


Politecnico di Torino  
Academic Year 2016/17  
01PPXMQ Mathematical Control Theory/Partial Differential Equations 

1st degree and Bachelorlevel of the Bologna process in Mathematics For Engineering  Torino 





Subject fundamentals
The notion of ''system'' is broadly considered a cornerstone of the rational approach to applied sciences, both from the epistemological and technological point of view. It is based on the construction and the investigation of mathematical models, for which advances mathematics are an indispensable tool. For an applied mathematician, especially interested in engineering problems, the ability of interpreting the phenomena of the real word in term of ''system'' is fundamental.
The course provides an introduction to methods, models and applications for ordinary and partial differential equations, mainly focusing on boundary value problems or initial value problems for linear equations of first and second order. It also introduces in a rigorous and rational framework the basic notions and the principal results of the mathematical theory related to the control of linear systems. Both the analysis and the design aspects are part of the training program. 
Expected learning outcomes
Ability to compute the controllability and observability indices of a system described by a linear model, and to investigate its stability properties.
Ability to compute, in simple situations, static or dynamic feedback laws, depending either on the state or the output, with the aim to improve the system performances. Ability to study simple optimization problem ﾍs associated to a linear system, and to determine their solutions in feedback form. Ability to distinguish the type of equation and to set up its solution in a classical or variational frame work. Ability to represent in simple cases solutions of some linear equations (the transport equation, the Laplace/Poisson equation, the heat equation, the wave equation). Ability to solve simple spectral problems. 
Prerequisites / Assumed knowledge
Differential and integral calculus for functions of n real variables. Linear algebra. Elements of functional analysis. Distributions. Fourier series. Lebesgue integral and L^p spaces.

Contents
Basic definitions: linear, semilinear, quasilinear, non linear equations. Example. Classical and weak solutions. The transport equation. First order linear equations, characteristics.
Second order linear equations. Classification. Initial and boundaryvalue problems. e Well posedness. Some results for the vibrating string. Laplace and Poisson equations: classical and distributional solutions. Physical interpretation. Dirichlet, Neumann and Robin boundary conditions. Uniqueness results. Geometrical meaning of the Laplace operator. Harmonic functions, the meanvalue property, the strong maximum principle, continuous dependance on the boundary value. The weak maximum principle, continuous dependance on the boundary value for the solution of the Poisson equation. Special cases: Poisson equation in the ndimensional ball, with polynomial data. The Dirichlet problem for the Laplace operator in the 2dimensional disc and in dimension n: Poisson formula for the solution. Regularity of harmonic functions. Radial harmonic functions in R^n. The fundamental solution for the Laplace operator.The solution of the Poisson equation in R^n and in a bounded open set through the Green’s function. Sobolev spaces. Definitions and properties of the space H^1(A), examples of H^1 functions. H^1 functions in dimension 1 are holdercontinuous. Counterexamples. Density of smooth functions in H^1. Generalizations to the speces H^k and W^{k,p}: definitions, main properties. Characterization of H^1(\R^n) by means of the Fourier transform. The spaces H^s for real values of s. Extension and trace theorems for H^1(A), in particular when A is a halfspace. The space H^{1/2}. Integration by parts in H^1(A). The space H^1_0(A). PoincaréFriedrichs and PoincaréWirtinger inequalities. Continuous and compact embedding results for H^s. The dual spaces of H^1 e H^1_0. Abstract notion of system. Basic facts about systems of ordinary differential equation, homogeneous and with forcing term, with constant coefficient. Linear equivalence. Jordan form, companion form. Stability and asymptotic stability of the equilibrium position. Liapunov functions, Liapunov first and second theorem, matrix Liapunov equation. Linear systems with inputs. Reachable sets. Controllability and observability. Kalman form. Stabilization. Brunowsky form. Asymptotic observers. BIBO stability. Pontrjagin maximum principle. Optimal time problem, quadratic regulator. Elliptic problems. Weak formulation of the DirichletNeumann problem for per linear elliptic second order operators. The LaxMilgram theorem. Wellposedness, properties of the weak solution, H^k regularity, counterexamples. The Stampacchia maximum principle. Spectral theory for elliptic selfadjoint problems. Eistence of a, ortonornal basis of eigenfucntions. The special case of the Laplace operator. Intepretation of Poincaré constant. Applications: the vibranting string, the square and circular membrane. Bessel functions. Solution to elliptic boundaryvalue problems in series of eigenfucntions. Parabolic problems. The heat equation, introduction and first properties. Heat conduction. Wellposed problems in one spacial dimension. The parabolic boundary. A simple example by separation of variables. Time dependent Sobolev spaces. Weak formulation of the CauchyDirichlet problem for the heat equation. Wellposedness. Apriori estimates. Representation of the solutions in series of eigenfunctions (FaedoGalerkin method). Regularity of the solution in the case of one spacial dimension. The heat kernel. The maximum principle for the CauchyDirichlet problem for the heat equation. Hyperbolic problems. The CauchyDirichlet problem for the wave equation in n variables. Weak formulation and wellposedness. Apriori estimates. Represe 
Delivery modes
Practical exercises are devoted to applications of the theoretical results presented in the lectures.
Simple simulations using MATLAB are possible. 
Texts, readings, handouts and other learning resources
Handnotes are prepared by the teacher, and are available on ``Portale della didattica del Politecnico'' for the students of the course.
Suggested book for the PDE part is: S. Salsa, Equazioni a derivate parziali: metodi, modelli e applicazioni, Springer, seconda edizione. Other related references: L.C. Evans. Partial differential equations, AMS 1998. P.E. Garabedian, Partial differential equations, John Wiley & Sons, 1964. H.F. Weinberger, A first course in partial differential equations, Blaishdell Publishing Company, 1965. A.N. Tihonov, A.A: Samarski, Equazioni della fisica matematica, Mir 1981. 
Assessment and grading criteria
The standard exam consists in a written test which includes theoretical questions and practical exercises on all the topics of the course. After a written test whose mark is not below 18/30, upon request of the professor or of the student, an oral interview may also take place: in this case the final mark of the exam takes into account both the written test and the oral interview. Otherwise, the mark of the written test is the final mark of the exam.

