On a class of diffeomorphic matching problems in one dimension (Q2706146)

The authors study the following optimal control problem, arising in the theory of matching problems: NEWLINE\[NEWLINE\begin{cases} \text{find }\phi^*\in \Hom^+,\text{ such that }\\ U_f(\phi^*)= \max\{U_f(\phi): \phi\in\Hom^+\},\end{cases}\tag{P}NEWLINE\]NEWLINE where \(\Hom^+\) is the set of all continuous strictly increasing functions \(\phi: [0,1]\to [0,1]\) such that \(\phi(0)= 0\) and \(\phi(1)= 1\); \(U_f\) is defined for all \(\phi\in \Hom^+\) by NEWLINE\[NEWLINEU_f(\phi)= \int^1_0 \sqrt{\dot\phi} f(\phi(x),x) dxNEWLINE\]NEWLINE (as \(\phi\) is differentiable a.e.), where \(f: [0,1]\times [0,1]\to \mathbb{R}_+\) is a measurable function. Under some appropriate sets of assumptions the authors prove an existence result for problem (P) as well as some regularity properties of the solutions of (P) (differentiability at some points \(x_0\) or even twice continuous differentiability).NEWLINENEWLINENEWLINEThe motivation for the study of the problem (P) is the so-called matching problem. In general a matching problem consists in finding, for two given functions \(\theta\) and \(\theta'\) defined on an interval \(I\subseteq\mathbb{R}\), a transformation \(\phi\in \Hom^+\) (called matching) such that \(\theta\) and \(\theta'\circ\phi\) are as close as possible in some choosen way. The control problem considered in the paper suits to the case of matching the shapes of two plane curves on the base of their silhouettes. Then one can consider \(f(y,x)=\Bigl|\cos{\theta(y)- \theta'(x)\over 2}\Bigr|\), where \([0,2\pi)\)-valued functions \(\theta\) and \(\theta'\) correspond to rotation angles of the unitary tangents to these shapes.