Optimal boundary control of heat transfer in a three-dimensional material: a hyperbolic model (Q2251875)

The authors of this important paper consider the hyperbolic model of heat transfer in a homogeneous isotropic field. The authors construct a class of solutions of this problem depending on a functional parameter by a proposed scheme. The Cauchy problem for system \((1,1^o)\) in the truncated cone \(Q\) with initial vector function \(\eta_\varphi (x)\) is considered. The authors construct a (generalized) solution \((T_\varphi, q_\varphi)\) of this problem with the use of Riemann matrices for one-dimensional hyperbolic operators. The use of the Lagrange method reduces the solution of a given problem to the solution of a system of Fredholm equations of the second kind with respect to the functions \(h_1, h_2\) and \(\lambda\) (\(\lambda\) is the Lagrange multiplier) and the kernels of the equations are constructed on the basis of entries of the Riemann matrices of an auxiliary one-dimensional hyperbolic operator.