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

On the Navier-Stokes equations with rotating effect and prescribed outflow velocity

We consider the equations of Navier-Stokes modeling viscous fluid flow past a moving or rotating obstacle in $\mathbb{R}^d$ subject to a prescribed velocity condition at infinity. In contrast to previously known results, where the prescribed velocity vector is assumed to be parallel to the axis of rotation, in this paper we are interested in a general outflow velocity. In order to use $L^p$-techniques we introduce a new coordinate system, in which we obtain a non-autonomous partial differential equation with an unbounded drift term. We prove that the linearized problem in $\mathbb{R}^d$ is solved by an evolution system on $L^p_{\sigma}(\mathbb{R}^d)$ for $1<p<\infty$. For this we use results about time-dependent Ornstein-Uhlenbeck operators. Finally, we prove, for $p\geq d$ and initial data $u_0\in L^p_{\sigma}(\mathbb{R}^d)$, the existence of a unique mild solution to the full Navier-Stokes system.Comment: 18 pages, to appear in J. Math. Fluid Mech. (published online first

Classical Solutions of Nonautonomous Riccati Equations Arising in Parabolic Boundary Control Problems, II

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Mathematical Phase Model of Neural Populations Interaction in Modulation of REM/NREM Sleep

Aim of the present study is to compare the synchronization of the classical Kuramoto system and the reaction - diffusion space time Landau - Ginzburg model, in order to describe the alternation of REM (rapid eye movement) and NREM (non-rapid eye movement) sleep across the night. These types of sleep are considered as produced by the cyclic oscillation of two neuronal populations that, alternatively, promote and inhibit the REM sleep. Even if experimental data will be necessary, a possible interpretation of the results has been proposed