1,462 research outputs found
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
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
Backstepping PDE Design: A Convex Optimization Approach
Abstract\u2014Backstepping design for boundary linear PDE is
formulated as a convex optimization problem. Some classes of
parabolic PDEs and a first-order hyperbolic PDE are studied,
with particular attention to non-strict feedback structures. Based
on the compactness of the Volterra and Fredholm-type operators
involved, their Kernels are approximated via polynomial
functions. The resulting Kernel-PDEs are optimized using Sumof-
Squares (SOS) decomposition and solved via semidefinite
programming, with sufficient precision to guarantee the stability
of the system in the L2-norm. This formulation allows optimizing
extra degrees of freedom where the Kernel-PDEs are included
as constraints. Uniqueness and invertibility of the Fredholm-type
transformation are proved for polynomial Kernels in the space
of continuous functions. The effectiveness and limitations of the
approach proposed are illustrated by numerical solutions of some
Kernel-PDEs
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.Linear convex control, Boundary control, HamiltonāJacobiāBellman equations, Optimal investment problems, Vintage capital
Maximum Principle for Boundary Control Problems Arising in 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 et al. [26, 27, 28, 29, 30]. 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.Linear convex control, Boundary control, HamiltonāJacobiāBellman equations, Optimal investment problems, Vintage capital
Optimal investment models with vintage capital: Dynamic Programming approach
The Dynamic Programming approach for a family of optimal investment models with vintage capital is here developed. The problem falls into the class of infinite horizon optimal control problems of PDE's with age structure that have been studied in various papers (see e.g. [11, 12], [30, 32]) either in cases when explicit solutions can be found or using Maximum Principle techniques. The problem is rephrased into an infinite dimensional setting, it is proven that the value function is the unique regular solution of the associated stationary Hamilton-Jacobi-Bellman equation, and existence and uniqueness of optimal feedback controls is derived. It is then shown that the optimal path is the solution to the closed loop equation. Similar results were proven in the case of finite horizon in [26][27]. The case of infinite horizon is more challenging as a mathematical problem, and indeed more interesting from the point of view of optimal investment models with vintage capital, where what mainly matters is the behavior of optimal trajectories and controls in the long run. The study of infinite horizon is performed through a nontrivial limiting procedure from the corresponding finite horizon problemsOptimal investment, vintage capital, age-structured systems, optimal control, dynamic programming, Hamilton-Jacobi-Bellman equations, linear convex control, boundary control
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