UDC 517.218+517.221+517.216+517.224

MATHEMATICS

Academician of the Academy of Sciences of the Azerbaijan SSR Z. I. KHALILOV, E. Dzh. ASLANOV

SOME VARIATIONAL PROBLEMS WITH ACCOUNT TAKEN OF THE COST OF CONTROL IN HILBERT SPACES

1. Denote by \(H\) a real Hilbert space with scalar product \((\, , \,)\) and norm \(\|\ \|\). The norms and scalar products of the Hilbert spaces \(H_1, H_2\), and \(H_3\) will be denoted by the corresponding subscript. Let \(\mathbf L_2 = \mathbf L_2(H,[0,T])\) be the Hilbert space of vector-functions with values in \(H\) (see \((^1)\)).

Theorem. Let \(D(L_2 \to L_2)\), \(F(H \to L_2)\), \(B(L_2 \to H_3)\), and \(C(H \to H_3)\) be linear bounded operators. Then for any prescribed pairs \(\xi_1 \in H_3\), \(\xi_2(t) \in L_2\) there exists a pair \(\eta \in L_2\), \(\zeta \in H\), minimizing the functional

\[ E(\eta,\zeta)=\|B\eta+C\zeta-\xi_1\|_3^2+ \int_0^T \|D\eta+F\zeta-\xi_2\|^2\,dt+ \alpha\int_0^T\|\eta\|^2\,dt+\gamma\|\zeta\|^2, \tag{1} \]

and \(\eta\) and \(\zeta\) are determined uniquely from the system of equations

\[ (B^*B+D^*D+\alpha)\eta+(B^*C+D^*F)\zeta=B^*\xi_1+D^*\xi_2, \]

\[ (C^*B+F^*D)\eta+(C^*C+F^*F+\gamma)\zeta=C^*\xi_1+F^*\xi_2. \tag{2} \]

Here and throughout the sequel \(\alpha,\gamma>0\).

For the proof of the theorem one uses the result from \((^2)\), if for \(H_1\) we take \(L_2 \times H\) with scalar product of the elements \(x=\langle\eta,\zeta\rangle\) and \(y=\langle\psi,\varphi\rangle\), defined by the formula \((x,y)_1=\alpha[\eta,\psi]+\gamma(\zeta,\varphi)\), and in \(H_2=H_1\times L_2\) introduce the scalar product of the elements \(z=\langle\langle\xi,\eta\rangle\rangle\) and \(r=\langle\langle\tau,\psi\rangle\rangle\) by the formula \((z,r)_2=(\xi,\tau)_3+[\eta,\psi]\), and as the operator \(K(H_1\to H_2)\) from \((^2)\) take the operator \(Kx=\langle\langle B\eta+C\zeta,D\eta+F\zeta\rangle\rangle\).

Then

\[ K^*z=\langle \alpha^{-1}(B^*\xi+D^*\eta),\gamma^{-1}(C^*\xi+F^*)\rangle . \]

2. We now consider the Cauchy problem for a differential equation in the Hilbert space \(H\)

\[ u'(t)+Au(t)=f(t),\qquad u(0)=\varphi,\qquad 0\le t\le T, \tag{3} \]

where \(A\) is a linear bounded self-adjoint operator in \(H\).

In the paper \((^2)\) (see also \((^3)\)) the problem was considered of finding a pair \(\langle f,\varphi\rangle\in H_1\) minimizing the functionals

\[ E_1(\langle f,\varphi\rangle)=\|u(T)-\psi\|^2+\alpha\int_0^T\|f(t)\|^2\,dt+\gamma\|\varphi\|^2, \]

\[ E_2(\langle f,\varphi\rangle)=\int_0^T\|u(t)-g(t)\|^2\,dt+\alpha\int_0^T\|f(t)\|^2\,dt+\gamma\|\varphi\|^2, \]

where \(\psi\in H\) and \(g(t)\in L_2\) are prescribed elements, and \(u(t)\) is the solution of the Cauchy problem (3) with unbounded self-adjoint linear operator \(A\).

We now consider analogous problems, where the “costs of control” remain the same, while into the “cost of deviation” we introduce the derivative of the solution \(u'(t)\). More precisely, the following is considered

Problem 1. Find a pair \(\langle f,\varphi\rangle\in H_1\) minimizing the functional

\[ E_1(\langle f,\varphi\rangle)=\|u(T)-\psi\|^2+\int_0^T \|u'(t)-h(t)\|^2\,dt+\alpha\int_0^T \|f(t)\|^2\,dt+\gamma\|\varphi\|^2, \tag{4} \]

where \(u(t)\) is the solution of problem (3), and \(\psi\in H\) and \(h(t)\in L_2\) are given elements. The problem has a solution, and a unique one, if in the theorem we put \(H_3=H\) and

\[ Bf=\int_0^T e^{-(T-s)A}f(s)\,ds,\qquad C\varphi=e^{-TA}\varphi, \]

\[ Df=f(t)-A\int_0^t e^{-(t-\tau)A}f(\tau)\,d\tau,\qquad F\varphi=-Ae^{-tA}\varphi. \tag{5} \]

It is not difficult, in particular, to establish that

\[ D^*Df=f(t)-\frac12 A\int_0^T \left[e^{-(2T-t-\tau)A}+e^{-|t-\tau|A}\right]f(\tau)\,d\tau, \]

\[ F^*Df=-\frac12 A\int_0^T \left[e^{-(2T-\tau)A}+e^{-\tau A}\right]f(\tau)\,d\tau, \]

\[ D^*F\varphi=-\frac12 A\left(e^{-(2T-t)A}+e^{-tA}\right)\varphi,\qquad F^*F\varphi=\frac12 A\left(I-e^{-2TA}\right)\varphi, \]

\[ B^*Bf=\int_0^T e^{-(2T-t-\tau)A}f(\tau)\,d\tau,\qquad C^*Bf=\int_0^T e^{-(2T-\tau)A}f(\tau)\,d\tau. \]

Then system (2), after some transformations, takes the form

\[ (1+\alpha)f(t)+\int_0^T W(t,\tau)f(\tau)\,d\tau+W(t,0)\varphi=G_1(t), \]

\[ (A+\gamma I)\varphi+\int_0^T W(0,\tau)f(\tau)\,d\tau+W(0,0)\varphi=G_1(0)-h(0), \tag{6} \]

where

\[ W(t,\tau)=\left(I-\frac12 A\right)e^{-(2T-t-\tau)A}-\frac12 Ae^{-|t-\tau|A}, \]

\[ G_1(t)=h(t)+e^{-(T-t)A}\psi-A\int_0^T e^{-(\tau-t)A}f(\tau)\,d\tau. \]

From (6), for \(t=0\), we obtain

\[ \varphi=(A+\gamma I)^{-1}\bigl[(1+\alpha)f(0)-h(0)\bigr]. \tag{7} \]

Therefore system (6) is replaced by the loaded integral equation

\[ (1+\alpha)f(t)+\int_0^T W(t,\tau)f(\tau)\,d\tau+(1+\alpha)W(t,0)(A+\gamma I)^{-1}f(0)= \]

\[ =G_1(t)+W(t,0)(A+\gamma I)^{-1}h(0). \tag{8} \]

Problem 2. Let \(h_1(t)\) and \(h_2(t)\) be given functions from \(L_2\). Find a pair \(\langle f,\varphi\rangle\in H_1\) minimizing the functional

\[ E_2(\langle f,\varphi\rangle)=\int_0^T \|u(t)-h_1(t)\|^2\,dt+\int_0^T \|u'(t)-h_2(t)\|^2\,dt+\alpha\int_0^T \|f(t)\|^2\,dt+\gamma\|\varphi\|^2, \tag{9} \]

where \(u(t)\) is the solution of the Cauchy problem (3).

Functional (9), as in problem 1, is a special case of functional (1), if in the theorem we put \(H_3=L_2\), define the operators \(D(L_2\to L_2)\), \(F(H\to L_2)\) as in (5), and define the operators \(B(L_2\to L_2)\) and \(C(H\to L_2)\) as follows:

so that

\[ Bf=\int_{0}^{t} e^{-(t-\tau)A} f(\tau)\,d\tau,\qquad C\varphi=e^{-tA}\varphi. \]

Then system (2) takes the form

\[ \begin{aligned} (1+\alpha)f(t)&+\int_{t}^{T}\int_{0}^{s} e^{-(2s-\tau-t)A} f(\tau)\,d\tau\,ds +\int_{0}^{T}\Psi(t,\tau)f(\tau)\,d\tau+{}\\ &+\int_{t}^{T} e^{-(2\tau-t)A}\varphi\,d\tau+\Psi(t,0)\varphi =G_{2}(t),\\ (A+\gamma I)\varphi&+\int_{0}^{T}\int_{0}^{s} e^{-(2s-\tau)A} f(\tau)\,d\tau\,ds +\int_{0}^{T}\Psi(0,\tau)f(\tau)\,d\tau+{}\\ &+\int_{0}^{T} e^{-2\tau A}\varphi\,d\tau+\Psi(0,0)\varphi =G_{2}(0)-h_{2}(0), \end{aligned} \tag{10} \]

where

\[ \Psi(t,\tau)=W(t,\tau)-e^{-(2T-t-\tau)A},\qquad G_{2}(t)=h_{2}(t)+\int_{t}^{T} e^{-(\tau-t)A}\bigl(h_{1}(\tau)-Ah_{2}(\tau)\bigr)\,d\tau . \]

This system is equivalent to the equation

\[ \begin{aligned} (1+\alpha)f(t)&+\int_{t}^{T}\int_{0}^{s} e^{-(2s-\tau-t)A} f(\tau)\,d\tau\,ds+{}\\ &+\int_{0}^{T}\Psi(t,\tau)f(\tau)\,d\tau +(1+\alpha)(A+\gamma I)^{-1}\int_{t}^{T} e^{-(2\tau-t)A}f(0)\,d\tau+{}\\ &+(1+\alpha)\Psi(t,0)(A+\gamma I)^{-1}f(0)=\\ &=G_{2}(t)+\Psi(t,0)(A+\gamma I)^{-1}h_{2}(0) +(A+\gamma I)^{-1}\int_{t}^{T} e^{-(2\tau-t)A}Ah_{2}(0)\,d\tau . \end{aligned} \tag{11} \]

1. Let us present one of the possible realizations of these variational problems. Let \(K(x,y)=R(x)R(y)\) be a continuous function for \(0\le x,y\le 1\), and

\[ \int_{0}^{1} R^{2}(\xi)\,d\xi=q<+\infty . \]

If, for each fixed \(t\), for functions \(u(t,x)\in L_{2}[0,1]\) one defines the operator \(A\) by the formula

\[ Au=(Au)(t,x)=\int_{0}^{1}R(x)R(y)u(t,y)\,dy, \tag{*} \]

then \(A\) will be a linear bounded self-adjoint operator mapping \(L_{2}[0,1]\) into itself.

Let a certain process be described by the Cauchy problem for the integro-differential equation

\[ \frac{\partial u(t,x)}{\partial t}+\int_{0}^{1}R(x)R(y)u(t,y)\,dy=f(t,x),\qquad u(0,x)=\varphi(x),\quad t\ge 0. \]

Here \(H=L_{2}[0,1]\), the operator \(A\) is defined by formula (*) and, for any \(v\in H=L_{2}[0,1]\),

\[ e^{-tA}v=v+\frac{e^{-tq}-1}{q}Av,\qquad (A+\gamma I)^{-1}v=\frac{1}{\gamma}v-\frac{1}{\gamma(\gamma+q)}Av. \]

The functionals \(E_{1}\) and \(E_{2}\) in this case take the form

\[ E_{1}(\langle f,\varphi\rangle)= \int_{0}^{1}|u(T,x)-\psi(x)|^{2}\,dx +\int_{0}^{T}\int_{0}^{1}\left|\frac{\partial u(t,x)}{\partial t}-h(t,x)\right|^{2}\,dx\,dt+ \]

\[ +\alpha \int_0^T\int_0^1 |f(t,x)|^2\,dx\,dt+\gamma\int_0^1|\varphi(x)|^2\,dx, \]

\[ \begin{aligned} E_2(\langle f,\varphi\rangle) &=\int_0^T\int_0^1 |u(t,x)-h_1(t,x)|^2\,dx\,dt +\int_0^T\int_0^1\left|\frac{\partial u(t,x)}{\partial t}-h_2(t,x)\right|^2\,dx\,dt \\ &\quad+\alpha\int_0^T\int_0^1 |f(t,x)|^2\,dx\,dt +\gamma\int_0^1|\varphi(x)|^2\,dx . \end{aligned} \]

Problem 1 becomes the problem of finding an optimal pair \(\langle f,\varphi\rangle\) that realizes the best (in a certain sense) approximation of the final result \(u(T,x)\) of the process \(u(t,x)\) to a prescribed function \(\psi(x)\). Its solution is found from the equations

\[ \begin{aligned} &(1+\alpha)f(t,x)+\int_0^T\int_0^1 \Phi(t,x,\tau,y,\tau)f(\tau,y)\,dy\,d\tau \\ &\quad +(1+\alpha)\int_0^1 P(t,x,y)f(0,y)\,dy \\ &\quad +\int_0^T\left\{\varepsilon(t,\tau)f(\tau,x)+\frac{1+\alpha}{\gamma T}\varepsilon(t,0)f(0,x)\right\}\,d\tau =G(t,x), \end{aligned} \tag{12} \]

\[ \varphi(x)=\frac{1}{\gamma}[(1+\alpha)f(0,x)-h(0,x)]- \frac{1}{\gamma(\gamma+q)}\int_0^1[(1+\alpha)f(0,y)-h(0,y)]R(x)R(y)\,dy, \tag{13} \]

where

\[ \Phi(t,x,y,\tau)=[U(t,\tau)-1/q]R(x)R(y), \]

\[ U(t,\tau)=(2-q)e^{-(2T-t-\tau)q}/2q-e^{-|t-\tau|q}/2, \]

\[ P(t,x,y)=[U(t,0)/(\gamma+q)-1/\gamma q]R(x)R(y),\qquad \varepsilon(t,\tau)=\varepsilon(t,0)=1, \]

\[ \begin{aligned} G(t,x) &=\int_0^1 P(t,x,y)h(0,y)\,dy +\frac{1}{q}\int_0^1\bigl(e^{-(T-t)q}-1\bigr)R(x)R(y)\psi(y)\,dy \\ &\quad-\int_t^T\int_0^1 e^{-(\tau-t)q}R(x)R(y)h(y,\tau)\,dy\,d\tau +h(t,x)+h(0,x)/\gamma+\psi(x). \end{aligned} \]

Problem 2 becomes the problem of finding an optimal pair \(\langle f,\varphi\rangle\) that realizes the best (in a certain sense) approximation of the process \(u(t,x)\) to the prescribed process \(h_1(t,x)\) over a certain time interval \((0,T)\). Its solution \(f(x,t)\) is found from (12), (13), where

\[ \Phi(t,x,\tau,y)=[Q(t,\tau)-\omega(t,\tau)/q]R(x)R(y),\qquad \omega(t,\tau)= \begin{cases} T-t, & 0\leqslant \tau\leqslant t,\\ T-\tau, & t\leqslant \tau\leqslant T, \end{cases} \]

\[ Q(t,\tau)=\bigl[(1-q^2)e^{-|t-\tau|q}-(1+q^2)e^{-(2T-t-\tau)q}\bigr]/2q^2,\qquad \varepsilon(t,\tau)=\omega(t,\tau), \]

\[ P(t,x,y)=[Q(t,0)/(\gamma+q)-\omega(t,0)/\gamma q]R(x)R(y), \]

\[ G(t,x)=\int_t^T\int_0^1\left[\frac{e^{-(\tau-t)q}-1}{q}h_1(\tau,y)-e^{-(\tau-t)q}h_2(\tau,y)\right]R(x)R(y)\,dy\,d\tau+ \]

\[ +\int_t^T h_1(\tau,x)\,d\tau+\int_0^1 P(t,x,y)h_2(0,y)\,dy+h_2(t,x)+\frac{\omega(t,0)}{\gamma}h_2(0,x). \]

Other problems of this type, concerning problems of best approximation to a prescribed temperature regime, are contained in \((^4\!-\!^7)\).

Institute of Cybernetics
Academy of Sciences of the Azerbaijan SSR

Received
11 VIII 1967

CITED LITERATURE

1. E. Hille, R. Phillips, Functional Analysis and Semi-Groups, IL, 1962.
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3. E. J. Aslanov, Tr. Inst. kibernetiki AN AzerbSSR, 5, Baku, 1968.
4. E. J. Aslanov, Dokl. AN AzerbSSR, 21, No. 7 (1965).
5. E. J. Aslanov, Izv. AN AzerbSSR, 5, ser. phys.-tech. and math. sciences (1965).
6. E. J. Aslanov, Materials of the Scientific-Technical Conference of Young Scientists, ser. phys.-math. sciences, Baku, 1967.
7. Z. I. Khalilov, DAN, 155, No. 4 (1964).