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Mathematics of Complexity and Dynamical Systems pp 102–125Cite as

Control of Non-linear Partial Differential Equations

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Article Outline

Glossary

Definition of the Subject

Introduction

Controllability

Stabilization

Optimal Control

Future Directions

Bibliography

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Abbreviations

State variables:

quantities describing the state of a system; in this note they will be denoted by u; in the present setting, u will be either a function defined on a subset of \({\mathbb{R}\times\mathbb{R}^n}\), or a function of time taking its values in an Hilbert space H.

Space domain:

the subset of \({\mathbb{R}^n}\) on which state variables are defined.

Partial differential equation:

a differential equation containing the unknown function as well as its partial derivatives.

State equation:

a differential equation describing the evolution of the system of interest.

Control function:

an external action on the state equation aimed at achieving a specific purpose; in this note, control functions they will be denoted by f; f will be used to denote either a function defined on a subset of \({\mathbb{R}\times\mathbb{R}^n}\), or a function of time taking its values in an Hilbert space F. If the state equation is a partial differential equation of evolution, then a control function can be:

  1. 1.

    distributed if it acts on the whole space domain;

  2. 2.

    locally distributed if it acts on a subset of the space domain;

  3. 3.

    boundary if it acts on the boundary of the space domain;

  4. 4.

    optimal if it minimizes (together with the corresponding trajectory) a given cost;

  5. 5.

    feedback if it depends, in turn, on the state of the system.

Trajectory:

the solution of the state equation u f that corresponds to a given control function f.

Distributed parameter system:

a system modeled by an evolution equation on an infinite dimensional space, such as a partial differential equation or a partial integro‐differential equation, or a delay equation; unlike systems described by finitely many state variables, such as the ones modeled by ordinary differential equations, the information concerning these systems is “distributed” among infinitely many parameters.

\({\mathbb{1}_A}\) denotes the characteristic function of a set \({A\subset\mathbb{R}^n}\), that is,

$$ \mathbb{1}_A(x)= \begin{cases} 1&x\in A \\ 0& x\in \mathbb{R}^n\setminus A \end{cases}$$

\({\partial_t\,,\;\partial_{x_i}}\) denote partial derivatives with respect to t and x i , respectively.

\({L^2(\Omega)}\) denotes the Lebesgue space of all real‐valued square integrable functions, where functions that differ on sets of zero Lebesgue measure are identified.

\({H^1_0(\Omega)}\) denotes the Sobolev space of all real‐valued functions which are square integrable together with their first order partial derivatives in the sense of distributions in Ω, and vanish on the boundary of Ω; similarly \({H^2(\Omega)}\) denotes the space of all functions which are square integrable together with their second order partial derivatives.

\({H^{-1}(\Omega)}\) denotes the dual of \({H^1_0(\Omega)}\).

\({\mathcal{H}^{n-1}}\) denotes the \({(n-1)}\)‐dimensional Hausdorff measure.

H denotes a normed spaces over ℝ with norm \({\|\cdot\|}\), as well as an Hilbert space with the scalar product \({\langle\cdot,\cdot\rangle}\) and norm \({\|\cdot\|}\).

\({L^2(0,T;H)}\) is the space of all square integrable functions \({f \colon [0,T]\to H}\); \({C([0,T];H)}\) (continuous functions) and \({H^1(0,T;H)}\) (Sobolev functions) are similarly defined .

Given Hilbert spaces F and H, \({\mathcal{L}(F,H)}\) denotes the (Banach) space of all bounded linear operators \( \Lambda \colon F\to H \) with norm \( \|\Lambda\|=\sup_{\|x\|=1}\|\Lambda x\| \) (when \({F=H}\), we use the abbreviated notation \( \mathcal{L}(H) \)); \( \Lambda^* \colon H\to F \) denotes the adjoint of Λ given by \( \langle \Lambda^*u,\phi\rangle=\langle u,\Lambda\phi\rangle \) for all \({u\in H}\), \({\phi\in F}\).

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Authors and Affiliations

  1. L.M.A.M., Université de Metz, Metz, France

    Fatiha Alabau‐Boussouira

  2. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Rome, Italy

    Piermarco Cannarsa

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  1. Fatiha Alabau‐Boussouira
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  2. Piermarco Cannarsa
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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© 2012 Springer-Verlag

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Alabau‐Boussouira, F., Cannarsa, P. (2012). Control of Non-linear Partial Differential Equations. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_8

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