We derive exact and approximate controllability conditions for the linear one-dimensional heat equation in an infinite and a semi-infinite domains. The control is carried out by means of the time-dependent intensity of a point heat source localized at an internal (finite) point of the domain. By the Green’s function approach and the method of heuristic determination of resolving controls, exact controllability analysis is reduced to an infinite system of linear algebraic equations, the regularity of which is sufficient for the existence of exactly resolvable controls. In the case of a semi-infinite domain, as the source approaches the boundary, a lack of L2-null-controllability occurs, which is observed earlier by Micu and Zuazua. On the other hand, in the case of infinite domain, sufficient conditions for the regularity of the reduced infinite system of equations are derived in terms of control time, initial and terminal temperatures. A sufficient condition on the control time, heat source concentration point and initial and terminal temperatures is derived for the existence of approximately resolving controls. In the particular case of a semi-infinite domain when the heat source approaches the boundary, a sufficient condition on the control time and initial temperature providing approximate controllability with required precision is derived.
This work describes a new study to achieve a combination of modified function projective synchronization between three different chaotic systems through adaptive control. Using the Lyapunov function theory, the asymptotic stability of the error dynamics is obtained and discussed. Further, we set some appropriate initial conditions for the state variables and assigning specific values to the parameters and obtain the graphical results, which shows the efficiencies of the new method. Finally, we summarized our work with conclusion and references.
Automation of data processing of contactless diagnostics (detection) of the technical condition of the majority of nodes and aggregates of railway transport (RWT) minimizes the damage from failures of these systems in operating modes. This becomes possible due to the rapid detection of serious defects at the stage of their origin. Basically, in practice, the control of the technical condition of the nodes and aggregates of the RWT is carried out during scheduled repairs. It is not always possible to identify incipient defects. Consequently, it is not always possible to warn personnel (machinists, repairmen, etc.) of significant damage to the RWT systems until their complete failure. The difficulties of obtaining diagnostic information is that there is interdependence between the main nodes of the RWT. This means that if physical damage occurs at any of the RWT nodes, in other nodes there can also occur malfunctions. As the main way to improve the efficiency of state detection of the nodes and aggregates of RWT, we see the direction of giving the adaptability property for an automated data processing system from various contactless diagnostic information removal systems. The global purpose can be achieved, in particular, through the use of machine learning methods and failure recognition (recognition objects). In order to improve the operational reliability and service life of the main nodes and aggregates of RWT, there are proposed an appropriate model and algorithm of machine learning of the operator control system of nodes and aggregates. It is proposed to use the Shannon normalized entropy measure and the Kullback-Leibler distance information criterion as a criterion of the learning effectiveness of the automated detection system and operator node state control of RWT. The article describes the application of the proposed method on the example of an automatic detection system (ADS) of the state of a traction motor of an electric locomotive. There are given the test data of the model and algorithm in the MATLAB environment.