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Explicit Construction of Piecewise Affine Mappings with Constraints

Waldemar Pompe (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

We construct explicitly piecewise affine mappings u:ℝ ⁿ → ℝ ⁿ with affine boundary data satisfying the constraint div u = 0. As an application of the construction we give short and direct proofs of the main approximation lemmas with constraints in convex integration theory. Our approach provides direct proofs avoiding approximation by smooth mappings and works in all dimensions n ≥ 2. After a slight modification of our construction, the constraint div u = 0 can be turned into det Du = 1, giving...

Explicit formulae of distributions and densities of characteristics of a dynamic advertising and pricing model

Kurt L. Helmes, Torsten Templin (2015)

Banach Center Publications

We analyze the optimal sales process of a stochastic advertising and pricing model with constant demand elasticities. We derive explicit formulae of the densities of the (optimal) sales times and (optimal) prices when a fixed finite number of units of a product are to be sold during a finite sales period or an infinite one. Furthermore, for any time t the exact distribution of the inventory, i.e. the number of unsold items, at t is determined and will be expressed in terms of elementary functions....

Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks*

Zhong-Jie Han, Gen-Qi Xu (2011)

ESAIM: Control, Optimisation and Calculus of Variations


In this paper, the stability of a Timoshenko beam with time delays in the boundary input is studied. The system is fixed at the left end, and at the other end there are feedback controllers, in which time delays exist. We prove that this closed loop system is well-posed. By the complete spectral analysis, we show that there is a sequence of eigenvectors and generalized eigenvectors of the system operator that forms a Riesz basis for the state Hilbert space. Hence the system satisfies the spectrum...

Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks*

Zhong-Jie Han, Gen-Qi Xu (2011)

ESAIM: Control, Optimisation and Calculus of Variations


In this paper, the stability of a Timoshenko beam with time delays in the boundary input is studied. The system is fixed at the left end, and at the other end there are feedback controllers, in which time delays exist. We prove that this closed loop system is well-posed. By the complete spectral analysis, we show that there is a sequence of eigenvectors and generalized eigenvectors of the system operator that forms a Riesz basis for the state Hilbert space. Hence the system satisfies the spectrum...

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