Normality of the maximum principle for nonconvex constrained Bolza problems

We consider a Bolza optimal control problem with state constraints. It is well known that under some technical assumptions every strong local minimizer of this problem satisfies first order necessary optimality conditions in the form of a constrained maximum principle. In general, the maximum principle may be abnormal or even degenerate and so does not provide a sufficient information about optimal controls. In the recent literature some sufficient conditions were proposed to guarantee that at least one maximum principle is nondegenerate, cf. [A.V. Arutyanov, S.M. Aseev, Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints, SIAM J. Control Optim. 35 (1997) 930–952; F. Rampazzo, R.B. Vinter, A theorem on existence of neighbouring trajectories satisfying a state constraint, with applications to optimal control, IMA 16 (4) (1999) 335–351; F. Rampazzo, R.B. Vinter, Degenerate optimal control problems with state constraints, SIAM J. Control Optim. 39 (4) (2000) 989–1007]. Our aim is to show that actually conditions of a similar nature guarantee normality of every nondegenerate maximum principle. In particular we allow the initial condition to be fixed and the state constraints to be nonsmooth. To prove normality we use J. Yorke type linearization of control systems and show the existence of a solution to a linearized control system satisfying new state constraints defined, in turn, by linearization of the original set of constraints along an extremal trajectory.     

Some properties of generating functions obtained by Amann-Conley-Zehnder reduction

Consider an initial Lagrangian submanifold Λ₀ ⊂ T* ℝ ⁿ that admits a global generating function and a Hamiltonian isotopy Φ _H ^t . Then, we provide a global generating function for the Lagrangian submanifold Λ _t = Φ _H ^t (Λ₀) realized by applying the so-called Amann-Conley-Zehnder reduction. When Λ₀ is the zero-section, we study in some detail the asymptotic behavior of such generating functions and give an approximation result. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 33, Suzdal Conference-2004, Part 1, 2005.     

Room temperature single-photon sources based on single colloidal nanocrystals in microcavities

Direct lithography of resist blends, embedding semiconductor colloidal nanocrystals (NCs) is an innovative way to achieve nanopositioning of NCs in quantum-confined optical resonators. In this work, we show a new appealing approach for the fabrication of single-photon sources operating at room temperature by localizing semiconductor colloidal NCs into vertical planar microcavities with lithographic techniques.     

Non‐Blinking Semiconductor Colloidal Quantum Dots for Biology,Optoelectronics and Quantum Optics

Twinkle, twinkle: The blinking of semiconductor colloidal nanocrystals is the main inconvenience of these bright nanoemitters. There are various approaches for obtaining non‐blinking nanocrystals, one of which is to grow a thick coat of CdS on the CdSe core (see picture). Applications of this method in the fields of optoelectronic devices, biologic labelling and quantum information processing are discussed.

     

Zero-sum state constrained differential games: existence of value for Bolza problem

We prove the existence of a lower semicontinuous value function for Bolza problem in differential games with state-constraints. As a byproduct, we obtain a new estimation of trajectories of a control system by trajectories with state constraints. This result which could be interesting by itself enables us to build a suitable strategy for constrained differential games. We also characterize the value function by means of viscosity solutions and give conditions under which the value function is locally Lipschitz continuous.Work supported by the European Community’s Human Potential Program under contract HPRN-CT-2002-00281, [Evolution Equations].     

Regularity of Solution Maps of Differential Inclusions Under State Constraints

Consider a differential inclusion under state constraints

$$L^{\infty }$$ estimates on trajectories confined to a closed subset,for control systems with bounded time variation

Mathematical Programming - The term ‘distance estimate’ for state constrained control systems refers to an estimate on the distance of an arbitrary state trajectory from the subset of...