A sequential quadratic Hamiltonian scheme to compute optimal relaxed controls

A new sequential quadratic Hamiltonian method for computing optimal relaxed controls for a class of optimal control problems governed by ordinary differential equations is presented. This iterative approach is based on the characterisation of optimal controls by means of the Pontryagin maximum principle in the framework of Young measures, and it belongs to the family of successive approximations schemes. The ability of the proposed optimisation framework to solve problems with regular and relaxed controls, including cases with oscillations and concentration effects, is demonstrated by results of numerical experiments. In all cases, the sequential quadratic Hamiltonian scheme appears robust and efficient, in agreement with convergence results of the theoretical investigation presented in this paper.

Sonorism

In this chapter, the use by twentieth-century composers of tone color, or timbre is explained with examples by those who made its use central to their compositional output. Poland, freed from the bonds of communism and the Soviet state, relaxed controls over the arts and in 1956 initiated the Warsaw Autumn festival where avant-garde Polish and Western music could be heard. Kazimierz Serocki cofounded the festival, contributing to the percussion canon his timbre-based sextet, Continuum. In the United States, the American composer George Crumb definitely had an ear for timbre coupled with a love for percussion evident in the works discussed. A young Polish/American composer, Marta Ptaszynska, created a number of works for both solo and ensemble percussion in the latter half of the century. Her work Siderals was conceived as an audio-visual, or mixed-media work utilizing ten percussionists, magnetic tape playback, and lighting. The three composers highlighted in this chapter approached the use of timbre in differing ways.

Existence of optimal controls for systems of controlled forward-backward doubly SDEs

We wish to study a class of optimal controls for problems governed by forward-backward doubly stochastic differential equations (FBDSDEs). Firstly, we prove existence of optimal relaxed controls, which are measure-valued processes for nonlinear FBDSDEs, by using some tightness properties and weak convergence techniques on the space of Skorokhod {\mathbb{D}} equipped with the S-topology of Jakubowski. Moreover, when the Roxin-type convexity condition is fulfilled, we prove that the optimal relaxed control is in fact strict. Secondly, we prove the existence of a strong optimal controls for a linear forward-backward doubly SDEs. Furthermore, we establish necessary as well as sufficient optimality conditions for a control problem of this kind of systems. This is the first theorem of existence of optimal controls that covers the forward-backward doubly systems.