8,583 research outputs found
Chandrasekhar equations for infinite dimensional systems. Part 2: Unbounded input and output case
A set of equations known as Chandrasekhar equations arising in the linear quadratic optimal control problem is considered. In this paper, we consider the linear time-invariant system defined in Hilbert spaces involving unbounded input and output operators. For a general class of such systems, the Chandrasekhar equations are derived and the existence, uniqueness, and regularity of the results of their solutions established
Chandrasekhar equations for infinite dimensional systems. II. Unbounded input and output case
AbstractA set of equations known as Chandrasekhar equations arising in the linear quadratic optimal control problem is considered. In this paper, we consider the linear time-invariant systems defined in Hilbert spaces involving unbounded input and output operators. For a general class of such systems, we derive the Chandrasekhar equations and establish the existence, uniqueness, and regularity results of their solutions
Mild solutions of semilinear elliptic equations in Hilbert spaces
This paper extends the theory of regular solutions ( in a suitable
sense) for a class of semilinear elliptic equations in Hilbert spaces. The
notion of regularity is based on the concept of -derivative, which is
introduced and discussed. A result of existence and uniqueness of solutions is
stated and proved under the assumption that the transition semigroup associated
to the linear part of the equation has a smoothing property, that is, it maps
continuous functions into -differentiable ones. The validity of this
smoothing assumption is fully discussed for the case of the Ornstein-Uhlenbeck
transition semigroup and for the case of invertible diffusion coefficient
covering cases not previously addressed by the literature. It is shown that the
results apply to Hamilton-Jacobi-Bellman (HJB) equations associated to infinite
horizon optimal stochastic control problems in infinite dimension and that, in
particular, they cover examples of optimal boundary control of the heat
equation that were not treatable with the approaches developed in the
literature up to now
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
Stochastic Optimal Control with Delay in the Control: solution through partial smoothing
Stochastic optimal control problems governed by delay equations with delay in
the control are usually more difficult to study than the the ones when the
delay appears only in the state. This is particularly true when we look at the
associated Hamilton-Jacobi-Bellman (HJB) equation. Indeed, even in the
simplified setting (introduced first by Vinter and Kwong for the deterministic
case) the HJB equation is an infinite dimensional second order semilinear
Partial Differential Equation (PDE) that does not satisfy the so-called
"structure condition" which substantially means that "the noise enters the
system with the control." The absence of such condition, together with the lack
of smoothing properties which is a common feature of problems with delay,
prevents the use of the known techniques (based on Backward Stochastic
Differential Equations (BSDEs) or on the smoothing properties of the linear
part) to prove the existence of regular solutions of this HJB equation and so
no results on this direction have been proved till now.
In this paper we provide a result on existence of regular solutions of such
kind of HJB equations and we use it to solve completely the corresponding
control problem finding optimal feedback controls also in the more difficult
case of pointwise delay. The main tool used is a partial smoothing property
that we prove for the transition semigroup associated to the uncontrolled
problem. Such results holds for a specific class of equations and data which
arises naturally in many applied problems
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
Stochastic Maximum Principle for Optimal Control ofPartial Differential Equations Driven by White Noise
We prove a stochastic maximum principle ofPontryagin's type for the optimal
control of a stochastic partial differential equationdriven by white noise in
the case when the set of control actions is convex. Particular attention is
paid to well-posedness of the adjoint backward stochastic differential equation
and the regularity properties of its solution with values in
infinite-dimensional spaces
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