Boundary Value Problem for Ordinary Differential Equations with Applications to Optimal Control

VPM (continuation method or homotopy method) may be considered as special development and modification of classical Newton’s method, see [1] [5], [16]. The main idea of VPM admits short formulation: reducing of a given problem to some IVP for ODE. This fundamental principle of mathematical physics is well known, widely used and effective because IVP for ODE may be considered as a routine computational problem. The VPM-scheme outline is presented here in pressed continuous form. It allows to combine simplicity of exposition of the main idea with conviniencies for numerical realization on the way to computational experiments. Some special VPM-schemes for solving BVP ODE are outlined below. It occurs convinient to produce numerical experiments within the approach in MAPLE program environment which permits to fulfil analytical and numerical-analytical calculations using MAPLE [15] computer algebra possibilities. By the way, instead of MAPLE, the MAXIMA program, for example, may be used. Serious computational difficulties, arising here, are adequate to complexity of the considered problem. Intention to solve PMP BVP arising in OC was the main impulse for providing the investigations. These results are of interest also for tutoring activity. Some numerics are presented in the paper. The elaborated software (Program) allows to solve regular BVP and some PMP BVP in OC, to search periodic solutions, limit cycles of ODE, to determine unknown parameters in nonlinear ODE, etc. Note that considered computational scheme does not coincide with previous ones.