×

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)
PDF BibTeX XML Cite