Optimal boundary control by the displacement of one endpoint of the string with the other endpoint being free in an arbitrarily large time interval. (English. Russian original) Zbl 1144.49023
Differ. Equ. 43, No. 10, 1403-1414 (2007); translation from Differ. Uravn. 43, No. 10, 1369-1381 (2007).
Summary: The optimal boundary control by either a displacement or an elastic boundary force at one endpoint of a string in an arbitrary large time interval \(T\) under the condition that the other endpoint is fixed was found in closed form in [the authors, Differ. Equ. 42, No. 11, 1633-1644 (2006); translation from Differ. Uravn. 42, No. 11, 1558–1570 (2006; Zbl 1131.49028), Differ. Equ. 42, No. 12, 1775–1786 (2006); translation from Differ. Uravn. 42, No. 12, 1699–1711 (2006; Zbl 1132.49032)].
The technique developed in [the authors, loc. cit.] proves to be insufficient for solving similar optimization problems with free second endpoint and should be substantially modified. In the present paper, we suggest a modification of the method in [the authors, loc. cit.], which permits one to compute and write out an explicit analytic expression for the optimal boundary control by the displacement of the endpoint \(x=0\) of the string for an arbitrary sufficiently large time interval \(T\); the second endpoint \(x=l\) of the string is free, and the control brings the vibration process of the string described by the generalized solution u(x, t) of the wave equation
\[ u_{tt}(x,t)-u_{xx}(x,t)=0 \]
from an arbitrary given initial state
\[ \{u(x,0)= \varphi(x), u_t(x,0)=\psi(x)\} \]
to an arbitrary terminal state
\[ \{u(x,T)= \widehat{\varphi}(x),\;u_t(x,T)= \widehat{\psi}(x)\}. \]
The technique developed in [the authors, loc. cit.] proves to be insufficient for solving similar optimization problems with free second endpoint and should be substantially modified. In the present paper, we suggest a modification of the method in [the authors, loc. cit.], which permits one to compute and write out an explicit analytic expression for the optimal boundary control by the displacement of the endpoint \(x=0\) of the string for an arbitrary sufficiently large time interval \(T\); the second endpoint \(x=l\) of the string is free, and the control brings the vibration process of the string described by the generalized solution u(x, t) of the wave equation
\[ u_{tt}(x,t)-u_{xx}(x,t)=0 \]
from an arbitrary given initial state
\[ \{u(x,0)= \varphi(x), u_t(x,0)=\psi(x)\} \]
to an arbitrary terminal state
\[ \{u(x,T)= \widehat{\varphi}(x),\;u_t(x,T)= \widehat{\psi}(x)\}. \]
MSC:
| 49N90 | Applications of optimal control and differential games |
| 35B37 | PDE in connection with control problems (MSC2000) |
| 35L05 | Wave equation |
| 74K05 | Strings |
References:
| [1] | Il’in, V.A. and Moiseev, E.I., Differ. Uravn., 2006, vol. 42, no. 11, pp. 1558–1570. |
| [2] | Il’in, V.A. and Moiseev, E.I., Differ. Uravn., 2006, vol. 42, no. 12, pp. 1699–1711. |
| [3] | Il’in, V.A., Differ. Uravn., 2000, vol. 36, no. 11, pp. 1513–1528. |
| [4] | Il’in, V.A., Differ. Uravn., 2000, vol. 36, no. 12, pp. 1670–1686. |
| [5] | Il’in, V.A. and Moiseev, E.I., Uspekhi Mat. Nauk, 2005, vol. 60, no. 6, pp. 89–114. · doi:10.4213/rm1678 |
| [6] | Il’in, V.A., Uspekhi Mat. Nauk, 1960, vol. 15, no. 2, pp. 97–154. |
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