This paper is concerned with the state-constrained optimal control of the threedimensional thermistor problem, a fully quasilinear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness and continuity for the state system are derived by employing maximal parabo...

In this paper we study the optimal control problem of the heat equation by a distributed control over a subset of the domain, in the presence of a state constraint. The latter is integral over the space and has to be satisfied at each time. Using for the first time the technique of alternative optimality systems in the context of optimal control of partial differential equations, we show that b...

We study a class of Markovian optimal stochastic control problems in which the controlled process Z is constrained to satisfy an a.s. constraint Z(T ) ∈ G ⊂ R P − a.s. at some final time T > 0. When the set is of the form G := {(x, y) ∈ R × R : g(x, y) ≥ 0}, with g non-decreasing in y, we provide a Hamilton-Jacobi-Bellman characterization of the associated value function. It gives rise to a sta...

In the above, ŷ = ŷ(t) and ŷT are the target trajectory and target state, and the functions u and y = y(t,u) are called the control and the state, respectively. (For the purpose of analysis, we will use an appropriate change of variables to subsume any mass matrices that appear into the matrices A, B, C and D.) We will focus on the case where there are no control or state constraints, and where...

We consider the infinite-horizon optimal portfolio liquidation problem for a von Neumann-Morgenstern investor in the liquidity model of Almgren (2003). Using a stochastic control approach, we characterize the value function and the optimal strategy as classical solutions of nonlinear parabolic partial differential equations. We furthermore analyze the sensitivities of the value function and the...

In this paper, the minimum time problem for differential systems of parabolic type with distributed control and control state constraints are considered. The minimum time problem is replaced by an equivalent one with fixed time and the necessary optimality conditions of time-optimal control are obtained by using the generalized Dubovitskii-Milyutin Theorem (see [1]).

We consider an Ito stochastic differential equation with delay, driven by brownian motion, whose solution, by an appropriate reformulation, defines a Markov process X with values in a space of continuous functions C, with generator L. We then consider a backward stochastic differential equation depending on X , with unknown processes (Y, Z), and we study properties of the resulting system, in p...

Viscosity solutions methods are used to pass to the limit in some penalization problems for rst order and second order, degenerate parabolic, Hamilton-Jacobi-Bellman equations. This characterizes the limit of the value functions of singularly perturbed optimal control problems for nonlinear deterministic systems and controlled degenerate diiusions, respectively. The results cover also cases whe...

This paper deals with optimal control problems of semilinear parabolic equations with pointwise state constraints and coupled integral state-control constraints. We obtain necessary optimality conditions in the form of a Pontryagin’s minimum principle for local solutions in the sense of Lp, p ≤ +∞.

This paper deals with state-constrained optimal control problems governed by semilinear parabolic equations. We establish a minimum principle of Pontryagin's type. To deal with the state constraints, we introduce a penalty problem by using Ekeland's principle. The key tool for the proof is the use of a special kind of spike perturbations distributed in the domain where the controls are de ned. ...