Optimization problems for the differential equation \(y''-qy=0\). II
These papers deal with the problem of finding \(\inf_ qy(1)\), where y satisfies \(y''-q(t)y=0\), \(0\leq t\leq 1\), \(y(0)=\pm 1\), \(y'(0)=a\), and \(\int^{1}_{0}| q(t)| dt=b\), with given parameters a and b. The problem (which is considered as an optimal control problem) is first transformed into an equivalent problem where the infimum is attained for some \(L_ 1\)-function. Then the necessary conditions for optimality (Pontryagin's maximum principle) are analyzed, and in this way the optimal solution(s) can be found. There is a number of cases (depending on a and b), and in each case inf y(1) is found as an explicit expression in a and b. In the first paper the case \(y(0)=1\) is treated and in the second the case \(y(0)=-1\). The results extend significantly some earlier results by \textit{M. Essén} [see ibid. 115, 278-304 (1986; Zbl 0603.34060)] obtained by other methods.
scientific article; zbMATH DE number 4129508