Semidefinite Optimization Estimating Bounds on Linear Functionals Defined on Solutions of Linear ODEs (Advances in Applied Mathematics and Mechanics, 2018)
In this paper, semidefinite optimization method is proposed to estimatebounds on linear functionals defined on solutions of linear ordinary differential equations(ODEs) with smooth coefficients. The method can get upper and lower boundsby solving two semidefinite programs, not solving ODEs directly. Its convergencetheorem is proved. The theorem shows that the upper and lower bounds series of linearfunctionals discussed can approach their exact values infinitely. Numerical resultsshow that the method is effective for the estimation problems discussed. In addition,in order to reduce calculation amount, Cheybeshev polynomials are applied to replaceTaylor polynomials of smooth coefficients in computing process.