We prove that if f is a real valued lower semicontinuous function on a Banach space X and if there exists a C^1, real valued Lipschitz continuous function on X with bounded support and which is not identically equal to zero, then f is Lipschitz continuous of constant K provided all lower subgradients of f are bounded by K. As an application, we give a regularity result of viscosity supersolutions (or subsolutions) of Hamilton-Jacobi equations in infinite dimensions which satisfy a coercive condition....

We consider the problem of internal regional controllability with output constraints. It consists in steering a hyperbolic system to a final state between two prescribed functions only on a subregion of the evolution system domain. This problem is solved by characterizing the optimal control in terms of a subdifferential associated with the minimized functional.

In this paper, an optimal vibration control problem for a nonlinear plate is considered. In order to obtain the optimal control function, wellposedness and controllability of the nonlinear system is investigated. The performance index functional of the system, to be minimized by minimum level of control, is chosen as the sum of the quadratic 10 functional of the displacement. The velocity of the plate and quadratic functional of the control function is added to the performance index functional as...

We consider the efficient and reliable solution of linear-quadratic optimal control problems governed by parametrized parabolic partial differential equations. To this end, we employ the reduced basis method as a low-dimensional surrogate model to solve the optimal control problem and develop a posteriori error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional. We show that our approach can be applied to problems involving...

A linear quadratic optimal control problem for a class of discrete distributed systems is analyzed. To solve this problem, we introduce an adequate topology and establish that optimal control can be determined though an inversion of the appropriate isomorphism. An example and a numerical approach are given.

Starting from Giaquinta’s counterexample [12] we introduce the class of splitting functionals being of $(p,q)$-growth with exponents $p\le q\<\infty$ and show for the scalar case that locally bounded local minimizers are of class $C^{1,\mu }$. Note that to our knowledge the only $C^{1,\mu }$-results without imposing a relation between $p$ and $q$ concern the case of two independent variables as it is outlined in Marcellini’s paper [15], Theorem A, and later on in the work of Fusco and Sbordone [10], Theorem 4.2.

Let $L:{ℝ}^N\times {ℝ}^N\to ℝ$ be a borelian function and consider the following problems$$inf\left\{F\left(y\right)={\int }_a^{b}L(y\left(t\right),y^{\text{'}}\left(t\right))\mathrm{d}t:y\in AC([a,b],{ℝ}^N),y\left(a\right)=A,y\left(b\right)=B\right\}\left(P\right)$$$$inf\left\{F^{...}\right\}$$

Let $L:{ℝ}^N\times {ℝ}^N\to ℝ$ be a Borelian function and consider the following problems $$inf\left\{F\left(y\right)={\int }_a^{b}L(y\left(t\right),y^{\text{'}}\left(t\right))\mathrm{d}t:y\in AC([a,b],{ℝ}^N),y\left(a\right)=A,y\left(b\right)=B\right\}\left(P\right)$$
 $$inf\left\{F^{**}\left(y\right)={\int }_a^b{L}^{**}(y\left(t\right),y^{\text{'}}\left(t\right))\mathrm{d}t:y\in AC([a,b],{ℝ}^N),y\left(a\right)=A,y\left(b\right)=B\right\}·\left(P^{**}\right)$$ We give a sufficient condition, weaker then superlinearity, under which $infF=inf{F}^{**}$ if L is just continuous in x. We then extend a result of Cellina on the Lipschitz regularity of the minima of (P) when L is not superlinear.

We prove higher integrability for the gradient of bounded minimizers of some variational integrals with anisotropic growth.

In this note we study the summability properties of the minima of some non differentiable functionals of Calculus of the Variations.

It is known that the vector stop operator with a convex closed characteristic $Z$ of class $C^1$ is locally Lipschitz continuous in the space of absolutely continuous functions if the unit outward normal mapping $n$ is Lipschitz continuous on the boundary $\partial Z$ of $Z$. We prove that in the regular case, this condition is also necessary.