The aim of the paper is to provide a linearization approach to the $\mathrm{See\; PDF}$-control problems. We begin by proving a semigroup-type behaviour of the set of constraints appearing in the linearized formulation of (standard) control problems. As a byproduct we obtain a linear formulation of the dynamic programming principle. Then, we use the $\mathrm{See\; PDF}$ approach and the associated linear formulations. This seems to be the most appropriate tool for treating $\mathrm{See\; PDF}$ problems in continuous and lower semicontinuous...

One shows that the linearized Navier-Stokes equation in $𝒪\subset R^d,d\ge 2$, around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller ${\displaystyle V(t,\xi )=\sum _{i=1}^N{V}_i\left(t\right){\psi }_i\left(\xi \right){\dot{\beta }}_i\left(t\right)}$, $\xi \in 𝒪$, where ${\left\{{\beta }_i\right\}}_{i=1}^N$ are independent Brownian motions in a probability space and ${\left\{{\psi }_i\right\}}_{i=1}^N$ is a system of functions on $𝒪$ with support in an arbitrary open subset ${𝒪}_0\subset 𝒪$. The stochastic control input ${\left\{V_i\right\}}_{i=1}^N$ is found in feedback form. One constructs also a tangential boundary noise controller which exponentially stabilizes in probability...

In this paper, we consider a Borel measurable function on the space of $m\times n$ matrices $f:M^{m\times n}\to \overline{ℝ}$ taking the value $+\infty$, such that its rank-one-convex envelope $Rf$ is finite and satisfies for some fixed $p>1$: $$-c_0\le Rf\left(F\right)\le {c(1+\parallel F\parallel }^p)\text{for}\text{all}F\in M^{m\times n},$$ where $c,c_0>0$. Let $Ø$ be a given regular bounded open domain of ${ℝ}^n$. We define on $W^{1,p}(Ø;{ℝ}^m)$ the functional $$I\left(u\right)={\int }_{Ø}f\left(\nabla u\left(x\right)\right)dx.$$ Then, under some technical restrictions on $f$, we show that the relaxed functional $\overline{I}$ for the weak topology of $W^{1,p}(Ø;{ℝ}^m)$ has the integral representation: $$\overline{I}\left(u\right)={\int }_{Ø}Q\left[Rf\right]\left(\nabla u\left(x\right)\right)dx,$$ where for a given function $g$, $Qg$ denotes...