Volume 9, Issue 3

- The paper presents results related to application of optimal control methods to investigation and solution of a class of inverse problems. Applications of these methods to particular inverse problems are considered.

- The nonlinear term of a semilinear elliptic problem can be uniquely recovered, assuming that for each locally supported right-hand term of the PDE the trace of the corresponding solution is known.

- The classical proof of the Trotter - Kato theorem also provides rate estimates for smooth initial data. We discuss the possibilities to reduce the so-called smoothness gap for special types of semigroups as, for instance, analytic semigroups. Furthermore, we show that the assumption of stability of the approximating semigroups can be relaxed if at the same time the consistency requirement is replaced by a stronger assumption. Finally, we give a proof for a ‘folk’ theorem for nonhomogeneous problems which repeatedly was proved under stronger assumptions than necessary.

- In the present paper we formulate and investigate a one-dimensional inverse acoustic problem in the form of a nonlinear system of Volterra integral equations. We prove the conditional stability of the inverse problem with explicitly given Lipschitz constant depending only on the depth l and on bounds of the norms of the unknown coefficient and inverse problem data. The proposed technique can be applied to prove the local well-posedness of the inverse problem in L 2 (0 , l ).

- The problem of dynamical reconstruction, of an unknown control acting upon a linear time-delay system, through inaccurate observations of phase states is discussed. A regularized solving algorithm functioning in real time mode is proposed. This algorithm is stable with respect to informational noise and computational errors.

- A method of finding solutions for the optimal compatibility problem in Hilbert spaces is suggested. The accuracy estimates for approximate solutions are presented and a constructive regularization algorithm for the ill-posed problem is described.

- Some fundamental results of the modern theory of optimum experimental design are extended here to address the problem of determining the optimal measurement scheduling, encountered while estimating unknown parameters in mathematical models described by partial differential equations from observations of the underlying physical phenomenon being modelled. Special emphasis is put on measurements realized by optimal motion of spatially-movable sensors for which we generalize the approach advanced by Rafajłowicz in his seminal paper [16] to the case of minimizing a general performance index defined on the Fisher information matrix related to the parameters to be identified. Since only the measurability of the resulting trajectories can be guaranteed, we also show how to ease this inconvenience by introducing a suitable parametrization of the set of admissible solutions. In the latter case, we also detail how to adapt standard sequential numerical algorithms of optimum experimental design so that they could be employed for computation of trajectories in particular situations.