“Multivalued Analysis” is the theory of set-valued maps (called multifonctions) and has important applications in many different areas. Multivalued analysis is a remarkable mixture of many different parts of mathematics such as point-set topology, measure theory and nonlinear functional analysis. It is also closely related to “Nonsmooth Analysis” (Chapter 5) and in fact one of the main motivations behind the development of the theory, was in order to provide necessary analytical tools for the study of problems in nonsmooth analysis. It is not a coincidence that the development of the two fields coincide chronologically and follow parallel paths. Today multivalued analysis is a mature mathematical field with its own methods, techniques and applications that range from social and economic sciences to biological sciences and engineering. There is no doubt that a modern treatise on “Nonlinear Functional Analysis” can not afford the luxury of ignoring multivalued analysis. The omission of the theory of multifunctions will drastically limit the possible applications.

SET-Valued Analysis

In this chapter we present some of the recent progresses done on the problem of controllability of partial differential equations (PDE). Control problems for PDE arise in many different contexts and ways. A prototypical problem is that of controllability. Roughly speaking it consists in analyzing whether the solution of the PDE can be driven to a given final target by means of a control applied on the boundary or on a subdomain of the domain in which the equation evolves. In an appropriate functional setting this problem is equivalent to that of observability which concerns the possibility of recovering full estimates on the solutions of the uncontrolled adjoint system in terms of partial measurements done on the control region. Observability/controllability properties depend in a very sensitive way on the class of PDE under consideration. In particular, heat and wave equations behave in a significantly different way, because of their different behavior with respect to time reversal. In this paper we first recall the known basic controllability properties of the wave and heat equations emphasizing how their different nature affects their main controllability properties. We also recall the main tools to analyze these problems: the so-called Hilbert uniqueness method (HUM), multipliers, microlocal analysis and Carleman inequalities. We then discuss some more recent developments concerning equations with low regularity coefficients, equations with potentials, bang-bang controls, etc. We also analyze the way control and observability properties depend on the norm and regularity of these coefficients, a problem which is also relevant when addressing nonlinear models. We then present some recent results on coupled models of wave–heat equations arising in fluid–structure interaction. We also present some open problems and future directions of research.

Controllability and Observability of Partial Differential Equations: Some Results and Open Problems

In this review, concerning parabolic equations, we give self-contained descriptions on derivations of Carleman estimates; methods for applications of the Carleman estimates to estimates of solutions and to inverse problems.Moreover limited to parabolic equations, we survey the previous and recent results in view of the applicability of the Carleman estimate. We do not intend to pursue any general treatments of the Carleman estimate itself but by showing it in a direct manner, we mainly aim to demonstrate the applicability of the Carleman estimate to the estimation of solutions and inverse problems.

https://iopscience.iop.org/article/10.1088/0266-5611/25/12/123013/pdf

Carleman estimates for parabolic equations and applications

Given $\alpha \in [0,2)$ and $f \in L^2 ((0,T)\times(0,1))$, we derive new Carleman estimates for the degenerate parabolic problem $w_t + (x^\alpha w_x) _x =f$, where $(t,x) \in (0,T) \times (0,1)$, associated to the boundary conditions $w(t,1)=0$ and $w(t,0)=0$ if $0 \leq \alpha <1$ or $(x^\alpha w_x)(t,0)=0$ if $1\leq \alpha <2$ The proof is based on the choice of suitable weighted functions and Hardy-type inequalities As a consequence, for all $0 \leq \alpha <2$ and $\omega\subset\subset(0,1)$, we deduce null controllability results for the degenerate one-dimensional heat equation $u_t - (x^\alpha u_x)_x = h \chi _\omega$ with the same boundary conditions as above

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Carleman Estimates for a Class of Degenerate Parabolic Operators

We prove an estimate of Carleman type for the one dimensional heat equation
$$ u_t - \left( {a\left( x \right)u_x } \right)_x + c\left( {t,x} \right)u = h\left( {t,x} \right),\quad \left( {t,x} \right) \in \left( {0,T} \right) \times \left( {0,1} \right), $$ where a(·) is degenerate at 0. Such an estimate is derived for a special pseudo-convex weight function related to the degeneracy rate of a(·). Then, we study the null controllability on [0, 1] of the semilinear degenerate parabolic equation
 $$ u_t - \left( {a\left( x \right)u_x } \right)_x + f\left( {t,x,u} \right) = h\left( {t,x} \right)\chi _\omega \left( x \right), $$ where (t, x) ∈(0, T) × (0, 1), ω=(α, β) ⊂⊂ [0, 1], and f is locally Lipschitz with respect to u.

Carleman estimates for degenerate parabolic operators with applications to null controllability

We prove null controllability results for the degenerate one-dimensional heat equation $$ u_t - (x^\alpha u_x)_x = f \chi _\omega , \quad x\in (0,1), \ \ t\in (0,T) .$$ As a consequence, we obtain null controllability results for a Crocco-type equation that describes the velocity field of a laminar flow on a flat plate.

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Null controllability of degenerate heat equations

* Introduction* Part 1. Weakly degenerate operators with Dirichlet boundary conditions* Controllability and inverse source problems* Notation and main results* Global Carleman estimates for weakly degenerate operators* Some Hardy-type inequalities (proof of Lemma 3.18)* Asymptotic properties of elements of $H^2 (\Omega) \cap H^1 _{A,0}(\Omega)$* Proof of the topological lemma 3.21* Outlines of the proof of Theorems 3.23 and 3.26* Step 1: computation of the scalar product on subdomains (proof of Lemmas 7.1 and 7.16)* Step 2: a first estimate of the scalar product: proof of Lemmas 7.2, 7.4, 7.18 and 7.19* Step 3: the limits as $\Omega^\delta \to \Omega$ (proof of Lemmas 7.5 and 7.20)* Step 4: partial Carleman estimate (proof of Lemmas 7.6 and 7.21)* Step 5: from the partial to the global Carleman estimate (proof of Lemmas 7.9-7.11)* Step 6: global Carleman estimate (proof of Lemmas 7.12, 7.14 and 7.15)* Proof of observability and controllability results* Application to some inverse source problems: proof of Theorems 2.9 and 2.11* Part 2. Strongly degenerate operators with Neumann boundary conditions* Controllability and inverse source problems: notation and main results* Global Carleman estimates for strongly degenerate operators* Hardy-type inequalities: proof of Lemma 17.10 and applications* Global Carleman estimates in the strongly degenerate case: proof of Theorem 17.7* Proof of Theorem 17.6 (observability inequality)* Lack of null controllability when $\alpha\geq 2$: proof of Proposition 16.5* Explosion of the controllability cost as $\alpha\to 2^-$ in space dimension $1$: proof of Proposition 16.7* Part 3. Some open problems* Some open problems* Bibliography* Index

Global Carleman Estimates for Degenerate Parabolic Operators With Applications

We consider the one-dimensional degenerate parabolic equation \begin{document}$u_t - (x^α u_x)_x =0 \;\;x∈(0,1),\ t ∈ (0,T) ,$ \end{document} controlled by a boundary force acting at the degeneracy point \begin{document}$ x=0$\end{document} . We study the reachable targets at some given time \begin{document}$T$\end{document} using \begin{document}$ H^1$\end{document} controls, studying the influence of the degeneracy parameter \begin{document}$ α ∈ [0,1)$\end{document} . First we obtain precise upper and lower bounds for the null controllability cost, proving that the cost blows up rationnally as \begin{document}$ \mathit{\alpha } \to {{\rm{1}}^{\rm{ - }}}$\end{document} and exponentially fast when \begin{document}$ \mathit{T} \to {{\rm{0}}^{\rm{ + }}}$\end{document} .   Next, thanks to the special structure of the eigenfunctions of the problem, we investigate and obtain (partial) results concerning the structure of the reachable states.   Our approach is based on the moment method developed by Fattorini and Russell [ 19 , 20 ]. To achieve our goals, we extend some of their general results concerning biorthogonal families, using complex analysis techniques developped by Seidman [ 48 ], Guichal [ 26 ], Tenenbaum-Tucsnak [ 49 ] and Lissy [ 35 , 36 ].

https://www.aimsciences.org/article/exportPdf?id=16945227-6bb2-4dc4-bd37-48212fc8f14c

The cost of controlling weakly degenerate parabolic equations by boundary controls

Motivated by several works on ordinary differential equations, we are interested in the asymptotic stability of intermittently controlled partial differential equations. We give a condition of asymptotic stability for second-order evolution equations uniformly damped by an on/off feedback. This result extends to the case of partial differential equations a previous result of R. A. Smith concerning ordinary differential equations.

Patrick Martinez

Papers

Carleman Estimates for a Class of Degenerate Parabolic Operators

Null controllability of degenerate heat equations

Global Carleman Estimates for Degenerate Parabolic Operators With Applications

The cost of controlling weakly degenerate parabolic equations by boundary controls

Asymptotic Stability for Intermittently Controlled Second-Order Evolution Equations