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The problem of the brachistochronic motion of a mechanical system composed of rigid bodies and variable-mass particles is solved. The laws of the time-rate of mass variation of the particles as well as relative velocities of the expelled (or gained) masses are assumed to be known. The system moves in an arbitrary field of known potential and nonpotential forces. Applying Pontryagin's minimum principle along with singular optimal control theory, a corresponding two-point boundary value problem is obtained. The appropriate numerical procedure based on the shooting method to solve the obtained two-point boundary value problem is presented. The considerations in the paper are illustrated by an example of determining the brachistochronic motion of a system composed of a rigid rod and two variable-mass particles attached to the rod.

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Machines operating in cycles and their properties have not been studied in depth in literature and, as such, are not well described by integral mathematical models. If the effects of their operation are actions on the given working surfaces under given constraints, then the quality of the affected surfaces can be described by reliability functions. In this manner, the operating capabilities of the platform can be determined. A majority of the published papers use a standard approach to the measured performances that depend on the machine's designed purposes. Such processes are described in [1] to [3] for the abrasive flow machines (AFM) with which material is hardened by randomly treating the working surface with abrasive particles with polymeric fillers, and dispersed within the flow media. The authors of [1] classified the work piece parameters into three groups based, among others, on the number of cycles (operations) and the machining time. Some of these parameters were determined experimentally in [2], in which the authors recognized that the parameters denoted as the creeping time and the cycles frequency have impact on the quality of the machining process. In [3], the authors experimentally prove that the aforementioned parameters influence the process. Common for all three papers is that they do not include hidden random effects caused by particles affecting the surfaces in cyclic operations, although such effects significantly influence the quality of the surface treatment. In all three papers, there is no mathematical modelling of the process.Another similar type of machine with cyclic operation affecting working surfaces is described in [4] as shot-peening (SP) platforms. They bombard a surface with spherical beads to increase the material fatigue strength. The physical modelling of the influence of the bead shapes on the performance of the surface hardening process is presented in [5]. Random surface effects due to bombing cycles are a result of the quality of the machine's performance. However, the connection between the effects and the particular operations is missing in • Proposed methodology for performances degradation caused by operations composed in cycles.• Using Gaussian probability distribution law to predict degradation measures.• Predicting the changes of probability dispersion based on modelling and experimental data.• Determination of an analytical model based on a hypothesis that the degrading effects are a function of the platform capacity, frequency of operations and the number of available cycles.

The paper deals with mathematical models of sequent salvos battle, of autonomous flight missiles (AFM) organized in the groups of combatants. Tactical integration of AFM system distance-controlled weapon is considered by performances of simultaneous approaches on targets, and continual battle models of guerilla and direct fire, are redesigned to the discrete-continual mixed model, for checking missiles sudden, and further salvos, attack effects. Superiority parameters, as well as losses and strengths of full, or the part of salvo battle, for the missiles groups as technology sub-systems combatants', is expressed by mathematical and simulation examples. Targets engagements capacities of the missiles battle unit, is conducted through designed scenarios and mathematically derived in the research. Model orientated on answers about employment of rapid reaction defending tactics, by distance missiles attacks.