Nonlinear boundary-value problems unsolved with respect to the derivative*/Asymptotic Stabilization of the Flexible Beam Oscillations**

*The mathematical description of numerous phenomena in many areas of electronics, theory of nonlinear oscillation, mechanics, biology and radio engineering leads to the necessity of investigating nonlinear partial differential equations. In a particular case, the latter equation leads to nonlinear differential equations unsolved with respect to the derivative. A linearization of such equations, leads to a linear differential-algebraic equation of the form
A(t)z0(t) = B(t)z(t) + f(t):
The monographs of A.M. Samoilenko, M.O. Perestiuk, V.P. Yakovets, O.O. Boichuk, as well as numerous works by foreign authors S. Campbell, J.R. Magnus, V.F. Chistyakov and others are devoted to the study of linear differential-algebraic equations using the central canonical form and perfect pairs and triples of matrices The difference of this report is that the finding of constructive conditions for existence and construction of solutions of nonlinear differential-algebraic boundary-value problems was done without using the central canonical form. This made it possible to study solutions of differential-algebraic boundary-value problems that depend on arbitrary continuous functions. The relevance of the studying nonlinear boundary-value problems unsolved with respect to the derivative is due to the fact that the study of a traditional problem resolved with respect to the derivative is sometimes difficult, in the case of obtaining nonlinearities not integrable in elementary functions. Thus this report is devoted to the study of the problems of finding constructive conditions for existence and construction of solutions of nonlinear boundary-value problems problems unsolved with respect to the derivative. The report will also deal with linear and nonlinear differential-algebraic boundaryvalue problems. The cases of degeneracy and nondegeneracy of differential-algebraic system will be investigated. The classification of nonlinear differential-algebraic boundary-value problems has been improved. The constructive solvability conditions and schemes for constructing solutions of nonlinear differential-algebraic boundary-value problems in critical and noncritical cases are found. Convergent iteration schemes for finding approximations to solutions of nonlinear differential-algebraic boundary value problems are constructed.

** This talk is devoted to the stabilization problem for exible beam oscillations. The considered mechanical system consists of a simply supported exible beam of the length l with collocated piezoelectric actuators and sensors and a controlled spring-mass attached at the point l0 of the beam. The beam's oscillations are modeled as the EulerBernoulli equation with interface and boundary conditions. Mathematical model is derived using Hamilton's principle. The equation of motion is presented in the form of an abstract dierential equation with control
ξ˙ = Aξ + Bu
in the Hilbert space X = H2(0, l)×L2(0, l)×R2 with fourth-order dierential operator A : D(A) → X. It is proved that the operator A is the innitesimal generator of a C0-semigroup in X. A feedback control is obtained in such a way that the total energy is non-increasing on the closedloop system trajectories. It is shown, based on Lyapunov's theorem, that the obtained feedback stabilizes the coupled distributed and lumped parameter system. The resolvent of the innitesimal generator is obtained and is proved to be compact. So, the trajectories of the closedloop system form a precompact set in X.
The main result is the proof that the innite-dimensional closedloop system has asymptotically stable trivial equilibrium under some natural assumptions on the mechanical structure. This result is obtained with the help of LaSalle's invariance principle. The spectral problem is studied for estimating the distribution of the beam's eigenfrequencies. The observation problem for linear system is investigated for nite-dimensional projections of the original dynamical system. A Luenberger-type observer is designed for the system derived by Galerkin's approximations. The observer gain parameters are dened in such a way that the observer properly estimates the complete state vector, i.e. the error dynamics has asymptotically stable trivial equilibrium. Results on the eigenfrequencies distribution and observer convergence are illustrated by numeric simulations.