Convergence results for two Lagrange-Newton-type methods of solving optimal control problems are presented. It is shown how the methods can be applied to a class of optimal control problems for nonlinear ODEs, subject to mixed control-state constraints. The first method reduces to an SQP algorithm. It does not require any information on the structure of the optimal solution. The other one is the shooting method, where information on the structure of the optimal solution is exploited. In each case,...

Many numerical simulations in (bilinear) quantum control use the monotonically convergent Krotov algorithms (introduced by Tannor et al. [Time Dependent Quantum Molecular Dynamics (1992) 347–360]), Zhu and Rabitz [J. Chem. Phys. (1998) 385–391] or their unified form described in Maday and Turinici [J. Chem. Phys. (2003) 8191–8196]. In Maday et al. [Num. Math. (2006) 323–338], a time discretization which preserves the property of monotonicity has been presented. This paper introduces a proof of...

Many inverse problems for differential equations can be formulated as optimal control problems. It is well known that inverse problems often need to be regularized to obtain good approximations. This work presents a systematic method to regularize and to establish error estimates for approximations to some control problems in high dimension, based on symplectic approximation of the Hamiltonian system for the control problem. In particular the work derives error estimates and constructs regularizations...

In this article we modify an iteration process to prove strong convergence and Δ- convergence theorems for a finite family of nonexpansive multivalued mappings in hyperbolic spaces. The results presented here extend some existing results in the literature.

The numerical minimization of the functional $\mathcal{F}\left(u\right)={\int }_{\mathrm{\Omega }}\varphi \left(x,{\nu }_u\right)\left|Du\right|+{\int }_{\partial \mathrm{\Omega }}\mu ud{\mathcal{H}}^{n-1}-{\int }_{\mathrm{\Omega }}\kappa udx$, $u\in BV\left(\mathrm{\Omega };\left\{-1,1\right\}\right)$ is addressed. The function $\varphi$ is continuous, has linear growth, and is convex and positively homogeneous of degree one in the second variable. We prove that $\mathcal{F}$ can be equivalently minimized on the convex set $BV\left(\mathrm{\Omega };\left[-1,1\right]\right)$ and then regularized with a sequence ${\left\{{\mathcal{F}}_{ϵ}\left(u\right)\right\}}_{ϵ}$, of stricdy convex functionals defined on $BV\left(\mathrm{\Omega };\left[-1,1\right]\right)$. Then both $\mathcal{F}$ and ${\mathcal{F}}_{ϵ}$, can be discretized by continuous linear finite elements. The convexity property of the functionals on $BV\left(\mathrm{\Omega };\left[-1,1\right]\right)$ is useful in the numerical minimization...

We address the numerical minimization of the functional $\mathcal{F}\left(v\right)={\int }_{\mathrm{\Omega }}\left|Dv\right|+{\int }_{\partial \mathrm{\Omega }}\mu vd{\mathcal{H}}^{n-1}-{\int }_{\mathrm{\Omega }}xvdx$, for $v\in BV\left(\mathrm{\Omega };\left\{-1,1\right\}\right)$. We note that $\mathcal{F}$ can be equivalently minimized on the larger, convex, set $BV\left(\mathrm{\Omega };\left[-1,1\right]\right)$ and that, on that space, $\mathcal{F}$ may be regularized with a sequence $\mathit{\left\{}\mathcal{F}{}_{ϵ}\right(v)=\int {}_{\mathrm{\Omega }}\sqrt{{ϵ}^2+{\left|Dv\right|}^2}+\int {}_{\partial \mathrm{\Omega }}\mu vd\mathcal{H}{}^{n-1}-\int {}_{\mathrm{\Omega }}xvdx\mathit{\}}{}_{ϵ}$of regular functionals. Then both $\mathcal{F}$ and ${\mathcal{F}}_{ϵ}$ can be discretized by continuous linear finite elements. The convexity of the functionals in $BV\left(\mathrm{\Omega };\left[-1,1\right]\right)$ is useful for the numerical minimization of $\mathcal{F}$. We prove the $\mathrm{\Gamma }-L^1\left(\mathrm{\Omega }\right)$-convergence of the discrete functionals to $\mathcal{F}$ and present a few numerical examples.

We investigate the minimum time transfer of a satellite around the Earth. Using an optimal control model, we study the controllability of the system and propose a geometrical analysis of the optimal command structure. Furthermore, in order to solve the problem numerically, a new parametric technique is introduced for which convergence properties are established.

We investigate the minimum time transfer of a satellite around the Earth. Using an optimal control model, we study the controllability of the system and propose a geometrical analysis of the optimal command structure. Furthermore, in order to solve the problem numerically, a new parametric technique is introduced for which convergence properties are established.

In this paper, we present an Uzawa-based heuristic that is adapted to certain type of stochastic optimal control problems. More precisely, we consider dynamical systems that can be divided into small-scale subsystems linked through a static almost sure coupling constraint at each time step. This type of problem is common in production/portfolio management where subsystems are, for instance, power units, and one has to supply a stochastic power demand at each time step. We outline the framework...

Differential evolution algorithm combined with chaotic pattern search(DE-CPS) for global optimization is introduced to improve the performance of simple DE algorithm. Pattern search algorithm using chaotic variables instead of random variables is used to accelerate the convergence of solving the objective value. Experiments on 6 benchmark problems, including morbid Rosenbrock function, show that the novel hybrid algorithm is effective for nonlinear optimization problems in high dimensional space....

A 3D-2D dimension reduction for −Δ1 is obtained. A power law approximation from −Δp as p → 1 in terms of Γ-convergence, duality and asymptotics for least gradient functions has also been provided.

In this paper, a new numerical method for solving the nonlinear constrained optimal control with quadratic performance index is presented. The method is based upon B-spline functions. The properties of B-spline functions are presented. The operational matrix of derivative (${𝐃}_{\phi }$) and integration matrix ($𝐏$) are introduced. These matrices are utilized to reduce the solution of nonlinear constrained quadratic optimal control to the solution of nonlinear programming one to which existing well-developed...