Existing online continuous-time parameter estimation laws provide exact (asymptotic/exponential or finite/fixed time) identification of dynamical linear/nonlinear systems parameters only if the external perturbations are vanishing to zero or independent with the regressor of the system. However, in real systems the disturbances are almost always nonvanishing and dependent with the regressor. In the presence of perturbations with such properties the abovementioned identification approaches ensure only boundedness of a parameter estimation error. The main goal of this study is to close this gap and develop a novel online continuous-time parameter estimator that guarantees the exact asymptotic identification of unknown parameters of linear systems in the presence of unknown but bounded perturbations and has relaxed convergence conditions. To achieve the aforementioned goal, it is proposed to augment the deeply investigated dynamic regressor extension and mixing (DREM) procedure with the novel instrumental variables (IV)-based extension scheme with averaging. Such an approach allows one to obtain a set of scalar regression equations with asymptotically vanishing perturbation if the initial disturbance that affects the plant is bounded and independent not with the system regressor, but with the IV. It is rigorously proved that a gradient estimation law designed on the basis of such scalar regressions ensures online unbiased asymptotic identification of the parameters of the perturbed linear systems if some weak independence and excitation assumptions are met. Theoretical results are illustrated and supported with adequate numerical simulations.

The restricted circular three-body problem is studied. All global families of periodic orbits adjacent to the libration points are found. A scenario for the evolution of orbits in the family is given. Chains of global families are highlighted; the chain begins at the triangular libration point, contains global families for the triangular and all collinear libration points, and ends with a family whose orbits are pressed against the main bodies. The evolution of global families in the chain associated with changes in the energy of the system is described. Planar and spatial orbits are studied.

We consider a class of well-known high-order trinomial linear difference equations and analyze the non-asymptotic behavior of their solutions under non-zero initial conditions from the unit box. It is shown that, for certain subsets of coefficients in the stability domain, there always exist initial conditions leading to \emph{peak}, a large deviation of solutions from the equilibrium position, and that these deviations may take arbitrarily large values. Various special cases are studied, numerical examples are presented.

This paper deals with a generalized mathematical model of a controlled belt con- veyor with a variable angle between the horizontal and belt planes. The model is defined using a system of four nonlinear differential equations with switching. It includes the linear movement of the conveyor belt, changes in the system momentum, axial and linear friction, the damping of the horizontal position of the conveyor, and the factors of smooth loading and instant unloading of cargo. Stabilization conditions are established for this model considering simulation components related to the nature of loading and unloading modes of the conveyor belt. A PID controller, a neuro-PID controller, and neural network controllers of recurrent and non-recurrent types are designed to control the angular position of the conveyor. Linear velocity control is implemented by introducing a sliding mode. Computational experiments are carried out and given an interpretation. The performance of the controllers mentioned above is comparatively analyzed.

The paper considers the geometry of the Modified Solid–Liquid–Vapor equation of thermodynamic state. This model describes a substance state in three phases. Thermodynamics states are points on Legendrian or Lagrangian manifolds in the corresponding contact or symplectic spaces in terms of differential geometry. The conditions of admissible states and the first order phase transition are got for the Modified Solid–Liquid–Vapor equation of state. The Lagrangian manifold, singularity curve and the phase transition curves are plotted for methane.