Francesca Bucci

Singular estimates and uniform stability of coupled systems of hyperbolic/parabolic PDEs

1.

The regulator problem with indefinite quadratic cost for boundary control systems: the finite horizon case☆

In this paper we study the finite horizon non-standard LQ-problem for an abstract dynamics, which models a large class of hyperbolic-like partial differential equations. We provide necessary/sufficient conditions for finiteness of the value function corresponding to the control problem. Sharpness of sufficient conditions is shown by means of counterexamples. The specific features of the finite, in contrast to infinite, horizon case are illustrated.

Exponential Decay Rates for Structural Acoustic Model with an Overdamping on the Interface and Boundary Layer Dissipation

We study uniform stability properties of a strongly coupled system of Partial Differential Equations of hyperbolic/parabolic type, which arises from the analysis and control of acoustic models with structural damping on an interface. A challenging feature of the present model is the presence of additional strong boundary damping which is responsible for lack of uniform stability of the free system ( overdamping phenomenon). It has been shown recently that by applying full viscous damping in the interior of the domain and suitable static damping on the interface, then the corresponding feedback system is uniformly stable. In this article we prove that uniform decay rates of solutions to the system can be achieved even if viscous damping is active just in an arbitrary thin layer near the interface.

A Dirichlet boundary control problem for the strongly damped wave equation

A boundary control problem is considered for the strongly damped wave equation, and it is solved by dynamic programming arguments.

Finite-dimensional attractor for a composite system of wave/plate equations with localized damping

The long-term behaviour of solutions to a model for acoustic–structure interactions is addressed; the system consists of coupled semilinear wave (3D) and plate equations with nonlinear damping and critical sources. The questions of interest are the existence of a global attractor for the dynamics generated by this composite system as well as dimensionality and regularity of the attractor. A distinct and challenging feature of the problem is the geometrically restricted dissipation on the wave component of the system. It is shown that the existence of a global attractor of finite fractal dimension—established in a previous work by Bucci et al (2007 Commun. Pure Appl. Anal. 6 113–40) only in the presence of full-interior acoustic damping—holds even in the case of localized dissipation. This nontrivial generalization is inspired by, and consistent with, the recent advances in the study of wave equations with nonlinear localized damping.

Absolute Stability of Feedback Systems in Hilbert Spaces

The problem of absolute stability of a feedback loop of an abstract differential system in Hilbert spaces is considered. Applications of Popov’s type frequency domain criteria and of the Kalman-Yakubovich Lemma for the construction of Lyapunov functions are illustrated, in two situations pertaining to distributed systems. Finally, a new criterion for absolute stability of a class of parabolic systems with boundary feedback is presented.

The Value Function of the Singular Quadratic Regulator Problem with Distributed Control Action

We study the regularity properties of the value function of a quadratic regulator problem for a linear distributed parameter system with distributed control action. No definiteness assumption on the cost functional is assumed. We study the regularity in time of the value function and also the space regularity in the case of a holomorphic semigroup system.

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 (0, ∞) becomes a reasonable problem for contemplation.)

A Theory of the Infinite Horizon LQ-Problem for Composite Systems of PDEs with Boundary Control

We study the infinite horizon linear-quadratic (LQ) 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 (PDEs) 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 in 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 com...

Stability of Holomorphic Semigroup Systems under Nonlinear Boundary Perturbations

In this paper we re-study, by a different approach, the absolute stability of a class of holomorphic semigroup systems with nonlinear boundary feedback. A frequency criterion for equi-asymptotic stability in the large of the equilibrium of the corresponding closed loop has been recently derived in [3], by means of the Lyapunov function method. That stability result requires that the ‘infinite-sector’ nonlinearities satisfy a suitable growth condition. In this paper we show that the previous restriction is unnecessary and that furthermore the result still holds true under a weaker frequency domain condition.