##
**Smart material structures. Modeling, estimation and control.**
*(English)*
Zbl 0882.93001

Research in Applied Mathematics. Chichester: Wiley. Paris: Masson. vi, 304 p. (1996).

The monograph is devoted to the study of smart material systems, a term containing systems known in the literature as controllable, adaptive, compliant, intelligent.

The authors themselves, especially the first one, are important contributors to the domain, with more than 50 papers quoted in the bibliography. The content is structured in 9 chapters presenting the results of other authors or their own ones.

The first chapter presents a number of smart materials of current interest in research and in applied engineering: piezoelectric and electrostrictive elements, magnetostrictive transducers, electrorheological fluids, shape memory alloys and fiber optic sensors.

The second and the third chapter are devoted to the presentation of structural modeling of beams, plates, shells and details for inclusion of both inactive and active dynamic effects of surface-mounted piezoceramics on these structures. Both chapters are of theoretical character where the movement and equilibrium equations are established and the corresponding boundary conditions in different systems of coordinates are formulated.

The fourth chapter is the central chapter of the volume and analyses well-formulated boundary problems for abstract structural models. The authors study the existence, the uniqueness and the continuous dependence of the solution on initial and boundary conditions for second-order in time partial differential equations describing the behaviour of material systems presented in the earlier chapters. For each analyzed model a variational formulation is given (weak formulation) which constitutes the mathematical framework for the determination of the material systems parameters, analyzed minutely in the following chapter, i.e. the fifth. The general problem studied in the fourth chapter refers to the second-order (in time) system of equations having the form \[ \ddot w(t)+A_1\dot w(t)+A_2w(t)=f(t),\text{ in }V^*,\;w(0)=w_0,\;\dot w(0)=w_1, \] where \(V^*\) is the dual space of \(V\) – the space of unknown functions with the scalar product \(\langle, \rangle\), dense in a complex Hilbert space \(H\), \(A_1\) and \(A_2\) are operators defined by the sesquilinear forms \(\sigma_1\) and \(\sigma_2\), \(\sigma_i:V\times V\to\mathbb{C}\), \(f\) is the input function \(f\in L^2(0,T)\). Under some hypotheses, to the above problem one attaches the weak variational form \[ \langle\ddot w(t),\phi\rangle+\sigma_2(\dot w(t),\phi)+\sigma_1(w(t),\phi)=\langle f(t),\Phi\rangle \] for all \(\Phi\in V\), \(w(0)=w_0\), \(w(1)=w_1\), for which the fundamental existence and uniqueness theorem is established. The solution is given by means of the semidiscrete Galerkin method. For the same problem, the authors give also a semigroup formulation and prove the equivalence of the solutions.

The fifth chapter analyses the parameter estimation and inverse problem for beams (impulse hammer excitation and piezoceramic patch sensing, piezoceramic patch excitation and accelerometer sensing, piezoceramic patch excitation and sensing) and plates for 5 experiments; each example is extensively studied achieving numerical and graphical results.

The sixth chapter is devoted to damage detection in smart material structures, where the presentation of the detection technique is followed by experimental and numerical results for beams.

In the last three chapters the authors study respectively the following problems: infinite-dimensional control problem and Galerkin approximation, implementation of finite-dimensional compensators and modeling and control in coupled systems. The principal results presented by the authors in these chapters are: (chapter 7) abstract formulation of the control problem, the infinite-dimensional linear quadratic regulator (LQR), finite horizon control problem, infinite horizon control problem, approximate LQR control problem with no exogenous input (in the finite and infinite case) and LQR control problems for systems with exogenous inputs; (chapter 8) continuous-time compensator problem, discrete-time compensator problem, control of circular plate vibrations, and implementation issues regarding partial differential equations based compensators; (chapter 9) modeling a structural acoustic system, noise attenuation in a three-dimensional cavity, noise attenuation in a two-dimensional model and acoustic units and scales.

From the above presentation of the contents of the monograph one sees that a large field of investigation in this new domain is treated. The authors are known specialists in this domain. The quality of the text is very well underlined by the good structure of the material, by the approach to treat each problem, theoretical and applied, by the high theoretical level of mathematical problems and by the special attention payed to numerical and experimental applications. The presentation of the material is clear, precise and rigorous at the same time. The book is a useful tool for all researchers (beginners or advanced) in this domain.

The authors themselves, especially the first one, are important contributors to the domain, with more than 50 papers quoted in the bibliography. The content is structured in 9 chapters presenting the results of other authors or their own ones.

The first chapter presents a number of smart materials of current interest in research and in applied engineering: piezoelectric and electrostrictive elements, magnetostrictive transducers, electrorheological fluids, shape memory alloys and fiber optic sensors.

The second and the third chapter are devoted to the presentation of structural modeling of beams, plates, shells and details for inclusion of both inactive and active dynamic effects of surface-mounted piezoceramics on these structures. Both chapters are of theoretical character where the movement and equilibrium equations are established and the corresponding boundary conditions in different systems of coordinates are formulated.

The fourth chapter is the central chapter of the volume and analyses well-formulated boundary problems for abstract structural models. The authors study the existence, the uniqueness and the continuous dependence of the solution on initial and boundary conditions for second-order in time partial differential equations describing the behaviour of material systems presented in the earlier chapters. For each analyzed model a variational formulation is given (weak formulation) which constitutes the mathematical framework for the determination of the material systems parameters, analyzed minutely in the following chapter, i.e. the fifth. The general problem studied in the fourth chapter refers to the second-order (in time) system of equations having the form \[ \ddot w(t)+A_1\dot w(t)+A_2w(t)=f(t),\text{ in }V^*,\;w(0)=w_0,\;\dot w(0)=w_1, \] where \(V^*\) is the dual space of \(V\) – the space of unknown functions with the scalar product \(\langle, \rangle\), dense in a complex Hilbert space \(H\), \(A_1\) and \(A_2\) are operators defined by the sesquilinear forms \(\sigma_1\) and \(\sigma_2\), \(\sigma_i:V\times V\to\mathbb{C}\), \(f\) is the input function \(f\in L^2(0,T)\). Under some hypotheses, to the above problem one attaches the weak variational form \[ \langle\ddot w(t),\phi\rangle+\sigma_2(\dot w(t),\phi)+\sigma_1(w(t),\phi)=\langle f(t),\Phi\rangle \] for all \(\Phi\in V\), \(w(0)=w_0\), \(w(1)=w_1\), for which the fundamental existence and uniqueness theorem is established. The solution is given by means of the semidiscrete Galerkin method. For the same problem, the authors give also a semigroup formulation and prove the equivalence of the solutions.

The fifth chapter analyses the parameter estimation and inverse problem for beams (impulse hammer excitation and piezoceramic patch sensing, piezoceramic patch excitation and accelerometer sensing, piezoceramic patch excitation and sensing) and plates for 5 experiments; each example is extensively studied achieving numerical and graphical results.

The sixth chapter is devoted to damage detection in smart material structures, where the presentation of the detection technique is followed by experimental and numerical results for beams.

In the last three chapters the authors study respectively the following problems: infinite-dimensional control problem and Galerkin approximation, implementation of finite-dimensional compensators and modeling and control in coupled systems. The principal results presented by the authors in these chapters are: (chapter 7) abstract formulation of the control problem, the infinite-dimensional linear quadratic regulator (LQR), finite horizon control problem, infinite horizon control problem, approximate LQR control problem with no exogenous input (in the finite and infinite case) and LQR control problems for systems with exogenous inputs; (chapter 8) continuous-time compensator problem, discrete-time compensator problem, control of circular plate vibrations, and implementation issues regarding partial differential equations based compensators; (chapter 9) modeling a structural acoustic system, noise attenuation in a three-dimensional cavity, noise attenuation in a two-dimensional model and acoustic units and scales.

From the above presentation of the contents of the monograph one sees that a large field of investigation in this new domain is treated. The authors are known specialists in this domain. The quality of the text is very well underlined by the good structure of the material, by the approach to treat each problem, theoretical and applied, by the high theoretical level of mathematical problems and by the special attention payed to numerical and experimental applications. The presentation of the material is clear, precise and rigorous at the same time. The book is a useful tool for all researchers (beginners or advanced) in this domain.

Reviewer: S.Zanfir (Craiova)

### MSC:

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

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

93B30 | System identification |

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