18 research outputs found
Optimal boundary control with critical penalization for a PDE model of fluid-solid interactions
We study the finite-horizon optimal control problem with quadratic
functionals for an established fluid-structure interaction model. The coupled
PDE system under investigation comprises a parabolic (the fluid) and a
hyperbolic (the solid) dynamics; the coupling occurs at the interface between
the regions occupied by the fluid and the solid. We establish several trace
regularity results for the fluid component of the system, which are then
applied to show well-posedness of the Differential Riccati Equations arising in
the optimization problem. This yields the feedback synthesis of the unique
optimal control, under a very weak constraint on the observation operator; in
particular, the present analysis allows general functionals, such as the
integral of the natural energy of the physical system. Furthermore, this work
confirms that the theory developed in Acquistapace et al. [Adv. Differential
Equations, 2005] -- crucially utilized here -- encompasses widely differing PDE
problems, from thermoelastic systems to models of acoustic-structure and, now,
fluid-structure interactions.Comment: 22 pages, submitted; v2: misprints corrected, a remark added in
section
Exponential decay properties of a mathematical model for a certain fluid-structure interaction
In this work, we derive a result of exponential stability for a coupled
system of partial differential equations (PDEs) which governs a certain
fluid-structure interaction. In particular, a three-dimensional Stokes flow
interacts across a boundary interface with a two-dimensional mechanical plate
equation. In the case that the PDE plate component is rotational inertia-free,
one will have that solutions of this fluid-structure PDE system exhibit an
exponential rate of decay. By way of proving this decay, an estimate is
obtained for the resolvent of the associated semigroup generator, an estimate
which is uniform for frequency domain values along the imaginary axis.
Subsequently, we proceed to discuss relevant point control and boundary control
scenarios for this fluid-structure PDE model, with an ultimate view to optimal
control studies on both finite and infinite horizon. (Because of said
exponential stability result, optimal control of the PDE on time interval
becomes a reasonable problem for contemplation.)Comment: 15 pages, 1 figure; submitte
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
Applications of Mathematical Models in Engineering
The most influential research topic in the twenty-first century seems to be mathematics, as it generates innovation in a wide range of research fields. It supports all engineering fields, but also areas such as medicine, healthcare, business, etc. Therefore, the intention of this Special Issue is to deal with mathematical works related to engineering and multidisciplinary problems. Modern developments in theoretical and applied science have widely depended our knowledge of the derivatives and integrals of the fractional order appearing in engineering practices. Therefore, one goal of this Special Issue is to focus on recent achievements and future challenges in the theory and applications of fractional calculus in engineering sciences. The special issue included some original research articles that address significant issues and contribute towards the development of new concepts, methodologies, applications, trends and knowledge in mathematics. Potential topics include, but are not limited to, the following: Fractional mathematical models; Computational methods for the fractional PDEs in engineering; New mathematical approaches, innovations and challenges in biotechnologies and biomedicine; Applied mathematics; Engineering research based on advanced mathematical tools
Mathematical Optimization Techniques
The papers collected in this volume were presented at the Symposium on Mathematical Optimization Techniques held in the Santa Monica Civic Auditorium, Santa Monica, California, on October 18-20, 1960. The objective of the symposium was to bring together, for the purpose of mutual education, mathematicians, scientists, and engineers interested in modern optimization techniques. Some 250 persons attended. The techniques discussed included recent developments in linear, integer, convex, and dynamic programming as well as the variational processes surrounding optimal guidance, flight trajectories, statistical decisions, structural configurations, and adaptive control systems. The symposium was sponsored jointly by the University of California, with assistance from the National Science Foundation, the Office of Naval Research, the National Aeronautics and Space Administration, and The RAND Corporation, through Air Force Project RAND
Mathematical control theory and Finance
Control theory provides a large set of theoretical and computational tools with applications in a wide range of fields, running from ”pure” branches of mathematics, like geometry, to more applied areas where the objective is to find solutions to ”real life” problems, as is the case in robotics, control of industrial processes or finance. The ”high tech” character of modern business has increased the need for advanced methods. These rely heavily on mathematical techniques and seem indispensable for competitiveness of modern enterprises. It became essential for the financial analyst to possess a high level of mathematical skills. Conversely, the complex challenges posed by the problems and models relevant to finance have, for a long time, been an important source of new research topics for mathematicians. The use of techniques from stochastic optimal control constitutes a well established and important branch of mathematical finance. Up to now, other branches of control theory have found comparatively less application in financial problems. To some extent, deterministic and stochastic control theories developed as different branches of mathematics. However, there are many points of contact between them and in recent years the exchange of ideas between these fields has intensified. Some concepts from stochastic calculus (e.g., rough paths) have drawn the attention of the deterministic control theory community. Also, some ideas and tools usual in deterministic control (e.g., geometric, algebraic or functional-analytic methods) can be successfully applied to stochastic control. We strongly believe in the possibility of a fruitful collaboration between specialists of deterministic and stochastic control theory and specialists in finance, both from academic and business backgrounds. It is this kind of collaboration that the organizers of the Workshop on Mathematical Control Theory and Finance wished to foster. This volume collects a set of original papers based on plenary lectures and selected contributed talks presented at the Workshop. They cover a wide range of current research topics on the mathematics of control systems and applications to finance. They should appeal to all those who are interested in research at the junction of these three important fields as well as those who seek special topics within this scope.info:eu-repo/semantics/publishedVersio