We consider a linear pursuit game of one pursuer and one evader whose motions are described by different-type linear discrete systems. Position of the evader satisfies phase constraints: y∈G, where G is a subset of Rn. We considered two cases: (1) controls of the players satisfy geometric constraints, and (2) controls of the players satisfy total constraints. Terminal set M is a subset of Rn and it is assumed to have a nonempty interior. Game is said to be completed if y(k)-x(k)∈M at some step k; thus, the evader has not the right to leave set G. To construct the control of the pursuer, at each step i, we use the value of the control parameter of the evader at the step i. We obtain sufficient conditions of completion of pursuit from certain initial positions of the players in finite time interval and construct a control for the pursuer in explicit form.
Heterogeneous parallel architecture (HPA) are inherently more complicated than their homogeneous counterpart. HPAs allow composition of conventional processors, with specialised processors that target particular types of task. However, this makes mapping and scheduling even more complicated and difficult in parallel applications. Therefore, it is crucial to use a robust modelling approach that can capture all the critical characteristics of the application and facilitate the achieving of optimal mapping. In this study, we perform a concise theoretical analysis as well as a comparison of the existing modelling approaches of parallel applications. The theoretical perspective includes both formal concepts and mathematical definitions based on existing scholarly literature. The important characteristics, success factors and challenges of these modelling approaches have been compared and categorised. The results of the theoretical analysis and comparisons show that the existing modelling approaches still need improvement in parallel application modelling in many aspects such as covered metrics and heterogeneity of processors and networks. Moreover, the results assist us to introduce a new approach, which improves the quality of mapping by taking heterogeneity in action and covering more metrics that help to justify the results in a more accurate way.
In this work, the null controllability problem for a linear system in ℓ2 is considered, where the matrix of a linear operator describing the system is an infinite matrix with λ∈ℝ on the main diagonal and 1s above it. We show that the system is asymptotically stable if and only if λ ≤- 1, which shows the fine difference between the finite and the infinite-dimensional systems. When λ ≤- 1 we also show that the system is null controllable in large. Further we show a dependence of the stability on the norm, i.e. the same system considered ℓ∞ is not asymptotically stable if λ = - 1.