Regularization of inclusions of differential equations solutions based on the kinematics of a vector field in stability problems

This paper presents the new results of computing the inclusions of sets of solutions of ordinary differential equations corresponding to perturbations acting on the solutions of the system. The regularization of algorithms for the inclusion of solutions is investigated. Inclusions of solutions are used to study the stability over a finite time interval under perturbations of the system parameters. The boundaries of solutions sets using methods that construct symbolic formulas that characterize the behavior of the system are computed. In this case, the Influence of permanent disturbances on the solutions is taken into account. It is proposed to use the parameters of the vector field of the problem in order to compensate for a strong growth of the computable solution boundaries, which is often encountered in many methods. This means the regularization of the problem of estimating inclusions of solution sets.

Algorithm for Finding Feedback in a Problem with Constraints for One Class of Nonlinear Control Systems

For a continuous nonlinear control system on a finite time interval with control constraints, where the right-hand side of the dynamics equations is linear in control and linearizable in the vicinity of the zero equilibrium position, we consider the construction of a feedback according to the Kalman algorithm. For this, the solution of an auxiliary optimal control problem with a quadratic functional is used by analogy with the SDRE approach.Since this approach is used in the literature to find suboptimal synthesis in optimal control problems with a quadratic functional with formally linear systems, where all coefficient matrices in differential equations and criteria can contain state variables, then on a finite time interval it becomes necessary to solve a complicated matrix differential Riccati equations, with state-dependent coefficient matrices. This circumstance, due to the nonlinearity of the system, in comparison with the Kalman algorithm for linear-quadratic problems, significantly increases the number of calculations for obtaining the coefficients of the gain matrix in the feedback and for obtaining synthesis with a given accuracy. The proposed synthesis construction algorithm is constructed using the extension principle proposed by V. F. Krotov and developed by V. I. Gurman and allows not only to expand the scope of the SDRE approach to nonlinear control problems with control constraints in the form of closed inequalities, but also to propose a more efficient computational algorithm for finding the matrix of feedback gains in control problems on a finite interval. The article establishes the correctness of the application of the extension principle by introducing analogs of the Lagrange multipliers, depending on the state and time, and also derives a formula for the suboptimal value of the quality criterion. The presented theoretical results are illustrated by calculating suboptimal feedbacks in the problems of managing three-sector economic systems.

Circuits with a mem-element: invariant manifolds control via pulse programmed sources

The paper considers the problem of controlling multistability in a general class of circuits composed of a linear time-invariant two-terminal (one port) element, containing linear R, L, C components and ideal operational amplifiers, coupled with one of the mem-elements (memory elements) introduced by Prof. L.O. Chua, i.e., memristors, memcapacitors, and meminductors. First, explicit expressions of the invariant manifolds of the circuit are directly given in terms of the state variables of the two-terminal element and the mem-element. Then, the problem of steering the circuit dynamics from an initial invariant manifold to a final one, and hence to potentially switch among different attractors of the circuit, is addressed by designing pulse shaped control inputs. The control inputs ensure that the transition between the initial and final manifolds is accomplished within a given finite time interval. Moreover, it is shown how the designed control inputs can be implemented by introducing independent voltage and current sources in the two-terminal element. Notably, it turns out that it is always possible to solve the considered control problem by using a unique independent source. Several examples are provided to illustrate how the proposed approach can be applied to different circuits with mem-elements and to highlight the influence of the features of the designed sources on the behavior of the controlled dynamics.

CONTROL SYSTEM DEPENDING ON A PARAMETER

A nonlinear control system depending on a parameter is considered in a finite-dimensional Euclidean space and on a finite time interval. The dependence on the parameter of the reachable sets and integral funnels of the corresponding differential inclusion system is studied. Under certain conditions on the control system, the degree of this dependence on the parameter is estimated. Problems of targeting integral funnels to a target set in the presence of an obstacle in strict and soft settings are considered. An algorithm for the numerical solution of this problem in the soft setting has been developed. An estimate of the error of the developed algorithm is obtained. An example of solving a specific problem for a control system in a two-dimensional phase space is given.

Time Lag-Based Modelling for Software Vulnerability Exploitation Process

With the increase in the discovery of vulnerabilities, the expected exploits occurred in various software platform has shown an increased growth with respect to time. Only after being discovered, the potential vulnerabilities might be exploited. There exists a finite time lag in the exploitation process; from the moment the hackers get information about the discovery of a vulnerability and the time required in the final exploitation. By making use of the time lag approach, we have developed a framework for the vulnerability exploitation process that occurred in multiple stages. The time lag between the discovery and exploitation of a vulnerability has been bridged via the memory kernel function over a finite time interval. The applicability of the proposed model has been validated using various software exploit datasets.