Admissible controls, modeling, and optimization for a new class of nonlinear stochastic delay systems

Abstract

The paper deals with several fundamental issues that have not been previously addressed in the modeling and optimization of nonlinear stochastic delay systems. For an example, consider the special case of a system with a delayed control term of the form f(u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> (t + θ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ), u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (t + θ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> )), where the delays θ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> <; 0 are different and f(·) is not the sum of functions of each of the controls separately. The class of adapted relaxed controls is no longer adequate as the class of admissible controls, at least in the sense that the infimum of the costs over this class is not the infimum over the class of ordinary controls, and the limit of convergent sequences might be meaningless. We deal with such issues of admissibility and optimization for a large class of systems that includes the above example. The appropriate extensions and the proofs are not obvious. The issues are crucial for the convergence of numerical approximations to optimal control problems, as well as for the optimization problem to be well-defined.