \sqrt{\frac{2T-s-t}{\pi}}\, \exp\left(-\frac{\xi^2}{4(2T-s-t)}\right) -\sqrt{\frac{|t-s|}{\pi}}\times \]
\[ \times \exp\left(-\frac{\xi^2}{4|t-s|}\right) -\frac{\xi}{4}\left[ \Phi\left(\frac{\xi}{2\sqrt{2T-s-t}}\right) -\Phi\left(\frac{\xi}{2\sqrt{|t-s|}}\right) \right]; \]
\[ G(x,t)=\int_t^T\int_{-\infty}^{+\infty} \frac{\exp\left(-\frac{(x-\eta)^2}{4(s-t)}\right)} {2\sqrt{\pi(s-t)}}\,g(s,\eta)\,d\eta\,ds . \]
Here
\[ \Phi(a)=2\pi^{-1/2}\int_0^a e^{-\tau^2}\,d\tau \]
is the probability integral.
The present article is, in content, close to the works \((^7,^8)\). Part of the results presented, concerning the problem of the best approximation to a prescribed temperature regime, is contained in \((^9,^10)\).
Institute of Cybernetics
Academy of Sciences of the Azerbaijan SSR
Received
5 IV 1966
REFERENCES
- E. Hille, R. Phillips, Functional Analysis and Semi-Groups, IL, 1962.
- T. Kato, Collection of Translations, Mathematics, 2 (4) (1958).
- P. E. Sobolevskii, Tr. Moscow Math. Soc., 10 (1961).
- H. Tanabe, Osaka Math. J., 11, No. 2 (1959).
- R. Bellman, I. Glicksberg, O. Gross, Some Questions in the Mathematical Theory of Control Processes, IL, 1962.
- E. Hille, Functional Analysis and Semi-Groups, IL, 1951.
- E. I. Halilov, DAN, 155, No. 4 (1964).
- E. I. Halilov, Dokl. Acad. Sci. Azerbaijan SSR, 20, No. 5 (1964).
- E. J. Aslanov, Dokl. Acad. Sci. Azerbaijan SSR, 21, No. 7 (1965).
- E. J. Aslanov, Izv. Acad. Sci. Azerbaijan SSR, 5, mathematical series (1965).