1,927 research outputs found
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
Max-plus fundamental solution semigroups for a class of difference Riccati equations
Recently, a max-plus dual space fundamental solution semigroup for a class of
difference Riccati equation (DRE) has been developed. This fundamental solution
semigroup is represented in terms of the kernel of a specific max-plus linear
operator that plays the role of the dynamic programming evolution operator in a
max-plus dual space. In order to fully understand connections between this dual
space fundamental solution semigroup and evolution of the value function of the
underlying optimal control problem, a new max-plus primal space fundamental
solution semigroup for the same class of difference Riccati equations is
presented. Connections and commutation results between this new primal space
fundamental solution semigroup and the recently developed dual space
fundamental solution semigroup are established.Comment: 17 pages, 3 figure
Feedback control of the acoustic pressure in ultrasonic wave propagation
Classical models for the propagation of ultrasound waves are the Westervelt
equation, the Kuznetsov and the Khokhlov-Zabolotskaya-Kuznetsov equations. The
Jordan-Moore-Gibson-Thompson equation is a prominent example of a Partial
Differential Equation (PDE) model which describes the acoustic velocity
potential in ultrasound wave propagation, where the paradox of infinite speed
of propagation of thermal signals is eliminated; the use of the constitutive
Cattaneo law for the heat flux, in place of the Fourier law, accounts for its
being of third order in time. Aiming at the understanding of the fully
quasilinear PDE, a great deal of attention has been recently devoted to its
linearization -- referred to in the literature as the Moore-Gibson-Thompson
equation -- whose mathematical analysis is also of independent interest, posing
already several questions and challenges. In this work we consider and solve a
quadratic control problem associated with the linear equation, formulated
consistently with the goal of keeping the acoustic pressure close to a
reference pressure during ultrasound excitation, as required in medical and
industrial applications. While optimal control problems with smooth controls
have been considered in the recent literature, we aim at relying on controls
which are just in time; this leads to a singular control problem and to
non-standard Riccati equations. In spite of the unfavourable combination of the
semigroup describing the free dynamics that is not analytic, with the
challenging pattern displayed by the dynamics subject to boundary control, a
feedback synthesis of the optimal control as well as well-posedness of operator
Riccati equations are established.Comment: 39 pages; submitte
Efficient Solution of Large-Scale Algebraic Riccati Equations Associated with Index-2 DAEs via the Inexact Low-Rank Newton-ADI Method
This paper extends the algorithm of Benner, Heinkenschloss, Saak, and
Weichelt: An inexact low-rank Newton-ADI method for large-scale algebraic
Riccati equations, Applied Numerical Mathematics Vol.~108 (2016), pp.~125--142,
doi:10.1016/j.apnum.2016.05.006 to Riccati equations associated with Hessenberg
index-2 Differential Algebratic Equation (DAE) systems. Such DAE systems arise,
e.g., from semi-discretized, linearized (around steady state) Navier-Stokes
equations. The solution of the associated Riccati equation is important, e.g.,
to compute feedback laws that stabilize the Navier-Stokes equations. Challenges
in the numerical solution of the Riccati equation arise from the large-scale of
the underlying systems and the algebraic constraint in the DAE system. These
challenges are met by a careful extension of the inexact low-rank Newton-ADI
method to the case of DAE systems. A main ingredient in the extension to the
DAE case is the projection onto the manifold described by the algebraic
constraints. In the algorithm, the equations are never explicitly projected,
but the projection is only applied as needed. Numerical experience indicates
that the algorithmic choices for the control of inexactness and line-search can
help avoid subproblems with matrices that are only marginally stable. The
performance of the algorithm is illustrated on a large-scale Riccati equation
associated with the stabilization of Navier-Stokes flow around a cylinder.Comment: 21 pages, 2 figures, 4 table
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