We consider an evolution equation similar to that introduced by Vese in [Comm. Partial Diff. Eq. 24 (1999) 1573–1591] and whose solution converges in large time to the convex envelope of the initial datum. We give a stochastic control representation for the solution from which we deduce, under quite general assumptions that the convergence in the Lipschitz norm is in fact exponential in time.

We consider an evolution equation similar to that introduced by Vese in [Comm. Partial Diff. Eq. 24 (1999) 1573–1591] and whose solution converges in large time to the convex envelope of the initial datum. We give a stochastic control representation for the solution from which we deduce, under quite general assumptions that the convergence in the Lipschitz norm is in fact exponential in time.

We consider an evolution equation similar to that introduced by Vese in [Comm. Partial Diff. Eq. 24 (1999) 1573–1591] and whose solution converges in large time to the convex envelope of the initial datum. We give a stochastic control representation for the solution from which we deduce, under quite general assumptions that the convergence in the Lipschitz norm is in fact exponential in time.

This article is devoted to the optimal control of state equations with memory of the form: $\dot{x}\left(t\right)=F(x\left(t\right),u\left(t\right),{\int }_0^{+\infty }A\left(s\right)x(t-s)\mathrm{d}s),t>0,$with initial conditions $x\left(0\right)=x,x(-s)=z\left(s\right),s>0$. Denoting by $y_{x,z,u}$ the solution of the previous Cauchy problem and:$v(x,z):={inf}_{u\in V}\left\{{\int }_0^{+\infty }{\mathrm{e}}^{-\lambda s}L(y_{x,z,u}\left(s\right),u\left(s\right))\mathrm{d}s\right\}$ where V is a class of admissible controls, we prove that v is the only viscosity solution of an Hamilton-Jacobi-Bellman equation of the form:$\lambda v(x,z)+H(x,z,{\nabla }_{x}v(x,z))+\langle D_{z}v(x,z),\dot{z}\rangle =0$ in the sense of the theory of viscosity solutions in infinite-dimensions of Crandall and Lions.

We show that solutions to some Hamilton-Jacobi Equations associated to the problem of optimal control of stochastic semilinear equations enjoy the hypercontractivity property.

We consider an optimal control problem for a system of the form $\dot{x}$ = f(x,u), with a running cost L. We prove an interior sphere property for the level sets of the corresponding value function V. From such a property we obtain a semiconcavity result for V, as well as perimeter estimates for the attainable sets of a symmetric control system.

En este artículo introducimos una nueva metodología para la generación de condiciones necesarias en problemas de optimización dinámicos.Denominamos a esta metodología la aproximación secuencial en contraposición a la aproximación puntual clásica y mostramos cómo obtener un principio de máximo puntual con este método.

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 setting.

The aim of the paper is to provide a linearization approach to the ${𝕃}^{\infty }$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 ${𝕃}^p$See PDF approach and the associated linear formulations. This seems to be the most appropriate tool for treating ${𝕃}^{\infty }$See PDF problems in continuous and lower semicontinuous...