### A Dirichlet problem for an H-system with variable H.

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A singular stochastic control problem in n dimensions with timedependent coefficients on a finite time horizon is considered. We show that the value function for this problem is a generalized solution of the corresponding HJB equation with locally bounded second derivatives with respect to the space variables and the first derivative with respect to time. Moreover, we prove that an optimal control exists and is unique

Rate-independent evolution for material models with nonconvex elastic energies is studied without any spatial regularization of the inner variable; due to lack of convexity, the model is developed in the framework of Young measures. An existence result for the quasistatic evolution is obtained in terms of compatible systems of Young measures. We also show as this result can be equivalently reformulated with probabilistic language and leads to the description of the quasistatic evolution in terms...

A linear-quadratic control problem with an infinite time horizon for some infinite dimensional controlled stochastic differential equations driven by a fractional Brownian motion is formulated and solved. The feedback form of the optimal control and the optimal cost are given explicitly. The optimal control is the sum of the well known linear feedback control for the associated infinite dimensional deterministic linear-quadratic control problem and a suitable prediction of the adjoint optimal system...

In this paper, we consider probability measures μ and ν on a d-dimensional sphere in ${\mathbf{R}}^{d+1},d\ge 1,$ and cost functions of the form $c(\mathbf{x},\mathbf{y})=l\left(\frac{{|\mathbf{x}-\mathbf{y}|}^{2}}{2}\right)$ that generalize those arising in geometric optics where $l\left(t\right)=-logt.$ We prove that if μ and ν vanish on $(d-1)$-rectifiable sets, if |l'(t)|>0,${lim}_{t\to {0}^{+}}l\left(t\right)=+\infty ,$ and $g\left(t\right):=t(2-t){\left({l}^{\text{'}}\left(t\right)\right)}^{2}$ is monotone then there exists a unique optimal map To that transports μ onto $\nu ,$ where optimality is measured against c. Furthermore, ${inf}_{\mathbf{x}}|{T}_{o}\mathbf{x}-\mathbf{x}|>0.$ Our approach is based on direct variational arguments. In the special case when $l\left(t\right)=-logt,$ existence of optimal maps...

Optimal nonanticipating controls are shown to exist in nonautonomous piecewise deterministic control problems with hard terminal restrictions. The assumptions needed are completely analogous to those needed to obtain optimal controls in deterministic control problems. The proof is based on well-known results on existence of deterministic optimal controls.

In this paper, we consider a class of infinite dimensional stochastic impulsive evolution inclusions driven by vector measures. We use stochastic vector measures as controls adapted to an increasing family of complete sigma algebras and prove the existence of optimal controls.

We consider the dynamic control problem of attaining a target position at a finite time T, while minimizing a linear-quadratic cost functional depending on the position and speed. We assume that the coefficients of the linear-quadratic cost functional are stochastic processes adapted to a Brownian filtration. We provide a probabilistic solution in terms of two coupled backward stochastic differential equations possessing a singularity at the terminal time T. We verify optimality of the candidate...

In this paper, we study optimal transportation problems for multifractal random measures. Since these measures are much less regular than optimal transportation theory requires, we introduce a new notion of transportation which is intuitively some kind of multistep transportation. Applications are given for construction of multifractal random changes of times and to the existence of random metrics, the volume forms of which coincide with the multifractal random measures.