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

Stationary Mean Field Games systems defined on networks

We consider a stationary Mean Field Games system defined on a network. In this framework, the transition conditions at the vertices play a crucial role: the ones here considered are based on the optimal control interpretation of the problem. We prove separately the well-posedness for each of the two equations composing the system. Finally, we prove existence and uniqueness of the solution of the Mean Field Games system

Dissipative solution to the Ericksen--Leslie system equipped with the Oseen--Frank energy

We analyze the Ericksen-Leslie system equipped with the Oseen?Frank energy in three space dimensions. The new concept of dissipative solutions is introduced. Recently, the author introduced the concept of measure-valued solutions to the considered system and showed global existence as well as weak-strong uniqueness of these generalized solutions. In this paper, we show that the expectation of the measure valued solution is a dissipative solution. The concept of a dissipative solution itself relies on an inequality instead of an equality, but is described by functions instead of parametrized measures. These solutions exist globally and fulfill the weak-strong uniqueness property. Additionally, we generalize the relative energy inequality to solutions fulfilling different nonhomogeneous Dirichlet boundary conditions and incorporate the influence of a temporarily constant electromagnetic field. Relying on this generalized energy inequality, we investigate the long-time behavior and show that all solutions converge for the large time limit to a certain steady state

Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs

This paper deals with linear-quadratic optimal control problems constrained by a parametric or stochastic elliptic or parabolic PDE. We address the (difficult) case that the state equation depends on a countable number of parameters i.e., on $\sigma_j$ with $j\in\N$, and that the PDE operator may depend non-affinely on the parameters. We consider tracking-type functionals and distributed as well as boundary controls. Building on recent results in [CDS1, CDS2], we show that the state and the control are analytic as functions depending on these parameters $\sigma_j$. We establish sparsity of generalized polynomial chaos (gpc) expansions of both, state and control, in terms of the stochastic coordinate sequence $\sigma = (\sigma_j)_{j\ge 1}$ of the random inputs, and prove convergence rates of best $N$-term truncations of these expansions. Such truncations are the key for subsequent computations since they do {\em not} assume that the stochastic input data has a finite expansion. In the follow-up paper [KS2], we explain two methods how such best $N$-term truncations can practically be computed, by greedy-type algorithms as in [SG, Gi1], or by multilevel Monte-Carlo methods as in [KSS]. The sparsity result allows in conjunction with adaptive wavelet Galerkin schemes for sparse, adaptive tensor discretizations of control problems constrained by linear elliptic and parabolic PDEs developed in [DK, GK, K], see [KS2]

Inverse Optimization with Noisy Data

Inverse optimization refers to the inference of unknown parameters of an optimization problem based on knowledge of its optimal solutions. This paper considers inverse optimization in the setting where measurements of the optimal solutions of a convex optimization problem are corrupted by noise. We first provide a formulation for inverse optimization and prove it to be NP-hard. In contrast to existing methods, we show that the parameter estimates produced by our formulation are statistically consistent. Our approach involves combining a new duality-based reformulation for bilevel programs with a regularization scheme that smooths discontinuities in the formulation. Using epi-convergence theory, we show the regularization parameter can be adjusted to approximate the original inverse optimization problem to arbitrary accuracy, which we use to prove our consistency results. Next, we propose two solution algorithms based on our duality-based formulation. The first is an enumeration algorithm that is applicable to settings where the dimensionality of the parameter space is modest, and the second is a semiparametric approach that combines nonparametric statistics with a modified version of our formulation. These numerical algorithms are shown to maintain the statistical consistency of the underlying formulation. Lastly, using both synthetic and real data, we demonstrate that our approach performs competitively when compared with existing heuristics

A contact covariant approach to optimal control with applications to sub-Riemannian geometry

We discuss contact geometry naturally related with optimal control problems (and Pontryagin Maximum Principle). We explore and expand the observations of [Ohsawa, 2015], providing simple and elegant characterizations of normal and abnormal sub-Riemannian extremals.Comment: A small correction in the statement and proof of Thm 6.15. Watch our publication: https://youtu.be/V04N9X3NxYA and https://youtu.be/jghdRK2IaU

Dissipative solution to the Ericksen-Leslie system equipped with the Oseen-Frank energy

We analyze the EricksenLeslie system equipped with the OseenFrank energy in three space dimensions. The new concept of dissipative solutions is introduced. Recently, the author introduced the concept of measure-valued solutions to the considered system and showed global existence as well as weak-strong uniqueness of these generalized solutions. In this paper, we show that the expectation of the measure valued solution is a dissipative solution. The concept of a dissipative solution itself relies on an inequality instead of an equality, but is described by functions instead of parametrized measures. These solutions exist globally and fulfill the weak-strong uniqueness property. Additionally, we generalize the relative energy inequality to solutions fulfilling different nonhomogeneous Dirichlet boundary conditions and incorporate the influence of a temporarily constant electromagnetic field. Relying on this generalized energy inequality, we investigate the long-time behavior and show that all solutions converge for the large time limit to a certain steady state