Multitime Dynamic Programming for Curvilinear Integral Actions

Abstract

Based on stochastic curvilinear integrals in the Cairoli-Walsh sense and in the Itô-Udrişte sense, we develop an original theory regarding the multitime stochastic differential systems. The first group of the original results refer to the complete integrable stochastic differential systems, the path independent stochastic curvilinear integral, the Itô-Udrişte stochastic calculus rules, examples of path independent processes, and volumetric processes. The second group of original results include the multitime Itô-Udrişte product formula, first stochastic integrals and adjoint multitime stochastic Pfaff systems. Thirdly, we formulate and we prove a multitime maximum principle for optimal control problems based on stochastic curvilinear integral actions subject to multitime Itô-Udrişte process constraints. Our theory requires the Lagrangian and the Hamiltonian as stochastic 1-forms.

Multivariate Optimal Control with Payoffs Defined by Submanifold Integrals

“…Reaching the above ideas, as well as the ideas developed throughout this paper, was possible after a consistent analysis of multivariate optimal control problems, from different points of views and more extensively than the preliminary approach initiated by Cesari [14] for Dieudonne-Rashevsky problems. For instance, the multivariate optimal control achieved new dimensions by considering other types of cost functionals (stochastic integrals [22], curvilinear-type integrals [23], or mixt payoffs containing both multiple or curvilinear integrals [24]), as well as various types of evolution dynamics (second order partial differential equations, nonholonomic constraints [25]), or different working techniques (multivariate dynamic programming [26], multivariate needle-shaped variations [24,27]). The applicative features of the multivariate Pontryagin's maximum principle were emphasized in [5], where the minimal submanifolds, the harmonic maps, or the Plateau problem were approached under this new light.…”

Controllability of Nonholonomic Black Holes Systems

“…We apply the Pontryagin minimum principle (for multitime minimum principle, see [21][22][23][24][25][26][27][28]). In general notation, we have x = (x i ), u = (u a ), p = (p i ), a = 1, 4; i = 1, 5,…”

“…To solve such a multitime optimum problem we use the techniques from our papers [21][22][23][24][25][26][27][28]. To simplify, let us show that the foregoing problem can be solved via controls of gradient type, i.e.…”

Multitime multiobjective variational problems via η-approximation method

“…The multitime control theory is related to the partial derivatives of dynamical systems and their optimization over multitime, also known as the multidimensional control problems, which have wide applications theoretically as well as numerically [1]. Multitime is the extension of single-time dynamic programming that contains m-dimensional evolution and path independent curvillinear integral functional was explained by Udriste and Tevy [38]. Further, a curvilinear integral type multitime multiobjective variational problems is studied in [30] and the results on Mond-Weir type duality is established by using (ρ, b)quasiinvexity.…”