Boundedness of optimal trajectories
The paper deals with the system \(\dot x=f(x,u)\), \(u\in U\subset R^ m\), \(x\in H\subset R^ n\), with the following optimality criterion: \[ I(x(\cdot),0,T)=\int^{T}_{0}F(x,u)dt\to \min,\quad T>0, \] where H is a closed set, U is compact, f and F are of class C in an open neighbourhood of H, u is a measurable function in U, x(0)\(\in X_ 0\subset R^ n\), x(T)\(\in X_ T\subset R^ n\), T is finite and fixed. In the standard theory of optimal control, a criterion based on the boundedness of the function f is often used in the proofs of the existence of an optimal control (i.e. \(\max_{u\in U}\| f(x,u)\|\) is bounded if \(\| x\| \to \infty\), where \(\| \cdot \|\) is the Euclidean norm). This paper gives some new criteria which are some generalizations of that one mentioned above. The criteria suggested in the paper can be used for the cases: a) fixed x(0) and x(T); b) fixed x(0) and free x(T); c) free x(0) and x(T).
scientific article; zbMATH DE number 3992267