Oriented measures with continuous densities and the bang-bang principle (Q1891774)

This paper is concerned with some extensions of the Lyapunov convexity theorem. A function \(f= (f_1,\dots, f_n)\) is said to verify the orientation condition \(\Delta\) on \([a, b]\) if, for each \(k\in \{1,\dots, n\}\), the determinant \(\text{det}(f^k(x_1),\dots, f^k(x_k))\) (where \(f^k= (f_1,\dots, f_k))\) is not equal to zero whenever \(x_i\in [a, b]\) for \(i= 1,\dots, k\) are distinct, and its sign is constant for each \(k\)-tuple \((x_1,\dots, x_k)\) such that \(a\leq x_1< x_2<\cdots< x_k\leq b\). The main result is the following. If \(f\) satisfies the orientation condition \(\Delta\), if \(f\) is continuous on \([a, b]\) with values in \(\mathbb{R}^n\), if \(\nu\in L^1(a, b)\), then there exist \(a\leq \alpha_1\leq\cdots\leq \alpha_n\leq b\) and \(a\leq \beta_1\leq\cdots\leq \beta_n\leq b\) such that \(\int^b_a f_j \chi_{E_\alpha}= \int^b_a f_j\nu= \int^b_a f_j \chi_{E_\beta}\), where \(\chi_{E_\alpha}\) and \(\chi_{E_\beta}\) denote characteristic functions, \(E_\alpha= \bigcup_{i\text{ odd}}[\alpha_i, \alpha_{i+ 1}]\) and \(E_\beta= \bigcup_{i\text{ even}} [\beta_i, \beta_{i+ 1}]\). The proof is based either on the implicit function theorem or on a global inversion theorem due to Cacciopoli. It is proved that, if \(f\) is regular enough and if the Wronskians \(W(f_1),\dots, W(f_1,\dots, f_n)\) do not vanish on \([a, b]\), then the orientation condition \(\Delta\) is satisfied. These results are applied to prove existence of bang-bang solutions to linear control systems of linear ordinary differential equations. Some generic existence theorems are also given for one-dimensional problems of the calculus of variations.