| Abstract |
In this study, we consider the stabilization problem of finding {y(x,t),u_1 (t),〖 u〗_2 (t)} for the loaded heat equation
{■(y_t (x,t)-y_xx (x,t)+α∙y(0,t)=0, {x,t}∊Q,@y(-π/2,t)=u_1 (t), y(π/2,t)=u_2 (t), y(x,0)=y_0 (x),)┤ (1)
where solution y(x,t) tends to the zero when t→∞, in the following way
‖y(x,t)‖_(L_2 (-π/2,π/2) )≤C_(0 ) e^(-σ_0 t). (2)
Here Q={x,t|-π/20},α∊C,σ_0 is a given positive number, y_0 (x)∊L_2 (-π/2, π/2) is a given function. The equation (1) is called loaded [1] – [3].
For the boundary value problem (1) on the semibar of width π with the nonhomogeneous Dirichlet boundary conditions and initial condition on the segment (-π/2, π/2) by the given function y_0 (x) we consider the auxiliary boundary value problem on the extended semibar of width 2π with the periodicity conditions (instead of Dirichlet conditions) and initial function z_0 (x) on the segment (-π, π). Further, we determine a function z_0 (x) as continuation of the given function y_0 (x).
Theorem on solvability of the stated inverse problem (1) – (2) is proved and the algorithm of approx construction of boundary controls is developed. Numeral calculations show efficiency of the offered algorithm.
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