We consider a finite-dimensional control system $\left(\Sigma \right)\dot{x}\left(t\right)=f(x\left(t\right),u\left(t\right))$, such that there exists a feedback stabilizer that renders $\dot{x}=f(x,k\left(x\right))$ globally asymptotically stable. Moreover, for with an output map and $1\le p\le q\le \infty$, we assume that there exists a ${𝒦}_{\infty }$-function such that $\parallel H\left(x_u\right){\parallel }_q\le {\alpha (\parallel u\parallel }_p)$, where is the maximal solution of ${\left(\Sigma \right)}_k\dot{x}\left(t\right)=f(x\left(t\right),k\left(x\left(t\right)\right)+u\left(t\right))$, corresponding to and to the initial condition . Then, the gain function $G_{(H,p,q)}$ of given by 14.5cm $$G_{(H,p,q)}\left(X\right)\stackrel{\mathrm{def}}{=}\underset{{\parallel u\parallel }_p=X}{sup}{\parallel H\left(x_u\right)\parallel }_q,$$ is well-defined. We call profile of for any ${𝒦}_{\infty }$-function which is of the same order of magnitude...

Given two measured spaces $(X,\mu )$ and $(Y,\nu )$, and a third space , given two functions  and , we study the problem of finding two maps $s:X\to Z$ and $t:Y\to Z$ such that the images $s\left(\mu \right)$ and $t\left(\nu \right)$ coincide, and the integral ${\int }_{X}u(x,s\left(x\right))d\mu -{\int }_{Y}v(y,t\left(y\right))d\nu$ is maximal. We give condition on and for which there is a unique solution.