We consider control problems governed by semilinear parabolic equations with pointwise state constraints and controls in an $L^p$-space ($p\<\infty$). We construct a correct relaxed problem, prove some relaxation results, and derive necessary optimality conditions.

We consider the weak closure $WZ$ of the set $Z$ of all feasible pairs (solution, flow) of the family of potential elliptic systems$$$$where $\Omega \subset {𝐑}^n$ is a bounded Lipschitz domain, $F_s$ are strictly convex smooth functions with quadratic growth and $S=\{\sigma measurable\mid {\sigma }_s\left(x\right)=0\text{or}1,s=1,\cdots ,s_0,{\sigma }_1\left(x\right)+\cdots +{\sigma }_{s_0}\left(x\right)=1\}$. We show that $WZ$ is the zero level set for an integral functional with the integrand $Qℱ$ being the $𝐀$-quasiconvex envelope for a certain function $ℱ$ and the operator $𝐀={\left(\text{curl,div}\right)}^m$. If the functions $F_s$ are isotropic, then on the characteristic cone $\Lambda$ (defined by the operator $𝐀$) $Qℱ$ coincides...

We consider the weak closure WZ of the set Z of all feasible pairs (solution, flow) of the family of potential elliptic systems $$\begin{array}{c}\text{div}\left(\sum _{s=1}^{s_0}{\sigma }_s\left(x\right)F_s^{\text{'}}(\nabla u\left(x\right)+g\left(x\right))-f\left(x\right)\right)=0\text{in}\Omega ,u=(u_1,\cdots ,u_m)\in H_0^1(\Omega ;{𝐑}^m),\sigma =({\sigma }_1,\cdots ,{\sigma }_{s_0})\in S,\end{array}$$ where Ω ⊂ Rn is a bounded Lipschitz domain, Fs are strictly convex smooth functions with quadratic growth and $S=\{\sigma measurable\mid {\sigma }_s\left(x\right)=0\text{or}1,s=1,\cdots ,s_0,{\sigma }_1\left(x\right)+\cdots +{\sigma }_{s_0}\left(x\right)=1\}$. We show that WZ is the zero level set for an integral functional with the integrand $Qℱ$ being the A-quasiconvex envelope for a certain function $ℱ$ and the operator A = (curl,div)m. If the functions Fs are isotropic, then on the characteristic cone...

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 its quasiconvex...

Multidimensional vectorial non-quasiconvex variational problems are relaxed by means of a generalized-Young-functional technique. Selective first-order optimality conditions, having the form of an Euler-Weiestrass condition involving minors, are formulated in a special, rather a model case when the potential has a polyconvex quasiconvexification.

The contribution is devoted to computations of the limit load for a perfectly plastic model with the von Mises yield criterion. The limit factor of a prescribed load is defined by a specific variational problem, the so-called limit analysis problem. This problem is solved in terms of deformation fields by a penalization, the finite element and the semismooth Newton methods. From the numerical solution, we derive a guaranteed upper bound of the limit factor. To achieve more accurate results, a local...

A class of parabolic initial-boundary value problems is considered, where admissible coefficients are given in certain intervals. We are looking for maximal values of the solution with respect to the set of admissible coefficients. We give the abstract general scheme, proposing how to solve such problems with uncertain data. We formulate a general maximization problem and prove its solvability, provided all fundamental assumptions are fulfilled. We apply the theory to certain Fourier obstacle type...

The existence of a solution to the dynamic contact of a body having a singular memory with a rigid undeformable support is proved under some weaker assumption than that in [3].

The problem of energy transfer in an -ladder network is considered. Using the maximum principle, an algorithm for constructing optimal control is proposed, where the cost function is the energy delivered to the network. In the case considered, optimal control exists. Numerical simulations were performed using Matlab.

We present some consequences of a deep result of J. Lindenstrauss and D. Preiss on $\Gamma$-almost everywhere Fréchet differentiability of Lipschitz functions on $c_0$ (and similar Banach spaces). For example, in these spaces, every continuous real function is Fréchet differentiable at $\Gamma$-almost every $x$ at which it is Gâteaux differentiable. Another interesting consequences say that both cone-monotone functions and continuous quasiconvex functions on these spaces are $\Gamma$-almost everywhere Fréchet differentiable....

We compute the quasiconvex envelope of certain functions defined on the space $M_{mn}$ of real $m\times n$ matrices. These functions are basically functions of a quadratic form on $M_{mn}$. The quasiconvex envelope computation is applied to densities that are related to the James-Ericksen elastic stored energy function.