##
**Control theory for partial differential equations: continuous and approximation theories. 1: Abstract parabolic systems.**
*(English)*
Zbl 0942.93001

Encyclopedia of Mathematics and Its Applications 74. Cambridge: Cambridge University Press. xxi, 644 p. (2000).

The book is a vast, detailed and rigorous treatise on the quadratic optimal control of linear parabolic PDEs. The abstract framework of the form
\[
\dot y= Ay+ Bu
\]
is under consideration. \(A\) is the generator of a strongly continuous semigroup on a Hilbert state space \(Y\) and \(B\) is an unbounded operator with degree of unboundedness up to the degree of unboundedness of \(A\); \(u\) is an \(L_2\) function of time. The unboundedness of \(B\) allows to handle boundary/point controls. Many applications to specific examples and a numerical theory are presented.

In chapter one, the linear quadratic optimal control problem with a finite time horizon is investigated using the variational approach of the author, i.e., the Riccati operator equation is obtained from the optimality data. One has regularity except possibly at the final time of the horizon, but stronger smoothing hypotheses on the operator of penalization of the final time will provide continuity also at the final time. Uniqueness of the optimal solution also follows under such assumptions.

Chapter two treats the same problem as chapter one but now the quadratic cost is defined over an infinite time interval. There is no terminal penalization but a Finite Cost Condition has to be introduced to guarantee existence of an optimal solution. Moreover, an additional detectability condition guarantees uniqueness of the solution to the Riccati operator equation. Once again the variational approach is used and the apparatus of the previous chapter is used so that the obtained optimal objects are made to tend to limiting objects when time becomes large in suitable function spaces.

Chapter three is devoted to the application of the results of the previous chapters to classical parabolic equations with Dirichlet boundary control (resp. Neumann boundary control) and interior (resp. interior or boundary) observation and to parabolic-like PDEs (damped Euler-Beam equation, Kelvin plate equation, damped wave equation, damped Kirchhoff equation) with mainly point control. Other cases are coupled PDEs modeling mechanical and thermic effects in particular thermoelastic plates with force or thermal control (boundary control). The case of the damped Euler-Bernoulli equation with boundary control is included.

Chapter four describes numerical approximation schemes to the problems of chapter two. They are based on discretization and convergence rates are obtained. Sometimes these rates are optimal.

Chapter five applies the results of chapter four to the practical situations from chapter five.

Chapter six generalizes the study to a min-max game when disturbances enter the dynamics (cf. \(H_\infty\) control). Once again the variational approach is used.

Each chapter ends with a list of references. Some background is introduced at the beginning and an index can be found at the end of the volume. The reader should be acquainted with functional analysis and semigroup theory.

A previous book of the authors treats similar topics [Differential and Algebraic Riccati Equations with Applications to Boundary-Point Control Problems: Continuous and Approximation Theory (1991; Zbl 0754.93038)] but many proofs were lacking and this treatise is a largely expanded version. Many results of the authors can be found here but they also mention other developments from other researchers.

This impressive volume is a superb achievement and will be a must for all those who are interested in the quadratic optimal control of parabolic PDEs and in general in the control of PDEs.

In chapter one, the linear quadratic optimal control problem with a finite time horizon is investigated using the variational approach of the author, i.e., the Riccati operator equation is obtained from the optimality data. One has regularity except possibly at the final time of the horizon, but stronger smoothing hypotheses on the operator of penalization of the final time will provide continuity also at the final time. Uniqueness of the optimal solution also follows under such assumptions.

Chapter two treats the same problem as chapter one but now the quadratic cost is defined over an infinite time interval. There is no terminal penalization but a Finite Cost Condition has to be introduced to guarantee existence of an optimal solution. Moreover, an additional detectability condition guarantees uniqueness of the solution to the Riccati operator equation. Once again the variational approach is used and the apparatus of the previous chapter is used so that the obtained optimal objects are made to tend to limiting objects when time becomes large in suitable function spaces.

Chapter three is devoted to the application of the results of the previous chapters to classical parabolic equations with Dirichlet boundary control (resp. Neumann boundary control) and interior (resp. interior or boundary) observation and to parabolic-like PDEs (damped Euler-Beam equation, Kelvin plate equation, damped wave equation, damped Kirchhoff equation) with mainly point control. Other cases are coupled PDEs modeling mechanical and thermic effects in particular thermoelastic plates with force or thermal control (boundary control). The case of the damped Euler-Bernoulli equation with boundary control is included.

Chapter four describes numerical approximation schemes to the problems of chapter two. They are based on discretization and convergence rates are obtained. Sometimes these rates are optimal.

Chapter five applies the results of chapter four to the practical situations from chapter five.

Chapter six generalizes the study to a min-max game when disturbances enter the dynamics (cf. \(H_\infty\) control). Once again the variational approach is used.

Each chapter ends with a list of references. Some background is introduced at the beginning and an index can be found at the end of the volume. The reader should be acquainted with functional analysis and semigroup theory.

A previous book of the authors treats similar topics [Differential and Algebraic Riccati Equations with Applications to Boundary-Point Control Problems: Continuous and Approximation Theory (1991; Zbl 0754.93038)] but many proofs were lacking and this treatise is a largely expanded version. Many results of the authors can be found here but they also mention other developments from other researchers.

This impressive volume is a superb achievement and will be a must for all those who are interested in the quadratic optimal control of parabolic PDEs and in general in the control of PDEs.

Reviewer: A.Akutowicz (Berlin)

### MSC:

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

93C20 | Control/observation systems governed by partial differential equations |

35K50 | Systems of parabolic equations, boundary value problems (MSC2000) |

93B28 | Operator-theoretic methods |

49N10 | Linear-quadratic optimal control problems |

49K20 | Optimality conditions for problems involving partial differential equations |

49J20 | Existence theories for optimal control problems involving partial differential equations |

35K90 | Abstract parabolic equations |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

74K20 | Plates |

74F05 | Thermal effects in solid mechanics |

93B36 | \(H^\infty\)-control |

74M05 | Control, switches and devices (“smart materials”) in solid mechanics |

93B40 | Computational methods in systems theory (MSC2010) |

74H15 | Numerical approximation of solutions of dynamical problems in solid mechanics |