Singular estimates and uniform stability of coupled systems of hyperbolic/parabolic PDEs (original) (raw)
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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.
Journal of Optimization Theory and Applications, 2008
We establish solvability of Riccati equations and optimal feedback synthesis in the context of Bolza control problem for a special class of control systems referred to in the literature as control systems with singular estimate. Boundary/point control problems governed by analytic semigroups constitute a very special subcategory of this class which was motivated by and encompasses many PDE control systems with both boundary and point controls that involve interactions of different types of dynamics (parabolic and hyperbolic) on an interface. We also discuss two examples from thermoelasticity and structure acoustics.
Systems & Control Letters, 2009
We consider a Bolza boundary control problem involving a fluid-structure interaction model. The aim of this paper is to develop an optimal feedback control synthesis based on Riccati theory. The model considered consists of a linearized Navier-Stokes equation coupled on the interface with a dynamic wave equation. The model incorporates convective terms resulting from the linearization of the Navier-Stokes equation around equilibrium. The existence of the optimal control and its feedback characterization via a solution to a Riccati equation is established. The main mathematical difficulty of the problem is caused by unbounded action of control forces which, in turn, result in Riccati equations with unbounded coefficients and in singular behavior of the gain operator. This class of problems has been recently studied via the so-called Singular Estimate Control Systems (SECS) theory, which is based on the validity of the Singular Estimate SE) [G. Avalos, Differential Riccati equations for the active control of a problem in structural acoustic, J. Optim. Theory, Appl. 91 (1996) 695-728; I. Lasiecka, Mathematical Control Theory of Coupled PDE's, in: NSF-CMBS Lecture Notes, SIAM, 2002. with Unbounded Controls; I. Lasiecka, A. Tuffaha, Riccati Equations for the Bolza Problem arising in boundary/point control problems governed by c0 semigroups satisfying a singular estimate, J. Optim. Theory Appl. 136 (2008) 229-246]
Decay Rates of Interactive Hyperbolic-Parabolic PDE Models with Thermal Effects on the Interface
Applied Mathematics and Optimization, 2000
We consider coupled PDE systems comprising of a hyperbolic and a parabolic-like equation with an interface on a portion of the boundary. These models are motivated by structural acoustic problems. A specific prototype consists of a wave equation defined on a three-dimensional bounded domain coupled with a thermoelastic plate equation defined on 0-a flat surface of the boundary ∂. Thus, the coupling between the wave and the plate takes place on the interface 0. The main issue studied here is that of uniform stability of the overall interactive model. Since the original (uncontrolled) model is only strongly stable, but not uniformly stable, the question becomes: what is the "minimal amount" of dissipation necessary to obtain uniform decay rates for the energy of the overall system? Our main result states that boundary nonlinear dissipation placed only on a suitable portion of the part of the boundary which is complementary to 0 , suffices for the stabilization of the entire structure. This result is new with respect to the literature on several accounts: (i) thermoelasticity is accounted for in the plate model; (ii) the plate model does not account for any type of mechanical damping, including the structural damping most often considered in the literature; (iii) there is no mechanical damping placed on the interface 0 ; (iv) the boundary damping is nonlinear without a prescribed growth rate at the origin; (v) the undamped portions of the boundary ∂ are subject to Neumann (rather than Dirichlet) boundary conditions, which is a recognized difficulty in the context of stabilization of wave equations, due to the fact that the strong Lopatinski condition does not hold.
Recent advences in averaged controllability of hyperbolic PDEs
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The aim of this memory is to study the averaged controllability for parameter-dependent systems. We discuss the notion of averaged control which has been introduced recently by E.Zauzua [12]. We apply the method HUM where it is based on uniqueness criteria (the direct and inverse inequalities) to some hyperbolic equations with an unknown parameter. First, we consider the problem of controllability for linear finite and infinite dimensional systems and we give the averaged rank condition for averaged controllability. Secondly, we study the averaged controllability of wave equation depending on a parameter when the control is applied on the boundary, by using the Hilbert Uniqueness Method (the method HUM) which has been introduced by J.L. Lions [8], where this method allows us to design the avraged control of our parameter-dependent wave equation. Finally, we study with the same argument of the HUM method the problem of vibrating plate equation depending on a parameter when the control is applied on the boundary also.