16,699 research outputs found
Maximum Principle for Linear-Convex Boundary Control Problems applied to Optimal Investment with Vintage Capital
The paper concerns the study of the Pontryagin Maximum Principle for an
infinite dimensional and infinite horizon boundary control problem for linear
partial differential equations. The optimal control model has already been
studied both in finite and infinite horizon with Dynamic Programming methods in
a series of papers by the same author, or by Faggian and Gozzi. Necessary and
sufficient optimality conditions for open loop controls are established.
Moreover the co-state variable is shown to coincide with the spatial gradient
of the value function evaluated along the trajectory of the system, creating a
parallel between Maximum Principle and Dynamic Programming. The abstract model
applies, as recalled in one of the first sections, to optimal investment with
vintage capital
Equilibrium points for Optimal Investment with Vintage Capital
The paper concerns the study of equilibrium points, namely the stationary
solutions to the closed loop equation, of an infinite dimensional and infinite
horizon boundary control problem for linear partial differential equations.
Sufficient conditions for existence of equilibrium points in the general case
are given and later applied to the economic problem of optimal investment with
vintage capital. Explicit computation of equilibria for the economic problem in
some relevant examples is also provided. Indeed the challenging issue here is
showing that a theoretical machinery, such as optimal control in infinite
dimension, may be effectively used to compute solutions explicitly and easily,
and that the same computation may be straightforwardly repeated in examples
yielding the same abstract structure. No stability result is instead provided:
the work here contained has to be considered as a first step in the direction
of studying the behavior of optimal controls and trajectories in the long run
Verification Theorems for Stochastic Optimal Control Problems via a Time Dependent Fukushima - Dirichlet Decomposition
This paper is devoted to present a method of proving verification theorems
for stochastic optimal control of finite dimensional diffusion processes
without control in the diffusion term. The value function is assumed to be
continuous in time and once differentiable in the space variable ()
instead of once differentiable in time and twice in space (), like in
the classical results. The results are obtained using a time dependent
Fukushima - Dirichlet decomposition proved in a companion paper by the same
authors using stochastic calculus via regularization. Applications, examples
and comparison with other similar results are also given.Comment: 34 pages. To appear: Stochastic Processes and Their Application
A theory of the infinite horizon LQ-problem for composite systems of PDEs with boundary control
We study the infinite horizon Linear-Quadratic problem and the associated
algebraic Riccati equations for systems with unbounded control actions. The
operator-theoretic context is motivated by composite systems of Partial
Differential Equations (PDE) with boundary or point control. Specific focus is
placed on systems of coupled hyperbolic/parabolic PDE with an overall
`predominant' hyperbolic character, such as, e.g., some models for
thermoelastic or fluid-structure interactions. While unbounded control actions
lead to Riccati equations with unbounded (operator) coefficients, unlike the
parabolic case solvability of these equations becomes a major issue, owing to
the lack of sufficient regularity of the solutions to the composite dynamics.
In the present case, even the more general theory appealing to estimates of the
singularity displayed by the kernel which occurs in the integral representation
of the solution to the control system fails. A novel framework which embodies
possible hyperbolic components of the dynamics has been introduced by the
authors in 2005, and a full theory of the LQ-problem on a finite time horizon
has been developed. The present paper provides the infinite time horizon
theory, culminating in well-posedness of the corresponding (algebraic) Riccati
equations. New technical challenges are encountered and new tools are needed,
especially in order to pinpoint the differentiability of the optimal solution.
The theory is illustrated by means of a boundary control problem arising in
thermoelasticity.Comment: 50 pages, submitte
Utility maximization with current utility on the wealth: regularity of solutions to the HJB equation
We consider a utility maximization problem for an investment-consumption
portfolio when the current utility depends also on the wealth process. Such
kind of problems arise, e.g., in portfolio optimization with random horizon or
with random trading times. To overcome the difficulties of the problem we use
the dual approach. We define a dual problem and treat it by means of dynamic
programming, showing that the viscosity solutions of the associated
Hamilton-Jacobi-Bellman equation belong to a suitable class of smooth
functions. This allows to define a smooth solution of the primal
Hamilton-Jacobi-Bellman equation, proving that this solution is indeed unique
in a suitable class and coincides with the value function of the primal
problem. Some financial applications of the results are provided
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