Univ. degli Studi di Firenze Dottorato di Ricerca in Matematica, Informatica, Statistica AA 2023-2024 Corso: Control theory for systems governed by Partial Differential Equations: an introduction Docenti: Francesca Bucci, Sergio Vessella PhD Programme in Mathematics, Computer Science, Statistics Academic Year 2023-2024 Course: Control theory for systems governed by Partial Differential Equations: an introduction Instructors: Francesca Bucci, Sergio Vessella LECTURES' DIARY (Diario delle lezioni) 1^ week Mer. 21 feb. 2024 (2 ore) -- Bucci F. The plan of the course: 1. Basic questions in mathematical control theory. Some insight on controllability of linear systems in R^n. 2. Representation of initial/boundary value problems for partial differential equations (PDE), possibly subjected to control actions: a functional-analytic perspective (---> one-parameter operator semigroups theory). 3. Focus on a couple of specific questions, such as e.g. (3a) exact controllability for the wave equation (whose proof entails establishing a suitable boundary regularity result), and (3b) uniform stability of a thermoelastic system. 4. A taste of Carleman estimates and the unique continuation property. Mathematical control theory: starting point (a control system), broad aim. Open- vs closed-loop controls. Major basic questions: controllability, stabilizability, observability (omitted, at the outset), optimal control problems (a special case: the Linear Quadratic problem). (Significant topics which will not be dealt with in the course, due to lack of time: optimal control problems with constraints, optimal impulse control, delay equations, equations with memory, systems on manifolds (---> geometric control), et alia.) Ven. 23 feb. 2024 (2 ore) -- Bucci F. Controllability of linear systems in R^n. The controllability matrix. The non-singularity of the controllability matrix Q_T for some T>0 ensures controllability (proved). The condition is also necessary for reachability of an arbitrary state b from a=0; see [Z, Part I, Prop. 1.2]. The rank condition; see [Z, Part I, Thm 1.2]. 2^ week Mer. 28 feb. 2024 (2 ore) -- Bucci F. Representation of initial-boundary value problems for linear evolutionary partial differential equations (PDE) as differential systems in Banach spaces. The heat equation in a bounded domain of R^n, with homogeneous boundary data. Modeling the presence of control actions: distributed controls, boundary controls and the Fattorini-Balakrishnan method; bounded vs unbounded control operator. Ven. 1 mar. 2024 (2 ore) -- Bucci F. Representation of initial-boundary value problems for PDE as differential systems in Banach spaces (cont'd). The Cauchy-Dirichlet problem for the wave equation in a bounded domain of R^n. Choosing a natural state space: energy/multiplier methods, conservation of energy. Introduction to the theory of (one-parameter, operator) strongly continuous semigroups in Banach spaces. C_0-semigroups: definition, notation; illustrative examples, the left translation semigroup in X=C_ub([0,infty)). C_0-groups, uniformly continuous semigroups. The left translation semigroup is not uniformly continuous. Characterization of uniformly continuous semigroups (without proof). 3^ week Mer. 6 mar. 2024 (2 ore) -- Bucci F. A roadmap to the Theory of operator Semigroups. (Prerequisite: The Bochner integral. See, e.g., the Appendix A in [L]. a) Asymptotic properties of semigroups: the sharp growth bound, growth rates of the semigroup. b) Mean value of the integral. c) The (infinitesimal) generator of a semigroup. Basic properties of semigroups. The Cauchy problem. d) Semigroups and spectral properties of their generators. Integral representation of the resolvent of the generator. e) Relevant subclasses of semigroups: semigroups of contraction. Generation results: the Hille-Yosida Theorem. Dissipative operators and the Lumer-Phillips Theorem. Corollaries: unitary operators and the Stone Theorem. f) Sectorial operators and analytic semigroups. Additional smoothing properties of the semigroup, aymptotic estimates; the sharp growth bound coincides with the spectral bound. Bibliographical references: [A_1], [B-DaP-D-M], [L_1], [P]. See also F. Bucci's Notes. Fri. 8 mar. 2024 (2 ore) -- Bucci F. The Department building in V.le Morgagni 67/A found closed owing to workers' strike; hence, the lecture was canceled. (The lost hours will be recovered in the subsequent weeks.) 4^ week Mer. 13 mar. 2024 (3 ore) -- Bucci F. Exact controllability of linear PDE. Lack of the property of exact controllability, parabolic vs hyperbolic equations. The Cauchy-Dirichlet problem for the wave equation: the issue of well-posedness, regularity of the mapping which associates to initial and boundary data the solution to the IBVP and its normal trace on the boundary. Major contributions due to: R. Sakamoto, J.-L. Lions, I. Lasiecka, R. Triggiani ('70s-'80s). The statement of a well-posedness result for initial data in L^2(Omega)xH^{-1}(Omega) and boundary datum in L^2(Sigma): interior and boundary regularity of the corresponding solutions, the enhanced regularity of the boundary traces (so called "hidden" regularity, by J.-L. Lions). Exact boundary controllability, the abstract viewpoint: the property of controllability as a set inclusion between images of linear operators, which is in turn equivalent to a suitable inequality involving the adjoint operators; see [Z, Theorem 2.2., p. 208]. Fri. 15 mar. 2024 (3 ore) -- Bucci F. Exact boundary controllability of the wave equation. Abstract reformulation of the boundary value problem as a control system y'=Ay+Bg; explicit computation of A and B, as well as of the respective adjoints. The interpretation of the abstract (reverse) inequality as a PDE estimate (from below) of the normal trace of the solutions to the Cauchy-Dirichlet problem for the wave equation, with homogeneous BC and initial data in H^1_0(Omega)xL^2(Omega). Literature: for the the first proof of the inequality credit must be given to the work of L.F. Ho (1986), subsequently improved by V. Komornik in 1987; see e.g. [K, Theorem 3.1, p. 36], or the Lecture Notes arXiv:2402.17894 by E. Zuazua. 5^ week Mer. 20 mar. 2024 (2 ore) -- Bucci F. Controllability of linear PDE (cont'd). Why the heat equation (with Dirichlet BC) is not exactly controllable; the weaker concepts of null and approximate controllability. The role of functional-analytic results pertaining to the images of linear operators, along with the respective PDE counterparts; the key (observability) inequality for the heat equation (proved in [Fursikov-Imanuvilov, 1996]. Another relevant question within Control Theory of PDE: stability of infinite-dimensional linear systems. The definitions of uniform exponential stability, uniform stability, strong stability, weak stability. The four concepts are equivalent in the finite dimensional setting. Uniform stability actually carries exponential decay rates; uniform vs strong stability (an illustrative example). Stability analysis of an established thermoelastic system (for the modeling, see [Lag, 1989]). Description of two variants of the coupled PDE system, with the (uncoupled) elastic equation either the Euler-Bernoulli or the Kirchhoff one. Ven. 22 mar. 2024 (2 ore) -- Bucci F. Uniform stability of a thermoelastic system (cont'd). Abstract reformulation: the realization of the bilaplacian in L^2(Omega) with hinged boundary conditions (dynamics of the elastic equation), its fractional powers of exponent 1/2 and 1/4 (roughly; for a formal exposition see [L_2, Ch. 4]). The natural energy spaces Y_0 and Y for the coupled system in the two cases gamma=0 and gamma>0, respectively. From the coupled PDE system to a first order abstract system in Y (or Y_0), the operator A that describes the whole dynamics. (Homework: determine D(A), compute A*, deduce the well-posedness for the Cauchy problem via a Corollary of the Lumer-Phllips Theorem pertaining to dissipative operators in Hilbert spaces.) If gamma=0, the C_0-semigroup generated by A is also analytic [Liu-Renardy, 1995], which implies uniform exponential stability (since the sharp growth bound equals the spectral bound, which is negative). In the case gamma>0 -- where the semigroup is not analytic -- the property of uniform stability can be established via the following equivalent condition: "there exists T>0 such that ||e^{TA}||<1" (the proof is omitted); then, PDE estimates are called for. An outline of the argument: derivation of a dissipativity identity for the total energy of the system, achieving that the integral of the total energy is bounded from above by the dissipation as final goal. Clever estimates of the kinetic and potential energy of the plate component as intermediate steps: the key role of an abstract multiplier introduced by G. Avalos and I. Lasiecka (1995-1996). Readers/students are referred to [Las-Tr, vol. I, Appendix 3J] for all details. 6^ week Mar. 26 mar. 2024 (2 ore) -- Vessella S. Introduction to Carleman estimates. Carleman estimates and unique continuation properties for differential equations. Carleman estimate for the Cauchy-Reimann operator and application to the proof of strong unique continuation for such operator. Mer. 27 mar. 2024 (2 ore) -- Vessella S. Illustration of Nirenberg's theorem on unique continuation for operators with constant coefficients in the principal part. Generalities on Carleman estimates for linear differential operators of order m. Stability of a Carleman estimate with respect to perturbations of order m-1 of the operator. Local nature of Carleman estimates. Setting up a Carleman estimate. Definition of the conjugate of an operator with respect to the weight function. Principal part of the conjugate operator. Brief mention of differential quadratic forms and the classical Hormander method. A very brief mention of pseudoconvexity conditions through elementary examples. BIBLIOGRAPHICAL REFERENCES [A_1] Paolo Acquistapace, Appunti di teoria dei controlli (2022); http://people.dm.unipi.it/acquistp/teocon.pdf [B-DaP-D-M] A. Bensoussan, G. Da Prato, M.C. Delfour, S.K. Mitter, Representation and control of infinite dimensional systems. Second edition. Systems & Control: Foundations & Applications. Birkhauser Boston, Inc., Boston, MA, 2007. xxviii+575 pp. [K] Vilmos Komornik, Exact Controllability and Stabilization: The Multiplier Method, Wiley, 1994. [Lag] John E. Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Applied and Numerical Mathematics 10, 1989. [Las-Tr] Irena Lasiecka and Roberto Triggiani, Control Theory for Partial Differential Equations, Continuous and Approximation Theories. Volume 1: Abstract Parabolic Systems, Volume 2: Abstract Hyperbolic-like Systems over a Finite Time Horizon, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, 2000. [L_1] Alessandra Lunardi, Introduzione alla teoria dei semigruppi, Dispense del Dipartimento di Matematica dell'Universita' di Parma (2015); https://people.dmi.unipr.it/alessandra.lunardi/ [L_2] Alessandra Lunardi, Interpolation theory, Third edition, Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 16. Edizioni della Normale, Pisa, 2018. xiv+199 pp. [P] A. Pazy, Semigroups of operators in Banach spaces, Lecture Notes in Math., 1017, Springer, Berlin, 1983. [Z] Jerzy Zabczyk, Mathematical control theory. An introduction, Second edition, Birkhauser/Springer, Cham, 2020, xxvi+336 pp.