Semiconcavity results have generally been obtained for optimal control problems in absence of state constraints. In this paper, we prove the semiconcavity of the value function of an optimal control ...problem with end-point constraints for which all minimizing controls are supposed to be nonsingular.
\par In this paper we investigate the existence and Lipschitz continuity of optimal trajectories for the autonomous Bolza problem in control theory. The main feature of our results is that they relax ...the usual fast growth condition for the Lagrangian. Furthermore, we show that optimal solutions do satisfy the maximum principle. \par
Let two second order evolution equations be coupled via the zero order terms, and suppose that thefirst one is stabilized by a distributed feedback. What will then be the effect of such a partial ...stabilization onthe decay of solutions at infinity? Is the behaviour of the first component sufficient to stabilize the second one?The answer given in this paper is that sufficiently smooth solutions decay polynomially at infinity, and that thisdecay rate is, in some sense, optimal. The stabilization result for abstract evolution equations is also applied tostudy the asymptotic behaviour of various systems of partial differential equations.
This paper is devoted to the study of the existence and uniqueness of the invariant measure associated to the transition semigroup of a diffusion process in a bounded open subset of ℝn. For this ...purpose, we investigate first the invariance of a bounded open domain with piecewise smooth boundary showing that such a property holds true under the same conditions that insure the invariance of the closure of the domain. A uniqueness result for the invariant measure is obtained in the class of all probability measures that are absolutely continuous with respect to Lebesgue's measure. A sufficient condition for the existence of such a measure is also provided.
Given a bounded domain \Omega in \mathbb{R}^2 with smooth boundary, the {\em cut locus} \overline \Sigma is the closure of the set of nondifferentiability points of the distance d from the boundary ...of \Omega. The normal distance to the cut locus, \tau(x), is the map which measures the length of the line segment joining x to the cut locus along the normal direction Dd(x), whenever x\notin \overline \Sigma. Recent results show that this map, restricted to boundary points, is Lipschitz continuous, as long as the boundary of \Omega is of class C^{2,1}. Our main result is the global Hölder regularity of \tau in the case of a domain \Omega with analytic boundary. We will also show that the regularity obtained is optimal, as soon as the set of the so-called {\em regular conjugate points} is nonempty. In all the other cases, Lipschitz continuity can be extended to the whole domain \Omega. The above regularity result for \tau is also applied to derive the Hölder continuity of the solution of a system of partial differential equations that arises in granular matter theory and optimal mass transfer.
The system of partial differential equations (formula omitted) arises in the analysis of mathematical models for sandpile growth and in the context of the Monge-Kantorovich optimal mass transport ...theory. A representation formula for the solutions of a related boundary value problem is here obtained, extending the previous two-dimensional result of the first two authors to arbitrary space dimension. An application to the minimization of integral functionals of the form (formula omitted) with f greater than or equal to 0, and h greater than or equal to 0 possibly non-convex, is also included. PUBLICATION ABSTRACT
We prove an estimate of Carleman type for the one dimensional heat equationut − (a(x)ux )x + c(t, x)u = h(t, x), (t, x) ∈ (0, T ) × (0, 1),where a(·) is degenerate at 0. Such an estimate is derived ...for a special pseudo-convex weight function related tothe degeneracy rate of a(·). Then, we study the null controllability on 0, 1 of the semilinear degenerate parabolicequationut − (a(x)ux )x + f (t,x,u) = h(t, x)χω(x),where (t, x) ∈ (0, T ) × (0, 1), ω = (α, β) ⊂⊂ 0, 1, and f is locally Lipschitz with respect to u.
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