Regularity of multipliers and second-order optimality conditions of KKT-type for semilinear parabolic control problems

A class of optimal control problems governed by semilinear parabolic equations with mixed constraints and a box constraint for control variable is considered. We show that if the separation condition is satisfied, then both optimality conditions of KKT-type and regularity of multipliers are fulfilled. Moreover, we show that if the initial value is good enough and boundary $\partial\Omega$ has a property of positive geometric density, then multipliers and optimal solutions are H\"{o}lder continuous.Comment: arXiv admin note: substantial text overlap with arXiv:2306.1029

Regularity of multipliers and optimal solutions in second-order optimality conditions for semilinear parabolic optimal control problems

A class of optimal control problems governed by semilinear parabolic equations with mixed pointwise are considered. We give some criteria under which Lagrange multipliers belong to $L^p$-spaces. We then establish first and second-order optimality conditions of KKT-type for the problem. Moreover, we show that if the initial value is good enough and boundary $\partial\Omega$ has a property of positive geometric density, then multipliers and optimal solutions are H\"{o}lder continuous

On almost subnormal subgroups in division rings

Let $D$ be a division ring with infinite center $F$, and $G$ an almost subnormal subgroup of $D^*$. In this paper, we show that if $G$ is locally solvable, then $G\subseteq F$. Also, assume that $M$ is a maximal subgroup of $G$. It is shown that if $M$ is non-abelian locally solvable, then $[D:F]=p^2$ for some prime number $p$. Moreover, if $M$ is locally nilpotent then $M$ is abelian.Comment: arXiv admin note: text overlap with arXiv:1809.0035

State Space Reduction on Wireless Sensor Network Verification Using Component-Based Petri Net Approach

With the recent advancement of Internet of Things, the applications of Wireless Sensor Networks (WSNs) are increasingly attracting attention from of both industry and research communities. However, since the deployment cost of a WSN is relatively large, one would want to make a logic model of a WSN and have the model verified beforehand to ensure that the WSN would work correctly and effectively once practically employed. Petri Net (PN) is very suitable to model a WSN, since PN strongly supports modeling concurrent and ad-hoc systems. However, verification of a PN-modeled system suffers from having to explore the huge state space of the system. In order to overcome it, in this paper we suggest a novel component-based approach to model and verify a PN-modeled WSN system. First of all, the original WSN system is divided into components, which can be further abstracted to reduce the model size. Moreover, when verifying the corresponding PN model produced from the abstracted WSN, we introduce a strategy of component-based firing, which can reduce the state space significantly. Compared to typical approach of PN-based verification, our method enjoys an impressive improvement of performance and resource consuming, as depicted in our experimental results

Asymptotically accurate and locking-free finite element implementation of first order shear deformation theory for plates

A formulation of the asymptotically exact first-order shear deformation theory for linear-elastic homogeneous plates in the rescaled coordinates and angles of rotation is considered. This allows the development of its asymptotically accurate and shear-locking-free finite element implementation. As applications, numerical simulations are performed for circular and rectangular plates, showing complete agreement between the analytical solution and the numerical solutions based on two-dimensional theory and three-dimensional elasticity theory.Comment: 27 pages, 7 figure