Optimal Moving Sensors for Parabolic Systems

This paper continues the investigations in SDS on observability issues motivated by environmental monitoring and related problems. Here the author introduces a specific class of scanning sensors that ensure solvability of the problem and can further lead to numerically robust techniques

Controllability of the Wave Equation with Pointwise Moving Control

The paper deals with approximate and exact controllability of the wave equation with interior pointwise control acting along the curve specified in advance in the system's spatial domain. The structure of the control input is dual to the structure of the observations describing the measurements of velocity and gradient of the solution of the dual system, obtained from the single pointwise moving sensor. A relevant formalization of such a control problem is discussed, based on transposition. The techniques used are based on the observability results involving a priori energy estimates of an instantaneous type

Observability of Parabolic Systems under Scanning Sensors

This paper continues the investigations in SDS on observability issues motivated by environmental monitoring and related problems. Here the author introduces a specific class of scanning sensors that ensure solvability of the problem and can further lead to numerically robust techniques

Source Localization Problem for Parabolic Systems

Two problems for the linear distributed parameter systems of parabolic type motivated by environmental monitoring are discussed: (1) Nonlinear localization problem: recover the location of an unknown single source on the basis of available observations. In general, the solution of this problem is set-valued and disconnected. (2) Identifiability problem: what types of observations are able to ensure enough information to restore the location point? An approach is given, based on the introduction of a suitable space of test-functions: in order to determine the unknown location, one has to analyze a proper system of algebraic equations. The latter can be constructed in advance. Sufficient conditions for identifiability are derived and the duality relations between the above nonlinear problems and the problems of open loop control and controllability for an associated adjoint linear system are established

Exact Observability of Nonstationary Hyperbolic Systems with Scanning Finite-Dimensional Observations

The paper considers a problem of exact observability for nonstationary linear hyperbolic systems with nonstationary interior observations. Three types of scanning observations are considered and the existence of measurement curves and set-valued maps for sensors that ensure exact observability is established. The proposed method is based on a priori energy estimates of instantaneous type for solutions. It allows to obtain required measurement curves that are continuous in a fixed interval or measurement maps that are lower semi-continuous in the interval with respect to Lebesgue measure