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

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 $L^2$ 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

Infinite Horizon and Ergodic Optimal Quadratic Control for an Affine Equation with Stochastic Coefficients

We study quadratic optimal stochastic control problems with control dependent noise state equation perturbed by an affine term and with stochastic coefficients. Both infinite horizon case and ergodic case are treated. To this purpose we introduce a Backward Stochastic Riccati Equation and a dual backward stochastic equation, both considered in the whole time line. Besides some stabilizability conditions we prove existence of a solution for the two previous equations defined as limit of suitable finite horizon approximating problems. This allows to perform the synthesis of the optimal control

On the Approximation of Operator-Valued Riccati Equations in Hilbert Spaces

In this work, we present an abstract theory for the approximation of operator-valued Riccati equations posed on Hilbert spaces. It is demonstrated here, under the assumption of compactness in the coefficient operators, that the error of the approximate solution to the operator-valued Riccati equation is bounded above by the approximation error of the governing semigroup. One significant outcome of this result is the correct prediction of optimal convergence for finite element approximations of the operator-valued Riccati equations for when the governing semigroup involves parabolic, as well as hyperbolic processes. We derive the abstract theory for the time-dependent and time-independent operator-valued Riccati equations in the first part of this work. In the second part, we prove optimal convergence rates for the finite element approximation of the functional gain associated with model one-dimensional weakly damped wave and thermal LQR control systems. These theoretical claims are then corroborated with computational evidence.Comment: Initial Releas

Estimation for Linear and Semi-linear Infinite-dimensional Systems

Estimating the state of a system that is not fully known or that is exposed to noise has been an intensely studied problem in recent mathematical history. Such systems are often modelled by either ordinary differential equations, which evolve in finite-dimensional state-spaces, or partial differential equations, the state-space of which is infinite-dimensional. The Kalman filter is a minimal mean squared error estimator for linear finite-dimensional and linear infinite-dimensional systems disturbed by Wiener processes, which are stochastic processes representing the noise. For nonlinear finite-dimensional systems the extended Kalman filter is a widely used extension thereof which relies on linearization of the system. In all cases the Kalman filter consists of a differential or integral equation coupled with a Riccati equation, which is an equation that determines the optimal estimator gain. This thesis proposes an estimator for semi-linear infinite-dimensional systems. It is shown that under some conditions such a system can also be coupled with a Riccati equation. To motivate this result, the Kalman filter for finite-dimensional and infinite-dimensional systems is reviewed, as well as the corresponding theory for both stochastic processes and infinite-dimensional systems. Important results concerning the infinite-dimensional Riccati equation are outlined and existence of solutions for a class of semi-linear infinite-dimensional systems is established. Finally the well-posedness of the coupling between a semi-linear infinite-dimensional system with a Riccati equation is proven using a fixed point argument

A note on the regularity of solutions of infinite dimensional Riccati equations

This note is concerned with the regularity of solutions of algebraic Riccati equations arising from infinite dimensional LQR and LQG control problems. We show that distributed parameter systems described by certain parabolic partial differential equations often have a special structure that smoothes solutions of the corresponding Riccati equation. This analysis is motivated by the need to find specific representations for Riccati operators that can be used in the development of computational schemes for problems where the input and output operators are not Hilbert-Schmidt. This situation occurs in many boundary control problems and in certain distributed control problems associated with optimal sensor/actuator placement