Given a Borel function ψ defined on a bounded open set Ω with Lipschitz boundary and $\varphi\in L^1(\partial\Omega,{\mathcal H}^{n-1})$ , we prove an explicit representation formula for the L 1 lower semicontinuous envelope of Mumford-Shah type functionals with the obstacle constraint $u^+\ge\psi$ ${\mathcal H}^{n-1}$ a.e. on Ω and the Dirichlet boundary condition $u=\varphi$ on $\partial\Omega$ .

The Linear-Quadratic (LQ) optimal control problem is studied for aclass of first-order hyperbolic partial differential equation modelsby using a nonlinear infinite-dimensional (distributed parameter) Hilbert state-spacedescription. First the dynamical properties of the linearized modelaround some equilibrium profile are studied. Next the LQ-feedbackoperator is computed by using the corresponding operator Riccatialgebraic equation whose solution is obtained via a relatedmatrix Riccati differential equation in the space variable. Then thelatter is applied to the nonlinear model, and the resultingclosed-loop system dynamical performances are analyzed.