The author starts this paper with a general review of the literature. This is a difficult task because of the great popularity of the subject, i.e. of the behavior and controllability of the Timoshenko beam. She mentions the controllability and stabilizability work on the same topic of Lagnese and his collaborators, which includes Irena Lasiecka and her students. Generally, most control theoreticians follow the state space analysis, whereby the relation between input and output of a controlled system is described through the state of the system, which in turn is dictated by the evolution process, described by the transformation semigroups. The beam is inhomogeneous. The control problems are greatly simplified if one assumes the separability of the forcing terms \(u_i (x, t)=g_i(x)f_i (t)\), \(i=1,2\), where the spacial profile \(g_i(x)\) is known, while the \(f_i(t)\) terms are unknown functions or generalized functions. Dirac delta (point load) and its first derivative (point moment) should not be excluded. Several dynamic models have been adopted under the name of Timoshenko’s beam. For example, V. I. Korobov, W. Krabs and G. M. Sklyar [J. Optimization Theory Appl. 107, No. 1, 51-68 (2000; Zbl 0971.93039)] considered the Timoshenko beam with the added effect of slow rotation, while Lasiecka and Lagnese dealt with control problems in their series of articles on the Timoshenko beam, using equations similar to this author, but somewhat simplified.
The reviewer appreciates the author’s inclusion of variable coefficients: \(\rho\)-density (the reviewer would prefer a different definition of density: \(A\rho(x)\), where \(A(x)\) is the cross-sectional area, which would be consistent with the variable moment of inertia of the cross-sectional area \(I(x)\), but this is only a minor detail); \(EI\) is the bending rigidity, \(K\) the shear rigidity, \(I_p\) the polar second moment of inertia. This detailed analysis is superior to the practice used by many mathematicians, and hated by engineers, of setting all of these variable coefficients equal to unity. Not only does such practice of equating all the coefficients to unity limit the results to beams with constant cross-sectional geometry, but also imposes severe limitations on choices of such cross-sections. But, in the author’s formulation, one can expand such an investigation to one also dealing with improvements in the design of the beam with respect to weight, or some form of stability, etc.…while at the same time satisfying controllability, observability, or other control criteria. The variables \(w,\varphi\) denote respectively: transverse displacement and slope due to bending at position \(x\) and time \(t\).
The author begins with the following mathematical model: \[ \begin{matrix} \rho w_{tt}- (Kw')'+ (K\varphi)'= g_1(x)f_1(t)\\ I_\rho \varphi_{tt} -(EI\varphi')' +K(\varphi-w') =0,\end{matrix} \begin{matrix} \quad('\equiv \partial/ \partial x),\;-\ell\geq x\geq 0,\;t\geq 0,\\ I_\rho= RK\neq EI,\end{matrix} \tag{1} \] with \(w(-\ell,t)= \varphi(-\ell,t) =0\) (clamped at the left end).
The boundary conditions at the right end \(\{x\in (0,t)\}\) are \(K\varphi- w_x=\alpha w_t\) and \(EI\varphi_x= -\beta\varphi_t\).
The energy terms are the traditional strain energy and kinetic energy terms plus the integral of \(K|\varphi -w'|^2\), which resembles variational formulas of K. Washizu in his virtual work interpretation of minimum energy principles.
The main results of the author consist of deriving explicit formulas for optimal control bringing the beam to rest at a specified time. The techniques are a continuation of previously introduced ones by this author. The introduction of the energy space produces a first-order operator formalism. This closed, dissipative and generally non-selfadjoint \(4\times 4\) matrix operator \(L_{\alpha\beta}\), obeying non-selfadjoint boundary conditions, is defined on an energy (state) space \(H\). Under some smoothness conditions, \(L_{\alpha\beta}\) is the generator of a strongly continuous semigroup of transformations. Since the resolvent is compact, it follows that its spectrum is discrete.
The author proceeds to give a detailed analysis of the properties of the spectral operator \(L_{\alpha \beta}\) and produces a set of generalized eigenfunctions of this operator with the following properties: They form a Riesz basis in the energy space \(H\), and so do the generalized eigenfunctions of the adjoint operator, and they are biorthogonal to each other in that space. If the spectrum is simple (no multiplicities of eigenvalues), then the families of non-harmonic exponentials \([\exp (i\lambda_n^{-\alpha} t)\cup \exp(i\lambda_n^{-\beta} t)]\) form the Riesz basis of: \(L^2(0,T+ \check T)\), where \(T=2 (\rho/K)^{1/2}\) and \(\check T=2(R \rho/EI)^{1/2}\). This is not surprising since the quantities \(2(K/\rho)^{1/2}\) and \(2(EI/R \rho)^{1/2}\) are proportional (there is a factor of \(2\pi)\) to the speeds of the shear and compression elastic waves, respectively. Thus a purely engineering argument produces such minimal lengths of time in which information can be communicated. This intuitive argument would suggest that the time period \(([0,T +\check T])\) could somehow be replaced by \(\max\{[0,T], [0,\check T]\}\) if these two branches \((\alpha\) and \(\beta)\) had common points. But the author excluded that possibility. The important result of this paper asserts that on such a time period, the Timoshenko beam is controllable, and the minimum time period allowing controllability is \([0,T+ \check T]\). Moreover she shows that infinitely many controls attaining this optimal result exist, and then she produces the exact controllability formulas for the control functions. Her proofs use the moment theorem to establish uniqueness, if the profiles \(g_1(x)\) and \(g_2(x)\) are non-trivial. But this review not only acknowledges that the specific problem solved here is important, but also that the techniques she used in the proofs are worthy of a careful study.
This is an excellent paper, establishing deep results concerning a specific case of controllability of the vibrating inhomogeneous Timoshenko beam. However, her technique is even more important, because, in the reviewer’s opinion, it can be easily extended to other cases in applied mechanics, and to more general physical systems.