## Boundary control of a string oscillating at one end, with the other end fixed and under the condition of the existence of finite energy.(English. Russian original)Zbl 1090.93531

Dokl. Math. 63, No. 3, 410-414 (2001); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 378, No. 6, 743-747 (2001).
From the text: We study the boundary control at the end $$x=0$$ of the oscillations of a string fastened at the end $$x=l$$ occurring over an arbitrary time interval $$T>0$$ and described by the generalized solution of the wave equation $$u_{tt}(x,t)-u_{xx}(x,t)=0$$ in a class that admits the existence of finite energy at any moment of time. Let $$Q_T=[0\leq x\leq l]\times [0\leq t\leq T]$$. The class $$\widehat W^1_2(Q_T)$$, is defined as the set of functions $$u(x,t)$$ of two variables that are continuous in the closed rectangle $$\overline Q_T$$ and have both generalized first-order partial derivatives $$u_x(x,t)$$ and $$u_t(x,t)$$ in $$Q_T$$, each of which belongs to the class $$L_2(Q_T)$$ and also to the class $$L_2[0\leq x\leq l]$$ for any $$t$$ in the interval $$[0,T]$$ and to the class $$L_2[0\leq t\leq T]$$ for any $$x$$ in the interval $$[0,l]$$.

### MSC:

 93C20 Control/observation systems governed by partial differential equations 35B37 PDE in connection with control problems (MSC2000)