Controllability problems for the heat equation with variable coefficients on a half-axis

Abstract

In the paper, the problems of controllability and approximate controllability are studied for the heat equation $w_t=\frac{1}{\rho}\left(kw_x\right)_x+\gamma w$ , $x>0$ , $t\in(0,T)$ , controlled by the Dirichlet boundary condition. Control is considered in $L^\infty(0,T)$ . It is proved that each initial state of this system is approximately controllable to any its end state in a given time $T>0$ .

Controllability Problems for the Heat Equation with Variable Coefficients on a Half-Axis Controlled by the Neumann Boundary Condition

“…Controllability problems for the heat equation with constant and variable coefficients were studied in a number of papers (see, e.g., [1-4, 9, 10, 13, 14, 17, 21-27]). However, there are only a few papers where these problems were investigated for the heat equation with constant coefficients on unbounded domains (see, e.g., [2,9,10,22,23]), and it seems these problems were investigated for the heat equation with variable coefficients on unbounded domains only in [11].…”

“…In [11], controllability problems for the heat equation with variable coefficients on a half-axis controlled by the Dirichlet boundary condition are studied. The general methods applied in the present paper are similar to those from paper [11].…”

“…In [11], controllability problems for the heat equation with variable coefficients on a half-axis controlled by the Dirichlet boundary condition are studied. The general methods applied in the present paper are similar to those from paper [11]. But for the case of the Neumann boundary condition, different spaces and operators are used that caused different technique of proofs of main results.…”

“…In the case of the heat equation, it has been proved that f and (•)P are bounded on [0, T ] (hence, [11]. That is why the integral operator in the right-hand side of (1.8) is not of the Hilbert-Schmidt type, and the Fredholm alternative is not applicable in the general case.…”

“…That is why the method of successive approximations has been used to construct a solution to (1.8) on [0, T ]. Then the Banach fixed-point theorem has been applied in L 2 -space on small intervals to prove the uniqueness of the solution [11]. This result is recalled in Lemma 3.5 below.…”