Stabilization of a non-linear distributed parameter vibratory system

The feedback stabilization problem for a non-linear elastic distributed parameter system including a destabilizing effect is studied. The model is formulated as an evolution equation in a Hilbert space. Semigroup theory is used to investigate an initial value problem for the equation. Moreover, a control system with a finite dimensional controller is constructed. It is then proved that the state of the overall system converges to zero with a desirable exponential decay rate as time tends to infinity.     

Existence and algorithm of solutions for mixed quasi-variational-like inclusions in Banach spaces

In this paper, we study a class of mixed quasi-variational-like inclusion with a bifunction in reflexive Banach spaces. By applying a minimax inequality due to the first author, some existence and uniqueness theorems of solutions for the mixed quasi-variational-like inclusions are proved. Next, by applying the auxiliary variational inequality technique, we suggest an iterative algorithm to compute the approximate solutions of the mixed quasi-variational-like inclusions. The convergence criteria of the algorithm is studied under quite mild conditions. Our results represent improvement and generalization of the previously known results in literature.     

Local stability of trusses in the context of topology optimization Part I: Exact modelling

The paper considers the problem of optimal truss topology design with respect to stress and local stability (i.e. buckling) constraints. In a context of topology optimization, the exact. management of buckling constraints is highly complex: member forces must satisfy functions which discontinuously depend on the design variables.New terminologies and an exact problem formulation are provided. It turns out that the classical constraints (equilibrium, stress) together with topological local buckling constraints do not necessarily guarantee the existence of a solution structure. We discuss a simple but typical example demonstrating this effect inherently contained in the problem. It is proved that the inclusion of slenderness constraints guarantees a solution. These additional constraints are motivated by practice and preserve the topology nature of the problem. Finally, an alternative formulation is developed serving as a basis for computational approaches. The numerical treatment is the topic of Part II.     

Numerical solutions of fourth order variational inequalities

We use variational inequality theory along with finite difference technique to obtain an approximation for the solution of a class of obstacle problem in elasticity, like those describing the equilibrium configuration of an elastic stretched over an elastic obstacle. The variational inequality formulation is used to discuss the problem of uniqueness and existence of the solution of the obstacle problems.     

Measurement optimization with sensitivity criteria for distributed parameter systems

We consider the problem of optimally designing sensors for observation of a class of distributed parameter systems. The design of sensors concerns the choice of measurement conditions so that the information provided by measurements is maximal. This problem has been posed as a deterministic optimal control problem for a system equation of the Riccati type which governs a filter covariance. In the present study we introduce a functional called a sensitivity criterion by extending the Fisher information matrix to function spaces. It is shown that maximizing this criterion leads to a suboptimal solution of the sensor design problem associated with an infinite-dimensional state estimation problem. The existence theorem for a type of measurement control problem is proved and some numerical results are presented.     

Existence of monotone and slow solutions for differential inclusions

The existence of slow and monotone solutions with respect to a continuous pre-order is proved for differential inclusions defined on a closed convex subset of R⁴. First we examine the ‘projected system’, and prove that it is equivalent to a differential variational inequality. We then establish the existence of monotone trajectories. Subsequently, we prove the existence of slow monotone solutions for a class of differential inclusions, satisfying a Nagumo-type condition. Finally we prove a convergence result.     

Admissible controls and attainable states for a class of nonlinear systems with general constraints

We consider nonlinear control systems where the control and the state variables are submitted to explicit constraints. This paper has two objectives. First, for a class of nonlinear systems with constraints, an existence result of nontrivial admissible controls and some of their interesting properties are proved. Then, we investigate the problem of local controllability in a neighbourhood of an equilibrium point, while observing state and control constraints along the whole trajectory. An iterative procedure is also given, which allows one to compute the steering admissible control function. This procedure is illustrated with a classical example.     

On almost constant contact stress distributions by shape optimization

This paper addresses the problem of finding shapes of contacting bodies avoiding undesirable stress concentrations. It has previously been shown that designing the shape of a rigid body in contact with a fixed linear elastic body by minimizing the equilibrium potential energy under an isoparametric constraint results in a uniform contact pressure distribution. As an extension of this result, it is shown here that when the shape of an elastic body in contact with a flat rigid foundation is chosen on the same premises, the uniform pressure distribution is found only if displacement gradients can be considered small. From the point of view of applications, an important conclusion is that this smallness holds in a case when linear elasticity is physically valid.     

Application of mathematica to the direct semi-numerical solution of finite element problems

Several numerical methods, such as the finite element method, reduce applied mechanics and additional engineering problems to systems of linear algebraic equations. It has been already suggested that the inclusion of a symbolic parameter in the corresponding numerical results leads to a generality and a wide applicability of these results. Here we suggest the direct solution of these equations by using the popular computer algebra system MATHEMATICA. Assuming the results expressed in a Taylor-Maclaurin series form with respect to the selected symbolic parameter, the whole problem is reduced to the solution of an appropriate number of systems of purely numerical linear equations. This can be achieved either inside MATHEMATICA or by using efficient external numerical routines. As an application the above modification of the finite element method was used in the classical problem of a tapered elastic beam. The obtained semi-numerical results by the finite element method were seen to be in agreement with the available theoretical results. Further possibilities are also suggested in brief.     

固定费率下城市出租车拥挤收费模型与算法

针对出租车自身的特殊性, 研究了实施拥挤收费政策过程中出租车是否也应进行拥挤收费。在收费费率已知条件下, 采用组合网络均衡模型描述了出租车拥挤收费前后对出行产生、模式划分、流量分配以及出行分布的影响, 构建了与之等价的最优化模型, 设计了求解模型的启发式算法, 比较了对出租车征收拥挤费前后的社会福利变化。算例表明, 收费前的社会福利总是明显大于收费后的, 并且二者之间的差距随着出行者时间价值(VOT)的增加而减小, 随着弹性灵敏参数的增加而增大。这一研究有助于为拥挤收费政策制定者提供科学的决策依据。     

Generalized -KKM type theorems in topological spaces with an application

In this paper, we first introduce a new concept of generalized L-KKM mapping and establish some new generalized L-KKM type theorems without any convexity structure in topological spaces. As an application, an existence theorem of equilibrium points for an abstract generalized vector equilibrium problem is proved in topological spaces. The results presented in this paper unify and generalize some known results in recent literature.     

Asymptotic behaviour of nonautonomous difference inclusions

Nonautonomous difference inclusions are formulated as difference inclusion cocycles, which generalize set-valued semigroups corresponding to autonomous difference inclusions, and the existence of a difference inclusion cocycle attractor is established under the assumption of upper semi-continuity of the cocycle mappings if the difference inclusion cocycle has an absorbing set. Several illustrative examples are provided.     

Pareto Effectiveness of Equilibria in Active Systems with the Distributed Control

Active systems with the distributed control (one active element and a few control centers) are considered. Questions for the existence of the Nash equilibrium of the game of centers in pure strategies are studied. It is shown that at any distribution of the incomes from the work of an active system among a few centers and the existence of at least one Pareto-ineffective Nash equilibrium, any Pareto-effective outcome can be implemented as a Nash equilibrium. It is proved that in the system consisting of two centers there always exists a Pareto-effective Nash equilibrium in pure strategies.     

Complex Optimal Control of a Thermally Stressed State of a Two-Component Body

New problems of optimal control for a thermally stressed state of a two-component solid body are considered in the paper. The body contains a thin and a weakly thermally penetrating inclusion. The existence of unique optimal controls is proved for every described case with quadratic cost functionals.     

Numerical analysis and simulations of contact problem with wear

This paper presents a quasistatic problem of an elastic body in frictional contact with a moving foundation. The model takes into account wear of the contact surface of the body caused by the friction. We recall existence and uniqueness results obtained in Sofonea et al. (2017). The main aim of this paper is to present a fully discrete scheme for numerical approximation together with an error estimation of a solution to this problem. Finally, computational simulations are performed to illustrate the mathematical model.     

An elastic hollow cylinder under axial tension containing a crack and two rigid inclusions of ring shape

This paper is concerned with the fracture of an axisymmetric hollow cylindrical bar containing rigid inclusions. The cylinder is under the action of uniformly distributed axial tension applied at infinity. The bar contains a ring-shaped crack at the symmetry plane whose surfaces are free of tractions and two ring-shaped rigid inclusions with negligible thickness symmetrically located on both sides of the crack. It is assumed that the material of the cylinder is linearly elastic and isotropic. The mixed boundary conditions of the problem lead the analysis to a system of three singular integral equations for crack surface displacement derivative and normal and shearing stress jumps on rigid inclusions. These integral equations are solved numerically and the stress intensity factors are calculated.     

On the monotonicity of higher-order interval iterations for bounding the inverse matrix

We are considering modifications of the higher-order interval Schulz methods for improving an initial inclusion for the inverse of a matrix. It is proved that the methods are strictly monotone in both bounds if this is the case for the first iteration step. A necessary and sufficient condition for the existence of such an initial inclusion is given as well as an easy-to-compute formula for it.     

Evolutionary subgradient inclusions with nonlinear weakly continuous operators and applications

In this paper we consider the first order evolutionary inclusions with nonlinear weakly continuous operators and a multivalued term which involves the Clarke subgradient of a locally Lipschitz function. First, we provide a surjectivity result for stationary inclusion with weakly–weakly upper semicontinuous multifunction. Then, we use this result to prove the existence of solutions to the Rothe sequence and the evolutionary subgradient inclusion. Finally, we apply our results to the non-stationary Navier–Stokes equation with nonmonotone and multivalued frictional boundary conditions.     

带Poisson跳的线性二次随机微分博弈及其在鲁棒控制中的应用

研究了一类带Poisson跳扩散过程的线性二次随机微分博弈,包括非零和博弈的Nash均衡策略与零和博弈的鞍点均衡策略问题．利用微分博弈的最大值原理,得到Nash均衡策略的存在条件等价于两个交叉耦合的矩阵Riccati方程存在解,鞍点均衡策略的存在条件等价于一个矩阵Riccati方程存在解的结论,并给出了均衡策略的显式表达及最优性能泛函值．最后,将所得结果应用于现代鲁棒控制中的随机H₂/H_∞控制与随机H_∞控制问题,得到了鲁棒控制策略的存在条件及显式表达,并验证所得结果在金融市场投资组合优化问题中的应用．     

Asymptotic Behavior and Stability for Solutions of a Biochemical Reactor Distributed Parameter Model

This paper deals with the dynamical analysis of a tubular biochemical reactor. The existence of nonnegative state trajectories and the invariance of the set of all physically feasible state values under the dynamical equation as well as the convergence of the state trajectories to equilibrium profiles are proved. In addition, the existence of multiple equilibrium profiles is analyzed. It is proved that, under physically meaningful conditions, the system has two stable and one unstable equilibrium profiles.