Numerical solutions to problems of the least squares type for ordinary differential equations
This paper is concerned with the problem of finding a solution of \(dx/dt=X(x,t)\), \(t\in [a,b]\) which minimizes a real functional \(v[x]=(g[x])^*g[x]\) locally, where x and X are n-dimensional vectors, g[x] a real m-dimensional functional, \({}^*\) denotes the transposition of vectors. Some articles have considered the problem by reducing it to the boundary value problem for the same differential equation with the condition \((g'(x)[\Phi_{(x)}])^*g(x)=0\). Here g' means the Fréchet derivative of g, and \(\Phi_{(x)}\) is the matrizant of the first variational equation of the above equation at x. In this direction, the authors combine their each preceding works to establish theorems on the local minimality of the solution and on applicable a-posteriori error estimation of the approximate solution. Furthermore, an illustrative example is given with finite Chebyshev series approximation. The approximate solution with more than 20 terms is shown to have a sharp bound for the norm of the difference from the exact solution by employing their theorem. At the same time, this error bound guarantees the local minimality of the exact solution for the original problem due to their another theorem.
scientific article; zbMATH DE number 3848521