We present a discrete analog of the recently introduced Hamilton– Pontryagin variational principle in Lagrangian mechanics. This unifies two, previously disparate approaches to discrete Lagrangian mechanics: either using the discrete Lagrangian to define a finite version of Hamilton’s action principle, or treating it as a symplectic generating function. This is demonstrated for a discrete Lagra...

a random walk on a lattice is one of the most fundamental models in probability theory. when the random walk is inhomogenous and its inhomogeniety comes from an ergodic stationary process, the walk is called a random walk in a random environment (rwre). the basic questions such as the law of large numbers (lln), the central limit theorem (clt), and the large deviation principle (ldp) are no...

Abstract We prove the stability of solitons Maxwell–Lorentz equations with extended charged rotating particle. The are solutions which correspond to uniform rotation To stability, we construct Hamilton–Poisson representation system. construction relies on Hamilton least action principle. constructed structure is degenerate and admits a functional family Casimir invariants. This allows us Lyapun...

We introduce a novel numerical method for a recently developed perspective Shape-from-Shading model. In order to discretise the corresponding partial differential equation (PDE), Prados et al. employed the dynamical programming principle yielding a Hamilton-Jacobi-Bellman equation. We reduce that model to its essential, namely to the underlying Hamilton-Jacobi equation. For this PDE, we propose...

In this paper, the analysis is focused on single-time optimal control problems based on simple integral cost functionals from Lagrangians whose order is smaller than the higher order of ODEs constraints. The basic topics of our theory include: variational differential systems, adjoint differential systems, Legendrian duality, single-time maximum principle. The main original results refer to the...

We present a formula for the viscosity subdiierential of the sum of two uniformly continuous functions on smooth Banach spaces. This formula is deduced from a new variational principle with constraints. We obtain as a consequence a weak form of Preiss' theorem for uniformly continuous functions. We use these results to give simple proofs of some uniqueness results of viscosity solutions of Hami...

In this paper, a design approach of an optimal control of nonlinear chemical processes is proposed. The idea consists in adopting the variational iteration method to solve the non-linear Hamilton equations derived from the minimum principle theory. These equations constitute a two-point boundary value problem with a coupled nature of solutions. Thus, by considering the correction functionals of...

The theory of non-equilibrium thermodynamics in the optimal control processes is established, by using the Pontryagin’s theory of optimal control and the Bellman’s theory of dynamic programming, for the first time. For this theory, we study the systems of nonlinear differential equations ̇⃗ x = f⃗ (x⃗, u⃗, t) of the state variables x⃗ under the condition that the control variables u⃗ be optimal. It is...

Part I of this paper introduced the notion of implicit Lagrangian systems and their geometric structure was explored in the context of Dirac structures. In this part, we develop the variational structure of implicit Lagrangian systems. Specifically, we show that the implicit Euler-Lagrange equations can be formulated using an extended variational principle of Hamilton called the Hamilton-Pontry...