Special bang-bang solutions for nonlinear control systems (Q1895857)

For a given solution \(x\), defined on an interval \([a, b]\), to the control system \(x''+ a(x, x')= ug(x, x')\), the authors prove the existence of two bang-bang solutions \(y\) and \(z\), with a finite number of switchings, satisfying \(x(a)= y(a)= z(a)\), \(x(b)= y(b)= z(b)\), \(x'(a)= y'(a)= z'(a)\), \(x'(b)= y'(b)= z'(b)\) and \(y(t)\leq x(t)\leq z(t)\). As an example, the result is applied to show the behavior of optimal solutions to the Bolza problem.