Subject: a groundwater filtration affects the strength and stability of underground and hydro-technical constructions. Research objectives: the study of one-dimensional problem of displacement of suspension by the flow of pure water in a porous medium. Materials and methods: when filtering a suspension some particles pass through the porous medium, and some of them are stuck in the pores. It is assumed that size distributions of the solid particles and the pores overlap. In this case, the main mechanism of particle retention is a size-exclusion: the particles pass freely through the large pores and get stuck at the inlet of the tiny pores that are smaller than the particle diameter. The concentrations of suspended and retained particles satisfy two quasi-linear differential equations of the first order. To solve the filtration problem, methods of nonlinear asymptotic analysis are used. Results: in a mathematical model of filtration of suspensions, which takes into account the dependence of the porosity and permeability of the porous medium on concentration of retained particles, the boundary between two phases is moving with variable velocity. The asymptotic solution to the problem is constructed for a small filtration coefficient. The theorem of existence of the asymptotics is proved. Analytical expressions for the principal asymptotic terms are presented for the case of linear coefficients and initial conditions. The asymptotics of the boundary of two phases is given in explicit form. Conclusions: the filtration problem under study can be solved analytically.

**Introduction. **The gained experience in the field of building automation and IoT technologies yields a new approach to the management of engineering subsystems that provides stated parameters of operation quality throughout the entire building lifecycle. This paper explores compensatory and predictive algorithms in the scope of the aforementioned approach to manifest control over building climate parameters utilizing IoT controllers. This research aims to improve the management efficiency of smart house engineering subsystems through the implementation of a control system (CS) capable to compensate disturbances and predict their variations using an IoT controller and an analytical server.

**Materials and methods. **In order to improve the quality of control, various algorithms based on analysis of data collected from controllers can be employed. The collected data about the object accumulated over the entire period of operation can be used to build a model for the purposes of predictive control. The predictive control allows forecasting the parameters having an effect on the object and compensating it beforehand under the inertia conditions. The continuous adaptation and adjustment of the CS model to operating conditions allows permanent optimizing the settings of the control algorithm ensuring the efficient operation of local control loops.

**Results. **The CS is based on an IoT controller and able to predict and compensate potential disturbances. The compensation algorithm is updated depending on the behavior of the object properties, quality of control and availability of data most suitable for identification.

**Conclusions. **The capabilities of the control system based on the IoT controller and generation of a compensatory and predictive control signal with the algorithm hosted at a cloud server are demonstrated on the indoor temperature control model. The following simulation models of the indoor temperature variation process are considered: model without CS, model with proportional plus integral controller with disturbance compensation and model with IoT controller-based CS with disturbance compensation. Structural and parametric identification of the models are accomplished by means of active experiment.

The mechanical-geometric model of the suspension filtering in the porous media is considered. Suspended solid particles of the same size move with suspension flow through the porous media - a solid body with pores - channels of constant cross section. It is assumed that the particles pass freely through the pores of large diameter and are stuck at the inlet of pores that are smaller than the particle size. It is considered that one particle can clog only one small pore and vice versa. Particles stuck in the pores remain motionless and form a deposit. The concentrations of suspended and retained particles satisfy a quasilinear hyperbolic system of partial differential equations of the first order, obtained as a result of macro-averaging of micro-stochastic diffusion equations. Initially the porous media contains no particles and both concentrations are equal to zero; the suspension supplied to the porous media inlet has a constant concentration of sus-pended particles. The flow of particles moves in the porous media with a constant speed, before the wave front the concentrations of suspended and retained particles are zero. Assuming that the filtration coefficient is small we construct an asymptotic solution of the filtration problem over the concentration front. Terms of the asymptotic expansions sat-isfy linear partial differential equations of the first order and are determined successively in an explicit form. It is shown that in the simplest case the asymptotics found matches the known asymptotic expansion of the solution near the concentration front.

The problem of filtering a suspension of tiny solid particles in a porous medium is con-sidered. The suspension with constant concentration of suspended particles at the filter inlet moves through the empty filter at a constant speed. There are no particles before the front; behind the front of the fluid flow solid particles interact with of the porous medium. The ge-ometric model of filtration without effects caused by viscosity and electrostatic forces is considered. Solid particles in suspension pass freely through large pores together with the fluid flow and are stuck in the pores that are smaller than the size of the particles. It is con-sidered that one particle can clog only one small pore and vice versa. The precipitated par-ticles form a fixed deposit increasing over time. The filtration problem is formed by the sys-tem of two quasi-linear differential equations in partial derivatives with respect to the con-centrations of suspended and retained particles. The boundary conditions are set at the filter inlet and at the initial moment. At the concentration front the solution of the problem is dis-continuous. By the method of potential the system of equations of the filtration problem is reduced to one equation with respect to the concentration of deposit with a boundary condi-tion in integral form. An asymptotic solution of the filtration equation is constructed near the concentration front. Terms of the asymptotic expansions satisfy linear ordinary differential equations of the first order and are determined successively in an explicit form. For verifi-cation of the asymptotics the comparison with known exact solutions is performed.

A classic pursuit problem is studied in which two material points - a Pursuer and a Pursued - move in a plane at constant speeds. The velocity vector of the Pursued does not change its direction and the velocity vector of the Pursuer turns and always aims at the Pursued. If the Pursuer moves at a higher speed, it will overtake the Pursued for any initial angle between velocity vectors. For example, a crane system simultaneously producing three movements: rotation, extension/retraction and luffing of the telescopic boom may seize the load moving uniformly in a straight line while the crane is standing motionless. In the coordinate system associated with the Pursuer, the length of the Pursued's trajectory is given by an integral, depending on two parameters: the ratio of the initial velocities of two points and the initial angle between them. The theorem on the asymptotic expansion of this integral Is formulated and proved under the assumption that the speed of the Pursuer is much greater than the speed of the Pursued. The first two nonzero terms of the asymptotic expansion provide fast convergence to the exact value of the integral because of the absence of the first- and the third-order terms of the asymptotics. The third nonzero term of the fifth order allows to determine the difference of trajectory lengths corresponding to the adjacent initial angles between the velocities of the points.

A classic pursuit problem is studied in which two material points - a Pursuer and a Pursued - move in a plane at constant speeds. The velocity vector of the Pursued does not change its direction and the velocity vector of the Pursuer turns and always aims at the Pursued. If the Pursuer moves at a higher speed, it will overtake the Pursued for any initial angle between velocity vectors.

The shape of the mechanical trajectory is established. The trajectory line rotates about the origin so that at the final meeting point the tangent line to the motion trajectory always coincides with the velocity vector of the Pursued. The two-parameter integral for the length of the pursuit curve is considered, its asymptotics up to the forth term is calculated under the assumption that the speed of the Pursuer is much greater than the speed of the Pursued. Rapid convergence of the asymptotics to the integral for the trajectory length is provided by the absence of the first and the third terms of the asymptotic expansion. Numerical computation of the trajectory length is compared with the asymptotic formulas. Calculations show that the resulting asymptotics is a good approximation of the integral for the trajectory length, and the fourth term in the asymptotic formulas significantly improves the approximation.

**Subject: **smart house maintenance requires taking into account a number of factors - resource conservation, mitigating working expenditures, safety enhancement, ensuring comfort of leisure and operation. Automation of such engineering systems networks as illumination, climate control, security and communication, may be achieved through utilization of contemporary technologies (e.g. IoT – Internet of Things). However, storing and processing the overwhelmingly massive corpora of data produced by the aforementioned systems poses a significant challenge. It is necessary to rationally manage the available big data during the stage of information modelling, due to the fact, that a building’s lifespan outlives most iterations of safety, comfort, and maintenance standards substantially.

**Materials and methods: **since smart houses may be classified as human-machine systems, the cybernetic approach will be considered as the base method of information system design and discovery. Instrumental methods are represented by set-theoretical modelling, automata theory and architectural principles of information management systems’ organization.

**Results: **an agile architecture of information system for smart house hardware management has been synthesized. The architecture encompasses several levels: client level, application level and data level; as well as three layers: presentation level, actuating devices layer and analytics layer. As proposed, the problem of growing volumes of information process by realtime message controller is attended by employment of sensors with configurable thresholds and actuating mechanisms, which implement control logic based on discrete automaton (namely, logical algorithm schemes). Multicircuit control system is suggested to be additionally enhanced with datamining module, DBMS, datamarts, and OLAP cube, which are jointly capable of processing large amount of data produced by hardware subsystems.

**Conclusions: **an information system for smart house hardware management, once built according to the proposed architecture, will enhance the quality of decision-making process, decrease operational costs of the smart house, due to the datamining-enabled control circuit. Suggested solution is recommended to be employed for the management of buildings and constructions, that utilize means of automation and IoT.