Abstract
In this paper, we present iterative and non-iterative methods for the solution of nonlinear optimal con-trol problems (NOCPs) and address the sufficient conditions for uniqueness of solution. We also studyconvergence properties of the given techniques. The approximate solutions are calculated in the form ofa convergent series with easily computable components. The efficiency and simplicity of the methods aretested on a numerical example.
Highlights
Numerical methods for solving optimal control problems are divided into two major classes: indirect methods and direct methods
We provide sufficient conditions for the convergence of Iterative Method (IM) solution series
Where p ∈ [0, 1] is the so-called homotopy parameter, Z(t, p) : [t0, tf ] × [0, 1] → Rn+1, and Z0 defines the initial approximation of the solution of (2.7)
Summary
We want to find the sufficient conditions for the uniqueness of solution in the space C([t0, tf ] → Rn+1) of real-valued continuous functions on the interval [t0, tf ]. Let Q : [t0, tf ] × Rn+1 → R be continuous and there exists a positive constant 0 < L, such that. Our proof is proceed as follows: Step 1. Let Br ⊂ C([t0, tf ] → Rn+1) be bounded; i.e., there exists a positive constant r such that Z θ ≤ r, ∀Z ∈ C([t0, tf ] → Rn+1). For the chosen value of θ, the operator Φ is a contraction map. By application of the Contraction Mapping Theorem Φ has a unique fixed point in C([0, T ] → Rn+1), which is the unique solution of the problem
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