A regularized solution to the Birkhoff interpolation problem
The authors study a relationship between the Birkhoff interpolation problem and a problem of linear control theory. Namely they consider the system \[ \dot{x}=Ax+bu,x(0)=x_0, \qquad y=cx, \] where \(A\) is \(n\times n,\) \(b\) is \(n\times 1,\) \(c\) is \(1\times n,\) \((x_0,u)\) supposed to be in \(\mathcal H=\mathbb R^n\times L_2[0,T].\) The basis functions \(l_{(i,j)}(s)\) are defined as \(cA^{j-1}e^{A(t_I-s)b}\chi_{(-\infty,t_I)}(s).\) As for the Birkhoff interpolation problem an incidence \((n\times n)\) matrix \(J\) with \(n\) unit entries \((i,j)\in E\) gives \(n\) constraints (for \((i,j)\in E\)) \[ \alpha_{(i,j)}=cA^{j-1}e^{At_I}x_0+\int_0^Tl_{(i,j)}(s)u(s)\,ds. \] The optimal problem is to find \[ \min_{(x_0,u)\in\mathcal H}I(x_0,u) \] subject to the \(n\) constraints from above, where \(I(x_0,u)\) is the cost function \(I(x_0,u)=\lambda x_0^{\prime}x_0+\int_0^Tu^2(t)\,dt.\) The main result: Birkhoff interpolation problem with incidence matrix \(J\) has a solution if and only if the solutions of the optimal problem from above are uniformly bounded for \(\lambda\in (0,T).\)
scientific article; zbMATH DE number 5012659