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This book presents cutting-edge contributions in the areas of control theory and partial differential equations. Over the decades, control theory has had deep and fruitful interactions with the theory of partial differential equations (PDEs). Well-known examples are the study of the generalized solutions of Hamilton-Jacobi-Bellman equations arising in deterministic and stochastic optimal control and the development of modern analytical tools to study the controllability of infinite dimensional systems governed by PDEs. In the present volume, leading experts provide an up-to-date overview of the connections between these two vast fields of mathematics. Topics addressed include regularity of the value function associated to finite dimensional control systems, controllability and observability for PDEs, and asymptotic analysis of multiagent systems. The book will be of interest for both researchers and graduate students working in these areas.



Some Remarks on the Dirichlet Problem for the Degenerate Eikonal Equation

In a bounded domain, we consider the viscosity solution of the homogeneous Dirichlet problem for the degenerate eikonal equation. We provide some sufficient conditions for the (local) Lipschitz regularity of such a function.
Paolo Albano

Lipschitz Continuity of the Value Function for the Infinite Horizon Optimal Control Problem Under State Constraints

This paper investigates sufficient conditions for Lipschitz regularity of the value function for an infinite horizon optimal control problem subject to state constraints. We focus on problems with a cost functional that includes a discount rate factor and allow time dependent dynamics and Lagrangian. Furthermore, our state constraints may be unbounded and with nonsmooth boundary. The key technical result used in our proof is an estimate on the distance of a given trajectory from the set of all its viable (feasible) trajectories (provided the discount rate is sufficiently large). These distance estimates are derived under a uniform inward pointing condition on the state constraint and imply, in particular, that feasible trajectories depend on initial states in a Lipschitz way with an exponentially increasing in time Lipschitz constant. As a corollary, we show that the value function of the original problem coincides with the value function of the relaxed infinite horizon problem.
Vincenzo Basco, Hélène Frankowska

Herglotz’ Generalized Variational Principle and Contact Type Hamilton-Jacobi Equations

We develop an approach for the analysis of fundamental solutions to Hamilton-Jacobi equations of contact type based on a generalized variational principle proposed by Gustav Herglotz. We also give a quantitative Lipschitz estimate on the associated minimizers.
Piermarco Cannarsa, Wei Cheng, Kaizhi Wang, Jun Yan

Observability Inequalities for Transport Equations through Carleman Estimates

We consider the transport equation \(\partial _t u(x,t) + H(t)\cdot \nabla u(x,t) = 0\) in \(\varOmega \times (0,T),\) where \(T>0\) and \(\varOmega \subset \mathbb R^d \) is a bounded domain with smooth boundary \(\partial \varOmega \). First, we prove a Carleman estimate for solutions of finite energy with piecewise continuous weight functions. Then, under a further condition which guarantees that the orbits of H intersect \(\partial \varOmega \), we prove an energy estimate which in turn yields an observability inequality. Our results are motivated by applications to inverse problems.
Piermarco Cannarsa, Giuseppe Floridia, Masahiro Yamamoto

On the Weak Maximum Principle for Degenerate Elliptic Operators

This paper provides an overview of some more or less recent results concerning the validity of the weak Maximum Principle for fully nonlinear degenerate elliptic equations. Special attention is devoted to the presentation of sufficient conditions relating the directions of degeneracy and the geometry of the possibly unbounded domain.
Italo Capuzzo Dolcetta

On the Convergence of Open Loop Nash Equilibria in Mean Field Games with a Local Coupling

The paper studies the convergence, as N tends to infinity, of a system of N weakly coupled Hamilton–Jacobi equations (the open loop Nash system) when the coupling between the players becomes increasingly singular. The limit equation is a mean field game system with local coupling.
Pierre Cardaliaguet

Remarks on the Control of Family of b–Equations

In this paper, we deal with the control of the viscous b–equation in a one-dimensional bounded domain. For \(b=2\) and \(b=0\), we get in particular the Camassa–Holm and the Burgers-\(\alpha \) equations, respectively. We prove that, for any real number b, we can steer the solution to the equation to zero at any given time, using a distributed control, locally supported in space, when the initial data are sufficiently small. Also, for \(b=0\), we prove the global null controllability for large time.
Enrique Fernández-Cara, Diego A. Souza

1-d Wave Equations Coupled via Viscoelastic Springs and Masses: Boundary Controllability of a Quasilinear and Exponential Stabilizability of a Linear Model

We consider the out-of-the-plane displacements of nonlinear elastic strings which are coupled through point masses attached to the ends and viscoelastic springs. We provide the modeling, the well-posedness in the sense of classical semi-global \(C^2\)-solutions together with some extra regularity at the masses and then prove exact boundary controllability and velocity-feedback stabilizability, where controls act on both sides of the mass-spring-coupling.
Günter Leugering, Tatsien Li, Yue Wang

A Semilinear Integro-Differential Equation: Global Existence and Hidden Regularity

Here we show a hidden regularity result for nonlinear wave equations with an integral term of convolution type and Dirichlet boundary conditions. Under general assumptions on the nonlinear term and on the integral kernel we are able to state results about global existence of strong and mild solutions without any further smallness on the initial data. Then we define the trace of the normal derivative of the solution showing a regularity result. In such a way we extend to integrodifferential equations with nonlinear term well-known results available in the literature for linear wave equations with memory.
Paola Loreti, Daniela Sforza

Lyapunov’s Theorem via Baire Category

Lyapunov’s theorem is a classical result in convex analysis, concerning the convexity of the range of nonatomic measures. Given a family of integrable vector functions on a compact set, this theorem allows to prove the equivalence between the range of integral values obtained considering all possible set decompositions and all possible convex combinations of the elements of the family. Lyapunov type results have several applications in optimal control theory: they are used to prove bang-bang properties and existence results without convexity assumptions. Here, we use the dual approach to the Baire category method in order to provide a “quantitative” version of such kind of results applied to a countable family of integrable functions.
Marco Mazzola, Khai T. Nguyen

Controllability Under Positivity Constraints of Multi-d Wave Equations

We consider both the internal and boundary controllability problems for wave equations under non-negativity constraints on the controls. First, we prove the steady state controllability property with nonnegative controls for a general class of wave equations with time-independent coefficients. According to it, the system can be driven from a steady state generated by a strictly positive control to another, by means of nonnegative controls, and provided the time of control is long enough. Secondly, under the added assumption of conservation and coercivity of the energy, controllability is proved between states lying on two distinct trajectories. Our methods are described and developed in an abstract setting, to be applicable to a wide variety of control systems.
Dario Pighin, Enrique Zuazua

Asymptotic Analysis of a Cucker–Smale System with Leadership and Distributed Delay

We extend the analysis developed in Pignotti and Reche Vallejo (J Math Anal Appl 464:1313–1332, 2018) [34] in order to prove convergence to consensus results for a Cucker–Smale type model with hierarchical leadership and distributed delay. Flocking estimates are obtained for a general interaction potential with divergent tail. We analyze also the model when the ultimate leader can change its velocity. In this case we give a flocking result under suitable conditions on the leader’s acceleration.
Cristina Pignotti, Irene Reche Vallejo

Global Non-negative Approximate Controllability of Parabolic Equations with Singular Potentials

In this work, we consider the linear \(1-d\) heat equation with some singular potential (typically the so-called inverse square potential). We investigate the global approximate controllability via a multiplicative (or bilinear) control. Provided that the singular potential is not super-critical, we prove that any non-zero and non-negative initial state in \(L^2\) can be steered into any neighborhood of any non-negative target in \(L^2\) using some static bilinear control in \(L^\infty \). Besides the corresponding solution remains non-negative at all times.
Judith Vancostenoble
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